Answer:
19) The domain is (12 , ∞) ⇒ answer C
17) The radical notation is [tex]\sqrt[7]{18y^{2} }[/tex]
10) The definition of number i is √(-1) ⇒ answer C
Step-by-step explanation:
19)
* Lets explain the meaning of the domain of the function
- The domain of any function is the values of x which makes the
function defined
- Examples:
# In the fraction the denominator con not be zero, then if the function
is a rational fraction then the domain is all the values of x except
the values whose make the denominator = 0
# In the even roots we can not put negative numbers under the radical
because there is no even roots for the negative number belonges to
the real numbers, then the domain is all the values of x except the
values whose make the quantity under the radical negative
* Now lets solve the question
∵ f(x) = 3 √(x - 12)
- To find the domain let (x - 12) greater than zero because there is
no square root for negative value
∵ x - 12 > 0 ⇒ add 12 to both sides
∴ x ≥ 12
∴ The domain is all values of x greater than 12
* The domain is (12 , ∞)
17)
* Lets talk about the radical notation
- The radical notation for the fraction power is:
the denominator of the power will be the radical and the numerator
of the power will be the power of the base
- Ex: [tex]x^{\frac{a}{b}}=\sqrt[b]{x^{a}}[/tex]
* Lets solve the problem
∵ (18 y²)^(1/7)
- The power 1/7 will be the radical over (18 y²)
∴ [tex](18y^{2})^{\frac{1}{7}}=\sqrt[7]{18y^{2}}[/tex]
* The radical notation is [tex]\sqrt[7]{18y^{2} }[/tex]
10)
* Lets talk about the imaginary number
- Because there is no even root for negative number, the imaginary
numbers founded to solve this problem
- It is a complex number that can be written as a real number multiplied
by the imaginary unit i, which is defined by i = √(-1) or i² = -1
- Ex: √(-5) = √[-1 × 5] = i√5
* The definition of number i is √(-1)
Given the stem and leaf plot, which of the following statements is true?
STEM LEAF
2 9
3 2 6 7
4 1 2
5 0
6
7 5
a) There are no outliers on the stem and leaf plot; b) the numbers 29 and 75 are the outliers on the stem and leaf plot; c) the number 75 is an outlier on the stem and leaf plot; d) the number 60 is the outlier on the stem and leaf plot.
Answer: c) The number 75 is an outlier on the stem and leaf plot.
Step-by-step explanation:
An outlier is a value in data which is extremely large or small from all the values. It appears detached from the other values in data.When we look in the given stem-leaf plot, there is no leaf attached to the stem with value 6.
It mean there is no value between 50 and 75.
It shows that the value of 75 is detached from the other values in the data.
⇒ The number 75 is an outlier on the stem and leaf plot.
Answer:
The answer is C
Step-by-step explanation:
Walgreens sells both Kleenex and NyQuil. In a random sample of 5000 purchases, it was found that 1200 contained Kleenex and 540 contained NyQuil. These are not independent because sick people are more likely to buy both and healthy people more likely to buy neither. In looking into that, it was found that 360 purchases contained both.
a. Make a two-way table to display this information. Fill in the table with the missing entries, and be sure your table has row and column totals.
b. If we randomly select a purchase that contained Kleenex, what is the probability it also contained NyQuil?
c. What percent of all purchases didn’t contain either product?
Thirty-five math majors, 23 music majors and 65 history majors are randomly selected from 586 math majors, 224 music majors and 254 history majors at the state university. What sampling technique is used?
Answer: Simple random sampling
Step-by-step explanation:
Simple random sampling is a technique of sampling in which an experimenter selects the group of samples or subjects randomly from a large group of population. Each sample is chosen randomly by chance and each entity of a population has equal possibility of being selected as sample.
According to the given situation, simple random sampling is the technique that should be used for sampling.
. Geometry The area of a triangular sign is 33 square meters. The base of the triangle is 1 meter less than double the altitude. Find the altitude and the base of the sign.
