12x-5y=-20
y=x+4
12x-5(x+4)=-20
12x-5x-20=-20
7x=0,
So, we get: x=0 and y=4
A recently published study shows that 50% of Americans adults take multivitamins regularly. Another recent study showed that 20.6% of American adults work out regularly. Suppose that these two variables are independent. The probability that a randomly selected American adult takes multivitamins regularly and works out regularly is _______.
Answer:
The probability that a randomly selected American adult takes multivitamins regularly and works out regularly is 0.103
Step-by-step explanation:
p(Americans adults take multivitamins regularly) = 50% = 0.50
p(American adults work out regularly) = 20.6 = 0.206
p(Americans adults take multivitamins regularly and American adults work out regularly) = 0.50* 0.206 = 0.103
Thus, p(Americans adults take multivitamins regularly and American adults work out regularly) = 0.103
Assume that you have $60 a week to spend on bottled water and chips. A bottle of water costs $2 and a bag of chips costs $3. If you buy 15 bottles of water, how many bags of chips can you purchase
Answer:
10 bags of chips
Step-by-step explanation:
15(2)+3x=60
30+3x=60
3x=30
x=10
Therefore, 10 bags of chips
Consider the following.
(−2, 4); (4, −1)
Find the slope of the line between the two points. (Enter an exact number as an integer, fraction, or decimal. If an answer is undefined, enter UNDEFINED.)
Answer:
The slope is -5/6
Step-by-step explanation:
To find the slope of a line given two points, we use the formula
m= (y2-y1)/(x2-x1)
= (-1-4)/(4--2)
=-5/(4+2)
=-5/6
Answer: the slope of the line between the two points is - 5/6
Step-by-step explanation:
Slope, m = change in value of y on the vertical axis / change in value of x on the horizontal axis.
change in the value of y = y2 - y1
Change in value of x = x2 -x1
y2 = final value of y
y 1 = initial value of y
x2 = final value of x
x1 = initial value of x
The line passes through (- 2, 4) and (4, - 1),
y2 = - 1
y1 = 4
x2 = 4
x1 = - 2
Slope,m = (- 1 - 4)/(4 - - 2) = - 5/6
QUESTION 1 (0.5 POINTS) In a large restaurant an average of 3 out of every 5 customers ask for water with their meal. A random sample of 10 customers is selected. What is the probability that less than 3 customers ask for water with their meal? a). 0.012 b). 0.055 c). 0.042 d). 0.011
Answer:
Option A) 0.012
Step-by-step explanation:
We are given the following information:
We treat customers asking for water as a success.
P(customers ask for water) = [tex]\dfrac{3}{5}[/tex] = 0.6
Then the number of customers follows a binomial distribution, where
[tex]P(X=x) = \binom{n}{x}.p^x.(1-p)^{n-x}[/tex]
where n is the total number of observations, x is the number of success, p is the probability of success.
Now, we are given n = 10
We have to evaluate:
[tex]P(x < 3) = P(x = 0) + P(x = 1) + P(x = 2) \\= \binom{10}{0}(0.6)^0(1-0.6)^{10} +\binom{10}{1}(0.6)^1(1-0.6)^{9} + \binom{10}{2}(0.6)^2(1-0.6)^{8}\\= 0.0001 +0.0015 + 0.0106\\=0.012[/tex]
0.012 is the probability that less than 3 customers ask for water with their meal.
The correct answer is option a) 0.012
The probability of success in each trial is 3 out of 5 (or 0.6), and the probability of failure is 2 out of 5 (or 0.4). Denoting the number of customers who ask for water as a binomial random variable X with n = 10 and p = 0.6.
The binomial probability formula is given by:
[tex]P(X=k) ={n \choose k}p^k(1-p)^{n-k}[/tex]
where:
n is the number of trials (10 in this case),
k is the number of successes (in this case, less than 3),
p is the probability of success (0.6), and
1−p is the probability of failure (0.4).
Probability that fewer than 3 customers ask for water needs to be found. This means we need to find:
P(X<3)=P(X=0)+P(X=1)+P(X=2)
Calculating each of these probabilities using the binomial formula:
For k=0:
[tex]P(X=0) ={10 \choose 0}(0.6)^0(0.4)^{10-0} = 1\times1\times0.4^{10} = 0.0001048576[/tex]
For k=1:
[tex]P(X=1) ={10 \choose 1}(0.6)^1(0.4)^{10-1} = 10\times0.6\times0.4^{9} = 0.001572864[/tex]
For k=2:
[tex]P(X=2) ={10 \choose 2}(0.6)^2(0.4)^{10-2} = 45\times0.36\times0.4^{8} = 0.010616832[/tex]
Now, we sum these probabilities to get the total probability that fewer than 3 customers ask for water:
P(X<3) = P(X=0) + P(X=1) + P(X=2)
P(X<3) = 0.0001048576 + 0.001572864 + 0.010616832
P(X<3) = 0.0122945536
Therefore rounding this value results in a probability of approximately 0.012.
A game at the fair involves a wheel with seven sectors. Two of the sectors are red, two of the sectors are purple, two of the sectors are yellow, and one sector is blue. Landing on the blue sector will give 3 points, landing on a yellow sector will give 1 point, landing on a purple sector will give 0 points, and landing on a red sector will give –1 point.
If you take one spin, what is your expected value?
What changes could you make to values assigned to outcomes to make the game fair? Prove that the game would be fair using expected values.
Answer:
Expected value: 3/7
Change landing on purple to -1.5
Step-by-step explanation:
Expected value
=3(1/7) + 1(2/7) + 0(2/7) - 1(2/7) = 3/7
Fair game: Expected value = 0
Change the value of 0 to -1.5
Answer:
3/7
change 0 to -1.5
Step-by-step explanation:
Got a 100
The times (in minutes) that several underwriters took to review applications for similar insurance coverage are 50, 230, 52, and 57. What is the median length of time required to review an application?