Answer: The altitude and the base of the sign are 6 meters and 11 meters respectively.
Step-by-step explanation:
Since we have given that
Area of triangular sign = 33 sq. meters
Let the altitude of the triangle be 'x'.
Let the base of the triangle be ' 2x-1'.
As we know the formula for "Area of triangle ":
[tex]Area=\dfrac{1}{2}\times base\times height\\\\33=\dfrac{1}{2}\times x(2x-1)\\\\33\times 2=2x^2-x\\\\66=2x^2-x\\\\2x^2-x-66=0\\\\2x^2-12x+11x-66=0\\\\2x(x-6)+11(x-6)=0\\\\(2x+11)(x-6)=0\\\\x=-\dfrac{11}{2},6\\\\x=-5.5,6[/tex]
Discarded the negative value of x for dimensions:
So, altitude of triangle becomes 6 meters
Base of triangle would be [tex]2(6)-1=12-1=11\ meters[/tex]
Functions f(x) and g(x) are shown below:
f(x) = 2cos(x)
g(x) = (3)sin(x+pi), a graph of sine function which starts at 0 comma 0 and decreases to the minimum of pi over 2 then increases to the maximum 3 pi over 2 then decreases to 2pi where the cycle repeats.
Using complete sentences, explain how to find the maximum value for each function and determine which function has the largest maximum y-value.
Please please please help!!!
Try this option:
the functions y=sin(x) and y=cos(x) are the standart functions; it means, the period of each is 2π, maximum value is '1', minimum value is '-1'.
Using the properties described above, to find the max. value for the function y=2cos(x): 2*1=2, where 2- the amplitude of the given function, 1- maximum of the standart function;
the max. value for the function y=3sin(x+π): 3*1=3, where 3 - the amplitude of the given function, 1- maximum of the standart function.
For more details see the attached picture.
In other words, common equation for such functions is y=A*sin(ωx+Ф), where A- amplitude, ω - frequency and Ф - initial phase of the given function.
Max. and min. values depend on A, all the points of the function repeat every 2π, including max. and min. values.
[tex]f(x)=2\cos x[/tex]
We know that the function f(x) is a cosine function and the maximum value is obtained by the function when this cosine function takes the maximum value.
We know that:
[tex]-1\leq \cos x\leq 1[/tex]
This means that the cosine function takes the maximum values as: 1
and when [tex]\cos x=1[/tex] then
[tex]f(x)=2\times 1\\\\i.e.\\\\f(x)=2[/tex]
i.e. Maximum value of function f(x)=2
The function g(x) is given by:[tex]g(x)=3\sin (x+\pi)[/tex]
Again the function g(x) will attain the maximum value when the function sine will takes the maximum value.
We know that the range of the sine function is: [-1,1]
This means that the maximum value of sine function is: 1
Hence, when [tex]\sin (x+\pi)=1[/tex] then,
[tex]g(x)=3\times 1\\\\i.e.\\\\g(x)=3[/tex]
i.e. Maximum value of function g(x)=3
The function g(x) has the largest maximum value.
( Since 3>2)
Expand the logarithm log 7(3x − 2)^2
[tex]\textbf{Transform}\\ \textrm{log} (7(3x -2)^2) \textbf{ into} \textrm{ log}(7) + \textrm{log}(3x-2)^2\\\\ \textbf{Expand} \\ \text{log}(3x-2)^2\\\\ \text{You can move 2 outside of }\text{log}(3x-2)^2\\\\ \textbf{Answer}\\ \text{log }7 + 2\text{ log}(3x-2)[/tex]
In a sample of 8 high school students, they spent an average of 28.8 hours each week doing sports with a sample standard deviation of 3.2 hours. Find the 95% confidence interval, assuming the times are normally distributed.
Answer:
[tex]26.12\:<\:\mu\:<\:31.48[/tex]
Step-by-step explanation:
Since the population standard deviation [tex]\sigma[/tex] is unknown, and the sample standard deviation [tex]s[/tex], must replace it, the [tex]t[/tex] distribution must be used for the confidence interval.
The sample size is n=8.
The degree of freedom is [tex]df=n-1[/tex], [tex]\implies df=8-1=7[/tex].
With 95% confidence level, the [tex]\alpha-level[/tex](significance level) is 5%.
Hence with 7 degrees of freedom, [tex]t_{\frac{\alpha}{2} }=2.365[/tex]. (Read from the t-distribution table see attachment)
The 95% confidence interval can be found by using the formula:
[tex]\bar X-t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n} } )\:<\:\mu\:<\:\bar X+t_{\frac{\alpha}{2}}(\frac{s}{\sqrt{n} } )[/tex].
The sample mean is [tex]\bar X=28.8[/tex] hours.
The sample sample standard deviation is [tex]s=3.2[/tex] hours.
We now substitute all these values into the formula to obtain:
[tex]28.8-2.365(\frac{3.2}{\sqrt{8} } )\:<\:\mu\:<\:28.8+2.365(\frac{3.2}{\sqrt{8} } )[/tex].
[tex]26.12\:<\:\mu\:<\:31.48[/tex]
We are 95% confident that the population mean is between 26.12 and 31.48 hours.
You find the following financial information about a company: net working capital = $735; fixed assets = $5,385; total assets = $8,190; and long-term debt = $4,281. What are the company's total liabilities?
Answer:
Company's total liabilities are $6,351.
Step-by-step explanation:
A company's financial information is given, we have to calculate the company's total liabilities.
Total assets = Fixed assets + current assets
Current assets = Total assets - Fixed assets
= $8,190 - $5,385
= $2,805
Networking capital = Current assets - Current liabilities
$735 = $2,805 - current liabilities
Current liabilities = $2,805 - $735
Current liabilities = $2,070
Total liabilities = Long term debt + current liabilities
= $4,281 + $2,070
= $6,351
Company's total liabilities are $6,351.
Use a Venn diagram to answer the question. A survey of 180 families showed that 67 had a dog; 52 had a cat; 22 had a dog and a cat; 70 had neither a cat nor a dog, and in addition did not have a parakeet; 4 had a cat, a dog, and a parakeet. How many had a parakeet only?
Answer:
There are 13 families had a parakeet only
Step-by-step explanation:
* Lets explain the problem
- There are 180 families
- 67 families had a dog
- 52 families had a cat
- 22 families had a dog and a cat
- 70 had neither a cat nor a dog, and in addition did not have a
parakeet
- 4 had a cat, a dog, and a parakeet (4 is a part of 22 and 22 is a part
of 67 and 520
* We will explain the Venn-diagram
- A rectangle represent the total of the families
- Three intersected circles:
C represented the cat
D represented the dog
P represented the parakeet
- The common part of the three circle had 4 families
- The common part between the circle of the cat and the circle of the
dog only had 22 - 4 = 18 families
- The common part between the circle of the dog and the circle of the
parakeet only had a families
- The common part between the circle of the cat and the circle of the
parakeet only had b families
- The non-intersected part of the circle of the dog had 67 - 22 - a =
45 - a families
had dogs only
- The non-intersected part of the circle of the cat had 52 - 22 - b =
30 - b families
had cats only
- The non-intersected part of the circle of the parakeet had c families
had parakeets only
- The part out side the circles and inside the triangle has 70 families
- Look to the attached graph for more under stand
∵ The total of the families is 180
∴ The sum of all steps above is 180
∴ 45 - a + 18 + 4 + 30 - b + b + c + a + 70 = 180 ⇒ simplify
- (-a) will cancel (a) and (-b) will cancel (b)
∴ (45 + 18 + 4 + 30 + 70) + (-a + a) + (-b + b) + c = 180
∴ 167 + c = 180 ⇒ subtract 167 from both sides
∴ c = 180 - 167 = 13 families
* There are 13 families had a parakeet only
A clothing store has just advertised at a bargain price, that a customer tries on a t-shirt is 0.40. a new range of t-shirts. The probability If a customer tries a t-shirt, the probability that they will buy it is 0.70 while if a customer does not try on a t-shirt, the probability that they will buy it is 0.15. A customer has just bought a t-shirt. Find the probability that the customer did not try on the t- shirt.