Answer:
The median is located at the 2.5th position, which is halfway between the values 52 and 57.
Step-by-step explanation:
The Median is referred to as the Middle term of a set of Data when ordered in Ascending/Descending order.
Consider the numbers 50, 230, 52, and 57
Arranging them in an Ascending Order
50, 52, 57, 230
The number of data in the set is even.
Dividing the data into two halves (50, 52 and 57, 230)
The median is halfway between the two halves.
The median is located at the 2.5th position, which is halfway between the values 52 and 57.
Credit sales are collected as follows:65 percent in the month of the sale.25 percent in the month after the sale.10 percent in the second month after the sale. The accounts receivable balance at the end of the previous quarter was $94,000 ($64,000 of which were uncollected December sales). a.Calculate the sale
Answer:
Step-by-step explanation:
a) November sales = (Total Uncollected Sales - Uncollected Sales from December) / Collection rate after two months
= ($121,100 - $87,300) / 0.20
November sales = $169, 000.00
b) December sales = Uncollected sales from December / Collection rate of the previous month sales,
Therefore: December sales = $87,300 / 0.45 = $194,000
December sales = $194,000.00
c) Each month's collection for the company are:
Collections for ech month = 0.20(Sales from 2 months ago) + 0.25(Last month's sales) + 0.55 (Current sales)
January collections = 0.20($169000.00) + 0.25($194,000.00) + 0.55($132,000)
January collections = $154,900.00
February collections = 0.20($194,000.00) + 0.25($132,000) + 0.55($149,000)
February collections = $153,750.00
March collections = 0.20($132,000) + 0.25($149,000) + 0.55($164,000)
March collections = $153,850.00
Seven blue and four red balls are to be arranged in order. How many ways can this be done if (1) The blue balls are distinguishable (e.g. numbered) as are the red balls. (2) Blue balls are distinguishable, but the red balls are identical. (3) The balls of each color are indistinguishable.
Answer:
1) n = 39916800
2) n = 1663200
3) n = 330
Step-by-step explanation:
1) If the blue balls are distinguishable as are the red balls
Then you can arrange these balls in the following ways, we must use a permutation
In totally we have 11 balls, then
n = 11P11
[tex]n = \frac{11!}{(11-11)!} = 11! = 39916800\\[/tex]
2) If Blue balls are distinguishable, but the red balls are identical
In this case, we need to do a correction due to the red balls are identical and we cannot identify the difference when we interchange two red balls
[tex]n = \frac{11!}{4!} = \frac{39916800}{24} = 1663200[/tex]
3) If the balls of each color are indistinguishable
We proceed equal to the before case but we include a correction due to blue balls also
[tex]n = \frac{11!}{4!*7!} = \frac{39916800}{24*5040} = 330[/tex]
An arc with a measure of 190° has an arc length of 40[tex]\pi[/tex]centimeters. What is the radius of the circle on which the arc sits?
Answer:
37.9 cm
Step-by-step explanation:
Arc length is:
s = 2πr (θ/360°)
where r is the radius and θ is the arc angle.
40π cm = 2πr (190°/360°)
20 cm = r (190°/360°)
r = 37.9 cm
Answer: the radius of the circle on which the arc sits is 3.8 cm
Step-by-step explanation:
The formula for determining the length of an arc is expressed as
Length of arc = θ/360 × 2πr
Where
θ represents the central angle.
r represents the radius of the circle.
π is a constant whose value is 3.14
From the information given,
θ = 190 degrees
Length of arc = 40π centimeters
Therefore,
40π = 190/360 × 2 × π × r
Dividing both sides of the equation by π, it becomes
40 = 380r/360
380r = 40 × 360 = 1440
r = 1440/380
r = 3.8 to the nearest tenth
Suppose we are interested in analyzing the weights of NFL players. We know that on average, NFL players weigh 247 pounds with a population standard deviation of 47 pounds. Suppose we take a sample of 30 new players and we find that the average weight from that sample is 237 pounds. We are interested in seeing if the weight of NFL players is decreasing. What is the standard error? What is the margin of error at 90% confidence? Using my sample of 30, what would be the 90% confidence interval for the population mean? If I wanted to control my margin of error and set it to 5 at 90% confidence, what sample size would I need to take instead of the 30? What are the null and alternative hypotheses? What is the critical value at 90% confidence? Calculate the test statistic (using the sample of 30 and NOT the answer from part d). Find the p-value. What conclusion would be made here at the 90% confidence level?
Answer:
[tex] SE= \frac{\sigma}{\sqrt{n}} = \frac{47}{\sqrt{30}}= 8.58[/tex]
[tex] ME= 1.64 *\frac{\sigma}{\sqrt{n}} = 1.64*\frac{47}{\sqrt{30}}= 14.073[/tex]
[tex] \bar X - ME = 237- 14.073 = 222.927[/tex]
[tex] \bar X + ME = 237+ 14.073 = 251.073[/tex]
[tex]n=(\frac{1.640(47)}{5})^2 =237.65 \approx 238[/tex]
So the answer for this case would be n=238 rounded up to the nearest integer
Null hypothesis:[tex]\mu \geq 247[/tex]
Alternative hypothesis:[tex]\mu <247[/tex]
[tex]z=\frac{237-247}{\frac{47}{\sqrt{30}}}=-1.165[/tex]
[tex]p_v =P(z<-1.165)=0.122[/tex]
If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't conclude that the true mean is lower than 247 at 10% of significance.