The probability that the customer did not try on the t- shirt is:
0.2432
Step-by-step explanation:It is given that the probability that a customer tries a t-shirt is: 0.40
After he will try
The probability that he will buy the t-shirt is: 0.70
The probability that he will not try t-shirt is: 1-0.40=0.60
If he will not try
then the probability of buying a t-shirt is: 0.15
This means that the total probability of Buying a t-shirt is:
0.40×0.70+0.60×0.15
= 0.37
It is given that the customer bought a t-shirt, we need to find the probability that he did not try the t-shirt:
The probability is given by:
[tex]\dfrac{0.60\times 0.15}{0.37}\\\\\\=\dfrac{0.09}{0.37}\\\\\\=0.2432[/tex]
The question asks for the conditional probability that a customer did not try on a t-shirt given that they bought it. This probability value can be calculated using the total probability theorem and Bayes' theorem. It is found to be approximately 0.159.
Explanation:This question falls under the topic of probability, specifically conditional probability and Bayes' theorem. The probability values are given in the problem statement but it seeks the probability that a customer did not try on a t-shirt given that they bought it.
Let A represent the event that a customer buys a t-shirt and B1 represent the event that a customer tries on a t-shirt whereas B2 represent the event that the customer did not try on a t-shirt.
We are given that P(B1) = 0.40, P(A|B1) = 0.70 and P(A|B2) = 0.15. We are asked to find P(B2|A).
To find P(B2|A), we use Bayes' theorem:
P(B2|A) = (P(B2) * P(A|B2)) / P(A)
We know that P(B1) + P(B2) = 1, so P(B2) = 1 - P(B1) = 1 - 0.40 = 0.60.
The probability that a customer tried a t-shirt is 0.40 and bought it is 0.70, that yields the joint probability P(A, B1) = P(B1) * P(A|B1) = 0.40 * 0.70 = 0.28. Similarly the joint probability P(A, B2) = P(B2) * P(A|B2) = 0.60 * 0.15 = 0.09.
P(A) is the probability that a customer buys a t-shirt and is calculated using the total probability theorem as P(A) = P(A, B1) + P(A, B2) = 0.28 + 0.09 = 0.37.
Substituting the values we calculated into Bayes' theorem, we find:
P(B2|A) = (0.60 * 0.15) / 0.37 ≈ 0.24.
So, the probability that a customer did not try the t-shirt given that they bought it is approximately 0.24.
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A city temperature is modeled as a normal random variable with mean and standard deviation both equal to 10 degrees Celsius. What is the probability that the temperature at a randomly chosen time will be less or equal to 15 degrees Celsius?
Answer: 0.6915
Step-by-step explanation:
Given : [tex]\text{Mean}=\mu=10^{\circ}C[/tex]
[tex]\text{Standard deviation}=\sigma=10^{\circ}C[/tex]
Since , the distribution follows a Normal distribution.
The formula to calculate the z-score is given by :-
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x=[tex]15^{\circ}C[/tex]
[tex]z=\dfrac{15-10}{10}=0.5[/tex]
The p-value = [tex]P(z\leq0.5)=0.6914625\approx0.6915[/tex]
Hence, the required probability : 0.6915
The probability that the temperature will be less than or equal to 15 degrees Celsius in a city with a mean temperature of 10 degrees Celsius and a standard deviation of 10 degrees Celsius is approximately 69.15%.
To find the probability that the temperature will be less than or equal to 15 degrees Celsius when the mean and standard deviation are both 10 degrees Celsius, we can calculate the z-score.
The z-score formula is:
z = (X - μ) / Σ
For X = 15, μ = 10, and Σ = 10, the z-score is:
z = (15 - 10) / 10 = 0.5
We then look up the z-score of 0.5 in a standard normal distribution table or use a calculator to find the cumulative probability corresponding to the z-score. The probability that the temperature is less than or equal to 15 degrees Celsius is approximately 0.6915 (or 69.15%).
Problem Page
The Bailey family and the Harris family each used their sprinklers last summer. The water output rate for the Bailey family's sprinkler was 15 L per hour. The water output rate for the Harris family's sprinkler was 40 L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 1325
. How long was each sprinkler used?
Answer:
Bailey family's sprinkler was used for 35 hours and Harris family's sprinkler was used for 20 hours.
Step-by-step explanation:
Set up a system of equations.
Let be "b" the time Bailey family's sprinkler was used and "h" the time Harris family's sprinkler was used.
Then:
[tex]\left \{ {{b+h=55} \atop {15b+40h= 1,325}} \right.[/tex]
You can use the Elimination method. Multiply the first equation by -15, then add both equations and solve for "h":
[tex]\left \{ {{-15b-15h=-825} \atop {15b+40h= 1,325}} \right.\\.............................\\25h=500\\\\h=\frac{500}{25}\\\\h=20[/tex]
Substitute [tex]h=20[/tex] into an original equation and solve for "b":
[tex]b+20=55\\\\b=55-20\\\\b=35[/tex]
A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. Imagine you stick your hand into the refrigerator and pull out a piece of fruit at random. What is the sample space for your action? ????={6, 5, 10, 3, 7, 11, 2}S={6, 5, 10, 3, 7, 11, 2} ????={apple, orange, banana, pear, peach, plum, mango}S={apple, orange, banana, pear, peach, plum, mango} ????={6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, 2 mangos}
Answer:
4th Option is correct.
Step-by-step explanation:
Given:
Number of apples in refrigerator = 6
Number of oranges in refrigerator = 5
Number of bananas in refrigerator = 10
Number of pears in refrigerator = 3
Number of peaches in refrigerator = 7
Number of plums in refrigerator = 11
Number of mangoes in refrigerator = 2
A piece is randomly taken out from refrigerator.
To find: Sample Space of the experiment
Sample Space : It is a set which contain all the possible outcome / results of the experiment.
So, here Sample Space = { 6 apples , 5 oranges , 10 bananas , 3 pears , 7 peaches , 11 plums , 2 mangoes }
Therefore, 4th Option is correct.
Solve the following system. (Use (x,y) format in a single answer space.)
x2 + y2 = 25
y2 - x2 = 7
The solution set is
Answer:
(3;4); (-3;-4); (3;-4); (-3;4)
Step-by-step explanation:
for more information see the attached picture.
Answer: The required solution set is
(x, y) = (3, 4), (-3, 4), (3, -4) and (-3, -4).
Step-by-step explanation: We are given to solve the following system :
[tex]x^2+y^2=25~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\y^2-x^2=7~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)[/tex]
We will be using the method of Elimination to solve the problem.
Adding equations (i) and (ii), we have
[tex](x^2+y^2)+(y^2-x^2)=25+7\\\\\Rightarrow 2y^2=32\\\\\Rightarrow y^2=16\\\\\Rightarrow y=\pm\sqrt{16}~~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]\\\\\Rightarrow y=\pm4.[/tex]
From equation (ii), we get
[tex](\pm4)^2-x^2=7\\\\\Rightarrow 16-x^2=7\\\\\Rightarrow x^2=16-7\\\\\Rightarrow x^2=9\\\\\Rightarrow x=\pm\sqrt9~~~~~~~~~~~~~~~~~~[\textup{taking square root on both sides}]\\\\\Rightarrow x=\pm3.[/tex]
Thus, the required solution set is
(x, y) = (3, 4), (-3, 4), (3, -4) and (-3, -4).