Step-by-step explanation:
For this case we have the following data given:
[tex] \bar X =237[/tex] represent the sample mean
[tex]\sigma = 47[/tex] represent the population deviation
[tex] n =30[/tex] represent the sample size selected
[tex]\mu_0 = 247[/tex] represent the value that we want to test.
The standard error for this case is given by:
[tex] SE= \frac{\sigma}{\sqrt{n}} = \frac{47}{\sqrt{30}}= 8.58[/tex]
For the 90% confidence the value of the significance is given by [tex] \alpha=1-0.9 = 0.1[/tex] and [tex] \alpha/2 = 0.05[/tex] so we can find in the normal standard distribution a quantile that accumulates 0.05 of the area on each tail and we got:
[tex] z_{\alpha/2}= 1.64[/tex]
And the margin of error would be:
[tex] ME= 1.64 *\frac{\sigma}{\sqrt{n}} = 1.64*\frac{47}{\sqrt{30}}= 14.073[/tex]
The confidence interval for this case would be given by:
[tex] \bar X - ME = 237- 14.073 = 222.927[/tex]
[tex] \bar X + ME = 237+ 14.073 = 251.073[/tex]
The margin of error is given by this formula:
[tex] ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (a)
And on this case we have that ME =5 and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=(\frac{z_{\alpha/2} \sigma}{ME})^2[/tex] (b)
Replacing into formula (b) we got:
[tex]n=(\frac{1.640(47)}{5})^2 =237.65 \approx 238[/tex]
So the answer for this case would be n=238 rounded up to the nearest integer
State the null and alternative hypotheses.
We need to conduct a hypothesis in order to check if the true mean is lower than 247 pounds, the system of hypothesis would be:
Null hypothesis:[tex]\mu \geq 247[/tex]
Alternative hypothesis:[tex]\mu <247[/tex]
Since we know the population deviation, is better apply a z test to compare the actual mean to the reference value, and the statistic is given by:
[tex]z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}[/tex] (1)
z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".
Calculate the statistic
We can replace in formula (1) the info given like this:
[tex]z=\frac{237-247}{\frac{47}{\sqrt{30}}}=-1.165[/tex]
P-value
Since is a left tailed test the p value would be:
[tex]p_v =P(z<-1.165)=0.122[/tex]
Conclusion
If we compare the p value and the significance level given [tex]\alpha=0.1[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can't conclude that the true mean is lower than 247 at 10% of significance.
Experiments on learning in animals sometimes measure how long it takes a gerbil to nd its way through a maze. The mean time is 14 seconds for one particular maze. A researcher thinks that playing soothing music will cause the gerbils to complete the maze slower. She measures how long each of 34 gerbils takes with a noise stimulus.
Complete Question
Experiments on learning in animals sometimes measure how long it takes a gerbil to nd its way through a maze. The mean time is 14 seconds for one particular maze. A researcher thinks that playing soothing music will cause the gerbils to complete the maze slower. She measures how long each of 34 gerbils takes with a noise stimulus.
She measures how long each of 34 gerbils takes with a noise stimulus. The sample mean is x = 16.5 seconds.
The alternative hypothesis for the significance test is
a. Ha:μ≠14
b. Ha:μ=16.5.
c. Ha:μ>14
Answer:
Option C is correct.
The alternative hypothesis for the significance test is Ha:μ>14
Step-by-step explanation:
The researcher sets up a maze and notes that the mean time to complete the maze is 14 seconds for gerbils.
She then theorizes that playing soothing music for the gerbils while they complete the maze slows the gerbils down.
Using a sample of 34 gerbils, she tests her hypothesis, and truly, the sample mean, when she plays soothing noise in the background, rises from 14 to 16.5 seconds; indicating that the noise stimulus indeed, slows down the gerbils.
Alternative hypothesis theorizes that there is a relationship between two variables. A relationship that is significant enough statistically, that when the hypothesis is introduced, it affects the result of the experiment just in the way that the conductors of the experimemt have predicted.
So, if the alternative hypothesis for this 'gerbils completing maze experiment' is represented by Ha, just like the researcher theorizes, when Ha is introduced, the gerbils become slower and the mean time for completing the maze is evidently higher than the mean.
Mathematically, it can be written as
Ha:μ>14
Again, this is interpreted as, when Ha is introduced (soothing music), the mean time for the gerbils completing the maze is higher than the original mean (14).
Hope this Helps!!!
If X=5 and 5=y, then x=y
Ar algebraic equality property justifies the above statement?
Answer:
Yes.
Step-by-step explanation:
According to the Transitive property of the Algebraic properties of equality, two values are said to be equal is they are differently equal to a corresponding third party. This is, If a=b and b=c then a=c.
Hence, if x=5 and 5=y, then following the transitive property of Algebraic properties of equality, x=y, hence the Algebraic property of equality Justifies the statement.
In Exercises 29 and 30, describe the possible echelon forms of the standard matrix for a linear transformation T . Use the notation of Example 1 in Section 1.2. 29. T W R3 ! R4 is one-to-one. 30. T W R4 ! R3 is onto.
Answer:
note:
solution is attached in word form due to error in mathematical equation. furthermore i also attach Screenshot of solution in word due to different version of MS Office please find the attachment
The pair of random variables (X,Y) is equally likely to take any of the four pairs of values (0,1), (1,0), (−1,0), (0,−1). Note that X and Y each have zero mean.
a) Find E[XY].
E[XY]=
b) YES or NO: For this pair of random variables (X,Y), is it true that Var(X+Y)=Var(X)+Var(Y)?
Select an option Yes No
c) YES or NO: We know that if X and Y are independent, then Var(X+Y)=Var(X)+Var(Y). Is the converse true? That is, does the condition Var(X+Y)=Var(X)+Var(Y) imply independence?