You want to have $600,000 when you retire in 10 years. If you can earn 8% interest compounded monthly, how much would you need to deposit now into the account to reach your retirement goal?
Answer:
$270,314.08
Step-by-step explanation:
The multiplier each month is 1+0.08/12 ≈ 1.0066667, so after 120 months, the amount is multiplied by (1.0066667)^120 ≈ 2.2196402. The amount needed is ...
$600,000/2.2196402 ≈ $270,314.08
To reach a retirement goal of $600,000 in 10 years with an 8% interest rate compounded monthly, you would need to deposit approximately $277,002.66 now.
Explanation:
In this case, we're using a formula to determine the amount needed to deposit today (P) for a future goal ($600,000) using an interest rate (r) of 8% compounded monthly for ten years. The formula to use is P = F / (1 + r/n)^(nt), where:
F is the future value of the investment ($600,000) r is the annual interest rate (8% or 0.08 as a decimal) n is the number of times that interest is compounded per unit t (12 times a year for our case) t is the time the money is invested for in years (10 years).
So, you need to plug these figures into the equation: P = 600,000 / (1 + 0.08/12)^(12*10). After doing the math, you would need to deposit around $277,002.66 now to reach your retirement goal of $600,000 in ten years given an 8% annual interest rate compounded monthly.
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The features of a Coverdell Education Savings Account include all of the following EXCEPT: (A) The contributions are deductible. (B) $2,000 is the maximum contribution in any one year. (C) Withdrawals are tax free. (D) Contributions are phased out for certain taxpayers who have adjusted gross income above a certain level.
Answer:
c
Step-by-step explanation:
32°
45°
48°
20°
FIND WZC
Answer:
48°
Step-by-step explanation:
In triangle CYZ, angles Y and C are complementary, so ...
(5x +22) + (12x) = 90
17x +22 = 90
17x = 68
x = 4
The angle of interest is 12x° , so has measure
12·4° = 48°
Answer:
m∠WZC=48°
Step-by-step explanation:
we know that
If ZC is an altitude
then
Triangle YZC is a right triangle
therefore
m∠CYW+m∠WZC=90° ----> by complementary angles
substitute
(5x+22)°+(12x)°=90°
17x=(90-22)°
x=4°
Find the measure of angle WZC
m∠WZC=(12x)°=12*4=48°
In a classroom of 32 Students , 14 are male . What % of the Class is female ????
Answer:
56.25%
Step-by-step explanation:
If 14 are male, that means that 32 - 14 = 18 are female. If 18/32 are female, do the division and then multiply by 100 to get the percentage.
18/32 = .5625
.5625 × 100% = 56.25%
Consider the equation of the line, c(c2 + 2)y = c − x where c > 0 is a constant. (a) Find the coordinates of the x-intercept and the y-intercept. x-intercept ( , ) y-intercept ( , ) (b) Find a formula for the area of the triangle enclosed between the line, the x-axis, and the y-axis. A(c) =
The equation of the line indicates that the x and y-intercept and the formula for the area of the triangle are;
x-intercept (c, 0)
y-intercept (0, 1/(c² + 7))
A(c) = c/(2·(c² + 7))
The steps used to find the x-intercept and the y-intercept are presented as follows;
The equation of the line c·(c² + 7)·y = c - x, can be expressed in the slope intercept form to find the coordinates of the x-intercept and the coordinates of the y-intercept as follows;
c·(c² + 7)·y = c - x
y = (c - x)/(c·(c² + 7))
y = c/(c·(c² + 7)) - x/(c·(c² + 7))
y = 1/((c² + 7)) - x/(c·(c² + 7))
The above equation is in the slope-intercept form, y = m·x + c
Where c is the y-coordinate of the y-intercept, and (0, c) ids the coordinate of the y-intercept; Therefore, the coordinates of the y-intercept is; (0, 1/((c² + 7)))