Select an option Yes No
Answer and Step-by-step explanation:
(a)
E[XY] = 1/4*0*1 + 1/4*1*0 + 1/4*-1*0 +1/4*0*-1 = 0
(b)
E[X] = 0, E[Y] = 0
Thus, Cov(X,Y) = E[XY] - E[X]E[Y] = 0
So, Var(X + Y) = Var(X) + Var(Y) is True
The answer is Yes
(c) No, the converse is not true
(a) The value of E[XY] is zero.
(b) Yes For this pair of random variables (X, Y) is true for the given relation.
(c) No, the converse is not true.
What is a variance?Variance is the value of the squared variation of the random variable from its mean value, in probability and statistics.
The given data in the problem will be;
(X, Y) is a random variable
pairs of values is given as (0,1), (1,0), (−1,0), (0,−1).
E[XY] =?
(a) The value of E[XY] is zero.
[tex]E[XY] = \frac{1}{4}\times 0\times1 +\frac{1}{4}\times 1\times 0 + \frac{1}{4} \times 1\times 0 +\frac{1}{4}\times0\times1 = 0[/tex]
Hence the value of E[XY] is zero.
(b) Yes or this pair of random variables (X, Y) is true for the given relation.
[tex]E[X] = 0, E[Y] = 0[/tex]
[tex]\rm Cov(X,Y) = E[X,Y] - E[X]E[Y] = 0Var(X + Y) = Var(X) + Var(Y)[/tex]
Hence ) Yes For this pair of random variables (X, Y) is true for the given relation.
(c) No, the converse is not true
X and Y are independent
[tex]Var(X+Y)=Var(X)+Var(Y).[/tex]
Hence ) No, the converse is not true
To learn more about the variance refer to the link;
https://brainly.com/question/7635845
8–8 If the derivative property of phasors is multiplication of the phasor by jω, the integral property of phasors is division of the phasor by jω. Use phasors and these properties to find the sinusoids in each of the following: υ 2 ( t ) = 1 100 d υ 1 ( t ) d t + 20 υ 1 ( t ) and υ1(t) = 10 cos(100t + 90°) V i 2 ( t ) = 10 ∫ i 1 ( t ) d t − 3 i 1 ( t ) and i1(t) = − 4 cos(5t) A
Answer:
solution attached below
Step-by-step explanation:
To find the sinusoids in the given equations, we can use the derivative and integral properties of phasors. For the first equation, υ2(t) = 1/100 dυ1(t)/dt + 20υ1(t), you can find the phasor representation of υ1(t) using the given equation υ1(t) = 10cos(100t + 90°) V and apply the derivative property of phasors. For the second equation, i2(t) = 10∫i1(t)dt - 3i1(t), you can find the phasor representation of i1(t) using the given equation i1(t) = -4cos(5t) A and apply the integral property of phasors. Therefore, the sinusoids in the given equations are υ2(t) ≈ 10cos(100t + 90°) + 2000cos(100t + 90°) V and i2(t) ≈ (4/5)cos(5t + 180°) + 12cos(5t + 180°) A·s.
Explanation:To find the sinusoids in the given equations, we can use the derivative and integral properties of phasors.
For the first equation, υ2(t) = 1/100 dυ1(t)/dt + 20υ1(t), we can start by finding the phasor representation of υ1(t) using the given equation υ1(t) = 10cos(100t + 90°) V:
υ1(t) = 10e^(j(100t + 90°))
Next, we differentiate υ1(t) with respect to t to find dυ1(t)/dt:
dυ1(t)/dt = -1000sin(100t + 90°) V/s
Using the derivative property of phasors (multiplying by jω), we have:
jωυ1(t) = j(100)(10)e^(j(100t + 90°)) = 1000ej(100t + 90°) V/s
Now, we can substitute these phasor representations back into the original equation:
υ2(t) = 1/100 (1000ej(100t + 90°) V/s) + 20(10e^(j(100t + 90°)) V
Simplifying, we get:
υ2(t) ≈ 10ej(100t + 90°) + 2000ej(100t + 90°) V
For the second equation, i2(t) = 10∫i1(t)dt - 3i1(t), we can find the phasor representation of i1(t) using the given equation i1(t) = -4cos(5t) A:
i1(t) = -4e^(j(5t + 180°))
Next, we can integrate i1(t) with respect to t to find ∫i1(t)dt:
∫i1(t)dt = (-4/5)e^(j(5t + 180°)) A·s
Using the integral property of phasors (dividing by jω), we have:
(1/jω)i1(t) = (-1/(j5))(4)e^(j(5t + 180°)) = (4/5)ej(5t + 180°) A·s
Substituting these phasor representations back into the original equation, we get:
i2(t) ≈ (4/5)ej(5t + 180°) - 3(-4)e^(j(5t + 180°)) A·s
Therefore, the sinusoids in the given equations are υ2(t) ≈ 10cos(100t + 90°) + 2000cos(100t + 90°) V and i2(t) ≈ (4/5)cos(5t + 180°) + 12cos(5t + 180°) A·s.
Mrs. Jones recorded the time, in minutes, she spends reading each day for two weeks. The results are shown. What is the IQR for each week? Week 1 Week 2 81 50 63 58 39 72 104 62 54 110 72 68 34 79 A. The IQR for Week 1 is 65, and the IQR for Week 2 is 76. B. The IQR for Week 1 is 63, and the IQR for Week 2 is 68. C. The IQR for Week 1 is 50, and the IQR for Week 2 is 54. D. The IQR for Week 1 is 31, and the IQR for Week 2 is 25.
Answer:
D.
The IQR for Week 1 is 31, and the IQR for Week 2 is 25.
Step-by-step explanation:
Answer: D.The IQR for week 1 is 31 and the IQR for week 2 is 25
Step-by-step explanation: When you divid a data set in groups of 4 and measure the bulk of the values, this is called Interquatile Range (IQR).
It's calculated as follows:
1) Put the data in order;
week 1 : 39 50 58 63 72 81 104
week 2 : 34 54 62 68 72 79 110
2) Find the median of each data set:
*Note: Median is the middle value of a set.
week 1 : Median 63
week 2 : Median 68
3) Find Q1, which is the median in the lower half of the set:
week 1: the lower set is 39 50 58.
The middle value is 50.
So Q1 = 50.
week 2: the lower set is 34 54 62
The middle value is 54.
So Q1 = 54.
4) Find Q3, which is the middle value of the upper half:
week 1 : the upper half is 72 81 104
Q2 = 81
week 2 : the upper half is 72 79 110
Q2 = 79
5) To determine IQR, subtract Q1 and Q3:
week 1: 81 - 50 = 31
week 2 : 79 - 54 = 25
In conclusion, the IQR for week 1 is 31 and for week 2 is 25.
Given the trinomial, what is the value of the coefficient B in the factored form?
2x^2 + 4xy – 48y^2 = 2(x + By)(x – 4y)
-12
12
-6
6
Answer:
6
Step-by-step explanation:
2x² + 4xy − 48y²
2 (x² + 2xy − 24y²)
2 (x − 4y) (x + 6y)
Yvonne bought a new computer and printer for college. The total cost was 2500, which she put on her new credit card that has an interest rate of 13.5%. She makes a $75.00 monthly payment. How many months will it take to pay off the credit card balance? Enter your answer as a whole number such as: 23.
Answer:
43 months
Step-by-step explanation:
The amortization formula is ...
A = P(r/n)/(1 -(1 +r/n)^(-nt))
We want to find the value of t for monthly payment A, financed amount P, interest rate r, and compounding monthly (n=12). Filling in the values, we have ...
75 = 2500(0.135/12)/(1 -(1 +0.135/12)^(-12t))
1 -(1.01125^(-12t) = 2500·0.135/(12·75) = 0.375
1 -0.375 = 1.01125^(-12t) . . . . . add 1.01125^(-12t) -0.375
Next, take logarithms and divide by the coefficient of t.
log(0.625)/(-12log(1.01125)) = t ≈ 3.50106 . . . . years
In months, that is ...
3.50106×12 ≈ 42.01
The balance will be not quite zero after 42 payments. (It will be about $0.94.) It will take 43 payments to pay off the credit card balance.
Answer:
42
I did the test and got it right hope this helps
There are many applications of exponentials and logarithms, including exponential growth and decay, half life, doubling time, Carbon dating, compound interest. Here are a couple of examples.
You find out that in the year 1800 an ancestor of yours invested 100 dollars at 6 percent annual interest, compounded yearly. You happen to be her sole known descendant and in the year 2005 you collect the accumulated tidy sum of _______________ dollars. You retire and devote the next 10 years of your life to writing a detailed biography of your remarkable ancestor.
Strontium-90 is a biologically important radioactive isotope that is created in nuclear explosions. It has a half life of 28 years. To reduce the amount created in a particular explosion by a factor 1,000 you would have to wait______________ years. Round your answer to the nearest integer.
Seeds found in a grave in Egypt proved to have only 53% of the Carbon-14 of living tissue. Those seeds were harvested ________________ years ago. The half life of Carbon-14 is 5,730 years.
Answer:
a) $15,406,443
b) 279.04 years = 279 years.
c) 5246.9 years = 5247 years.
Step-by-step explanation:
Compound interest
The final amount obtainable, A, from saving an initial amount, P, compounded at a rate of r in t number of years is given as
A = P (1 + r)ᵗ
A = ?
P = $100
r = 6% = 0.06
t = 2005 - 1800 = 205
A = 100 (1 + 0.06)²⁰⁵ = 100 × 154064.43 = $15406443
b) Radioactivity
Let the initial amount of Strontium be A
After 1 half life,
Amount remaining is A/2
After two half lives,
Amount remaining = A/2²
After 3 half lives,
Amount remaining = A/2³
After n half lives,
Amount remaining = A/2ⁿ
So, for this question,
(A)/(A/2ⁿ) = 1000
2ⁿ = 1000
In 2ⁿ = In 1000
n = (In 1000)/(In 2)
n = 9.966
1 half life = 28 years
n half lives = n × 28 = 9.966 × 28 = 279.04 years.
c) Carbon dating
The general relation of amount left to amount of Carbon-14 that is started with follows a first order rate of decay kinetics like every radioactive decay
A = A₀ e⁻ᵏᵗ
A = amount of Carbon-14 left at any time
A₀ = initial amount of Carbon-14
k = rate constant = (In 2)/(half life) = 0.693/5730 = 0.000121 /year.
(A/A₀) = 53% = 0.53
e⁻ᵏᵗ = 0.53
-kt = In 0.53 = -0.6349
t = 0.6349/k = 0.6349/0.000121 = 5246.9 years = 5247 years
Using the principles of compound interest, radioactive decay and carbon dating, we calculate that the investment made in 1800 would equal about $8,680,204 in 2005, that it would take approximately 280 years to reduce the amount of Strontium-90 by a factor 1000, and that the seeds were harvested around 8463 years ago.
Explanation:The sum you would've collected in the year 2005 from a 100 dollars investment at 6 percent annual interest, compounded yearly since the year 1800 depends on the principle of compound interest. The formula to calculate compound interest is A = P(1 + r/n)^(nt), where A is the total sum, P is the principal amount, r is the annual rate of interest, n is the number of compounding periods a year, and t is the time in years. In your case, we have P = 100, r = 0.06, n = 1 (compounded yearly), and t = 2005 - 1800 = 205 years. So, A = 100(1 + 0.06/1)^(1*205) which equals approximately $8,680,204.
Strontium-90, with a half life of 28 years, would require waiting for a certain amount of years to reduce by a factor of 1000. For half life calculations, use the formula N = N0*(1/2)^(t/h), where N is the final amount, N0 is the initial amount, t is time and h is the half life. We want N0/N = 1000, so log2(1000) = t/28, which gives t = 28*log2(1000) = approximately 280 years.
The carbon-14 dating principle lets us calculate the age of the seeds. Using log2(1/0.53) = t/5730 to get t = 5730 * log2(1/0.53), we find the seeds were harvested about 8463 years ago.
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Verify that y1(t) =t2 and y2(t) =t−1 are two solutions of the differential equation t2y−2y=0 for t > 0.Then show that y=c1t2 +c2t−1 is also a solution of this equation for any c1 and c2.
Answer:
Step-by-step explanation:
Consider first
[tex]y_1(t) = t^2\\[/tex]
We differentiate this two times to get
[tex]y_1'(t) = 2t\\y_1"(t) =2[/tex]
Substitute in the given equation
[tex]t^2 (2) -2(t^2 =0[/tex]
Hence satisfied
Consider II equation
[tex]y_2(t) = t^{-1} \\y_2'(t) = - t^{-2}\\y_2"(t) = -2 t^{-3}[/tex]
Substitute in the given equation to get
[tex]t^2 (-2 t^{-3})+2 t^{-1} = 0[/tex]
Hence satisfied
Together if we have
[tex]y = c_1 t^2 +c_2 t^{-1}[/tex]
being linear combination of two solutions
automatically this also will satisfiy the DE Or
this is a solution to the given DE
Since this expression is identically zero for any values of c1 and c2, we can conclude that y=c1t2 +c2t−1 is also a solution of the differential equation for t > 0.
Verifying that y1(t) =t2 is a solution of the differential equation t2y−2y=0:
Substituting y1(t) =t2 into the differential equation, we get:
t2(t2) - 2(t2) = 0
t4 - 2t2 = 0
2t2(t2 - 1) = 0
Since t > 0, t2 - 1 = 0
t2 = 1
t = 1 or t = -1
However, t > 0, so the only valid solution is t = 1. Therefore, y1(t) =t2 is indeed a solution of the differential equation for t > 0.
Verifying that y2(t) =t−1 is a solution of the differential equation t2y−2y=0:
Substituting y2(t) =t−1 into the differential equation, we get:
t2(t−1) - 2(t−1) = 0
t3 - t - 2t + 2 = 0
t3 - 3t + 2 = 0
(t - 1)(t2 - 2t + 2) = 0
Since t > 0, t2 - 2t + 2 = 0
(t - 1)(t - 1) = 0
t = 1 or t = 1
However, t > 0, so the only valid solution is t = 1. Therefore, y2(t) =t−1 is indeed a solution of the differential equation for t > 0.
Showing that y=c1t2 +c2t−1 is also a solution of this equation for any c1 and c2:
Substituting y=c1t2 +c2t−1 into the differential equation, we get:
t2(c1t2 +c2t−1) - 2(c1t2 +c2t−1) = 0
c1t4 +c2t3 - 2c1t2 - 2c2t + 2 = 0
For each of the following pairs of Treasury securities (each with $ 1 comma 000 par value), identify which will have the higher price: a. A three-year zero-coupon bond or a five-year zero-coupon bond? b. A three-year zero-coupon bond or a three-year 4 % coupon bond? c. A two-year 5 % coupon bond or a two-year 6 % coupon bond?
Answer:
Step-by-step explanation:
A. The three-year zero coupon bond, because the future value is receiver sooner, thus the present value is higher.
B. The three year 4% coupon bond because it pays interest payments, whereas the zero coupon bond is pure discount bond.
C. The 6% coupon bond because the coupon (interest) payments are higher.
Final answer:
In summary, a three-year zero-coupon bond has a higher price than a five-year zero-coupon bond due to its shorter maturity. A three-year 4% coupon bond will have a higher price than a three-year zero-coupon bond because it provides interest payments. A two-year 6% coupon bond will have a higher price than a two-year 5% coupon bond due to its higher interest payments.
Explanation:
The Treasury security prices are determined by several factors, including maturity and coupon rates. In answering the questions presented by the student, we compare the prices of different types of Treasury securities based on these attributes.
A three-year zero-coupon bond will have a higher price than a five-year zero-coupon bond because it has a shorter time to maturity, meaning less time for interest rates to affect its price.Between a three-year zero-coupon bond and a three-year 4% coupon bond, the coupon bond will generally have a higher price since it provides periodic interest payments in addition to the repayment of par at maturity.Comparing a two-year 5% coupon bond with a two-year 6% coupon bond, the bond with the higher coupon rate (6%) will typically have a higher price, as it offers more substantial periodic payments.Let u solve cu = 0. Show that any derivative, say w = uxt, also solves cw = 0. In cu = 0, u = u(t, x) do the change of variables (ξ, η) specified below, to find the equation for v(ξ, η). Is it vξξ − c 2vηη = 0? (a) Translation ξ = t − T, η = x − y where y, T are fixed. (b) Dilation ξ = at, η = ax for any constant a. (c) Find the change of variables (ξ, η) = (?, ?) such that v(ξ, η) satisfies 1v = vξξ −vηη = 0
Answer:
See the pictures attached
Step-by-step explanation:
.................................
Option C: The sum of the infinite geometric series is [tex]\frac{15}{2}[/tex]
Explanation:
The sum of infinite geometric series can be determined using the formula,
[tex]S_{\infty}=\frac{a}{1-r}[/tex]
Substituting the values [tex]a=5[/tex] and [tex]r=\frac{1}{3}[/tex] in the above formula, we have,
[tex]S_{\infty}=\frac{5}{1-\frac{1}{3} }[/tex]
Simplifying the denominator by taking LCM.
Thus, we have,
[tex]S_{\infty}=\frac{5}{\frac{2}{3} }[/tex]
Simplifying, we get,
[tex]S_{\infty}=5\times{\frac{3}{2} }[/tex]
Multiplying, we get,
[tex]S_{\infty}={\frac{15}{2} }[/tex]
Thus, the sum of the infinite geometric series is [tex]\frac{15}{2}[/tex]
Hence, Option C is the correct answer.
Answer:
C
Step-by-step explanation:
Sum = a/(1 - r)
= 5/(1 - ⅓)
= 5/(⅔)
= 5 × 3/2
= 15/2
= 7.5
(6 pts) The average age of CEOs is 56 years. Assume the variable is normally distributed. If the SD is four years, find the probability that the age of randomly selected CEO will be between 50 and 55 years old.
Answer:
The probability that the age of a randomly selected CEO will be between 50 and 55 years old is 0.334.
Step-by-step explanation:
We have a normal distribution with mean=56 years and s.d.=4 years.
We have to calculate the probability that a randomly selected CEO have an age between 50 and 55.
We have to calculate the z-value for 50 and 55.
For x=50:
[tex]z=\frac{x-\mu}{\sigma}=\frac{50-56}{4}=\frac{-6}{4}= -1.5[/tex]
For x=55:
[tex]z=\frac{x-\mu}{\sigma}=\frac{55-56}{4}=\frac{-1}{4}=-0.25[/tex]
The probability of being between 50 and 55 years is equal to the difference between the probability of being under 55 years and the probability of being under 50 years:
[tex]P(50<x<55)=P(x<55)-P(x<50)\\\\P(50<x<55)=P(z<-0.25)-P(z<-1.5)\\\\P(50<x<55)=0.40129-0.06681\\\\P(50<x<55)=0.33448[/tex]
Final answer:
The probability that the age of a randomly selected CEO will be between 50 and 55 years old is approximately 0.3345.
Explanation:
To find the probability that the age of a randomly selected CEO will be between 50 and 55 years old, we need to calculate the z-scores for both values and then use the standard normal distribution table.
Step 1: Calculate the z-score for 50 years old:
z = (50 - 56) / 4 = -1.5
Step 2: Calculate the z-score for 55 years old:
z = (55 - 56) / 4 = -0.25
Step 3: Use the standard normal distribution table to find the area to the left of each z-score:
P(z < -1.5) = 0.0668
P(z < -0.25) = 0.4013
Step 4: Calculate the probability between the two z-scores:
P(-1.5 < z < -0.25) = P(z < -0.25) - P(z < -1.5) = 0.4013 - 0.0668 = 0.3345
Therefore, the probability that the age of a randomly selected CEO will be between 50 and 55 years old is approximately 0.3345.
A local department store is going out of business and is selling every item for 40% off the original price.The discount will be taken at the register. Gene buys 2 pairs of shorts for $18.77 each,one polo shirt for $21.87, and a pair of shoes for $34.24. Gene wants to know approximately how much he will pay. A) use your own words to describe the big-picture ideas in the scenario. B) use your own words to list out each important detail in the scenario. C) Estimate (round first) what Gene will pay for the items after the discount is taken.
Answer:
Gene would pay approximately $56.19 for his total purchase of two pairs of shorts, one polo shirt, and one pair of shoes.
Step-by-step explanation:
First identify what you know:
1) Every item in the store is 40% off the ticketed price (original price)
2) Gene buys two pairs of shorts for $18.77 each.
3) Gene buys one polo shirt for $21.87.
4) Gene buys one pair of shoes for $34.24.
5) Discount is taken at the register.
So, assuming that each price given, IS the original price, we need to figure out exactly how much Gene paid with the 40% discount. We can solve this in two different ways.
1) Individually calculate 40% of each price of each item, and then add them up to find the total discounted price of all the items.
2) Add all the original items and calculate 40% off the original price, and subtract that amount from the original price.
I find it simplest to add up the original prices, and find the total, and then calculate the discounted (40%) total.
Original total = (2 x $18.77) + ($21.87) + ($34.24)
Original Price = ($37.54) + ($21.87) + ($34.24)
Original Total Price: $93.65
Now, we know that without the discount, the total for ALL the items would be $93.65. Now, we just need to find out what 40% of 93.65 is! TO do this, simply multiply 93.65 by 40% OR multiply 93.65 by 0.40.
93.65 x 0.40 = 37.46
So, 40% of $93.65 is $37.46. Now we just need to subtract that percentage to find the discounted price!
$93.65 - $37.46 = $56.19
Gene paid approximately $56.19 for two pairs of short, one polo shirt, and one pair of shoes, at a discount of 40% off.
Hope this helps! :)
to compound a custom solution, you are asked to add 10mg of drug to a 500ml bag of 5% dextrose injection. the drug is available in pre mixed bottles of 0.5% solution. how many ml should you add
Answer:
2 mL
Step-by-step explanation:
If the drug is mixed in a solution of 0.5%, it means that for each ml of the solution, there is 0.5% or 0.005 grams of the drug. In order to get 10 mg of the drug, the volume required is:
[tex]0.005\frac{g}{mL}*V = 10*10^{-3} g\\ V= 2\ mL[/tex]
You should add 2 mL of the solution.
*Note that the volume of the bag should not be used since the amount of drug needed was specified in weight and not in concentration.
A study of consumer smoking habits includes 194194 people in the 18-22 age bracket (4848 of whom smoke), 142142 people in the 23-30 age bracket (3333 of whom smoke), and 8989 people in the 31-40 age bracket (2020 of whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 18-22 or does not smoke.
Answer:
0.8753
Step-by-step explanation:
Total Population=194+142+89=425
Population of those who smoke=48+33+20=101
Population of non smokers=425-101=324
Population of those in 18-22 bracket=194
Population of nonsmoker's in 18-22 age bracket=194-48=146
Probability of getting someone who is age 18-22 = [TeX]\frac{194}{425}[/TeX]
Probability of getting someone who does not smoke = [TeX]\frac{324}{425}[/TeX]
Let the event that the person is in age bracket 18-22 be A
Let the event that the person does not smoke be B
Therefore, the probability of getting someone who is age 18-22 or does not smoke
=Probability of getting someone who is age 18-22 + Probability of getting someone who does not smoke - Probability of those who do not smoke and are in the age bracket 18-22
P(A or B)=P(A)+P(B)-P(A and B)
=[TeX]\frac{194}{425}+\frac{324}{425}-\frac{146}{425}[/TeX]
[TeX]=\frac{194+324-146}{425}=\frac{372}{425}=0.8753[/TeX]
The Maclaurin series for sin−1(x) is given by sin−1(x) = x + [infinity] n = 1 1 · 3 · 5 (2n − 1) 2 · 4 · 6 (2n) x2n+1 2n + 1 . Use the first five terms of the Maclaurin series above to approximate sin−1 3 7 . (Round your answer to eight decimal places.)
Answer:
0.44290869
Step-by-step explanation:
The Maclaurin series for sin⁻¹(x) is given by
sin⁻¹(x) = x + [tex]x^{\alpha } _{n=1}[/tex] [tex]\frac{1.3.5...(2n - 1)}{2.4.6...(2n)} * \frac{x^{2n + 1} }{2n + 1}[/tex]
Use the first five terms of the Maclaurin series above to approximate sin⁻¹ [tex]\frac{3}{7}[/tex]. (Round your answer to eight decimal places.)
Answer
sin⁻¹(x) = x + [tex]x^{\alpha } _{n=1}[/tex] [tex]\frac{1.3.5...(2n - 1)}{2.4.6...(2n)} * \frac{x^{2n + 1} }{2n + 1}[/tex]
in the above equation [tex]x^{\alpha } _{n=1}[/tex] summation from n=1 to ∞
we are estimating this for the first 5 terms as follows
sin⁻¹(x) = x + [tex]\frac{1}{2} * \frac{x^{3} }{3}[/tex] + [tex]\frac{1*3}{2*4} * \frac{x^{5} }{5}[/tex] + [tex]\frac{1*3*5}{2*4*6} * \frac{x^{7} }{7}[/tex] + [tex]\frac{1*3*5*7}{2*4*6*8} * \frac{x^{9} }{9}[/tex]
sin⁻¹(x) = x + [tex]\frac{x^{3} }{6}[/tex] + [tex]\frac{3x^{5} }{40}[/tex] +[tex]\frac{15x^{7} }{336}[/tex] + [tex]\frac{105x^{9} }{3456}[/tex]
now to get
sin⁻¹([tex]\frac{3}{7}[/tex]) substitute
hence,
sin⁻¹([tex]\frac{3}{7}[/tex]) = [tex]\frac{3}{7} + \frac{\frac{3}{7} ^{3} }{6} + \frac{3 * \frac{3}{7} ^{5} }{40} + \frac{15* \frac{3}{7} ^{7} }{336} + \frac{105* \frac{3}{7} ^{9} }{3456}[/tex]
sin⁻¹([tex]\frac{3}{7}[/tex]) = 0.42857142 + 0.01311953 + 0.00108437 + 0.00011855 + 0.00001482
= 0.44290869
what is the domain and range on this graph?
Answer:
The answer to your question is Domain (-∞, ∞) Range [-4, ∞)
Step-by-step explanation:
The Domain is the set of all possible values of the independent variable (x).
The Range is the set of all the possible values of the dependent variable when substitute the domain in the function.
On the graph, we find the domain looking at the x-axis
On a graph, we find the range, looking at all the y-axis
In this graph, x has values from -infinite to infinite, then, the domain is (-∞, ∞).
In this graph, y has values from -4 to infinite, then, the range is [-4, ∞)
If every student is independently late with probability 10%, find the probability that in a class of 30 students: a) nobody is late, 4.2% 8.0% 17.4% 33.3% unanswered b) exactly 1 student is late. 3.33% 5.25% 7.75% 14.1%
Answer:
a) 4.2%
b) 14.1%
Step-by-step explanation:
a) 0.9³⁰ = 0.0423911583
b) 30C1 × 0.1 × 0.9²⁹ = 0.1413038609
Answer:
a.) 4.2%
b.) 14.1%
Step-by-step explanation:
We solve using the probability distribution formula for selection and this formula uses the combination formula for estimation.
When choosing a random selection of "r" items from a sample of "n" items, The formula is generally denoted by:
P(X=r) = nCr × p^r × q^n-r.
Where p = probability of success
q= probability of failure.
From the given question,
n = number of samples =30,
p = Probability that a student is late = 10% = 0.1,
q=0.9
a.) when no student is late, that is when r = 0, then
P(X=0) = 30C0 × 0.1^0 × 0.9^30
P(X=0) = 0.0424 = 4.24 ≈ 4.2%
b.) when exactly one student is late, that is when r=1, then
P(X=1) = 30C1 × 0.1¹ × 0.9^29
P(X=1) = 0.1413 = 14.13 ≈ 14.1%