The coordinate of the x-intercept can be found by plugging in y = 0, in the above equation to get;
0 = 1/((c² + 7)) - x/(c·(c² + 7))
x/(c·(c² + 7)) = 1/((c² + 7))
x = (c·(c² + 7))/((c² + 7))
(c·(c² + 7))/((c² + 7)) = c
x = c
Therefore coordinates of the x-intercept is; (c, 0)
The triangle enclosed by the line and the x-axis and y-axis is a right triangle, therefore;
The positive x and y-values of the x-intercept and y-intercept indicates that the area of the triangle is the product half the distance from the origin to the y-intercept and the distance from the origin to the x-intercept
Area = (1/2) × (1/((c² + 7)) - 0) × (c - 0)
(1/2) × (1/((c² + 7))) × (c) = c/(2·(c² + 7)
Area of the triangle, A(c) = c/(2·(c² + 7)
The complete question found through search can be presented as follows;
Consider the equation of the line c·(c² + 7)·y = c - x where c > 0 is a constant
(a) Find the coordinates of the x-intercept and the y-intercept
x-intercept ( , )
y-intercept ( , )
(b) Find a formula for the area of the triangle enclosed between the line, the x-axis and the y-axis
An old picture frame has dimensions 33 inches by 24 inches. What one length must be cut from each dimension so that the ratio of the shorter side to the longer side is 2/3?
Final answer:
To achieve a 2:3 ratio for the picture frame's dimensions, 6 inches must be cut from both the length and width.
Explanation:
To resize the picture frame while maintaining a specific ratio between its sides, we need to establish a proportion based on the desired ratio of 2:3 (shorter side to the longer side). Let's denote the amount that needs to be cut from the shorter side as x inches, and the amount to be cut from the longer side as y inches. After cutting, the dimensions of the frame will be (33 - y) inches by (24 - x) inches.
We want to set up a proportion to reflect the desired ratio: (24 - x)/(33 - y) = 2/3. To solve for x and y, we can use the property that cross-multiplying the terms in a proportion gives us an equality: 3(24 - x) = 2(33 - y). This simplifies to 72 - 3x = 66 - 2y. We also know that the amount cut off from both dimensions should be the same, thus x = y. Substituting y for x gives us 72 - 3x = 66 - 2x. Solving this equation, we get x = 6 inches. So, 6 inches must be cut from both the length and the width to achieve the 2:3 ratio.
If (-3, y) lles on the graph of y = 3*, then y =
The line y = 3 means that x = 0.
The point (-3, y) tells me that x = -3 when y is 3.
So, y = 3 completes the point (-3, 3).
The Royal Fruit Company produces two types of fruit drinks. The first type is 55% pure fruit juice, and the second type is 100% pure fruit juice. The company is attempting to produce a fruit drink that contains 70% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 240 pints of a mixture that is 70%
pure fruit juice?
Answer:
First type of fruit drinks: 160 pints
Second type of fruit drinks: 80 pints
Step-by-step explanation:
Let's call A the amount of first type of fruit drinks. 5.5% pure fruit juice
Let's call B the amount of second type of fruit drinks. 100% pure fruit juice
The resulting mixture should have 70% pure fruit juice and 240 pints.
Then we know that the total amount of mixture will be:
[tex]A + B = 240[/tex]
Then the total amount of pure fruit juice in the mixture will be:
[tex]0.55A + B = 0.7 * 240[/tex]
[tex]0.55A + B = 168[/tex]
Then we have two equations and two unknowns so we solve the system of equations. Multiply the first equation by -1 and add it to the second equation:
[tex]-A -B = -240[/tex]
[tex]-A -B = -240[/tex]
+
[tex]0.55A + B = 168[/tex]
--------------------------------------
[tex]-0.45A = -72[/tex]
[tex]A = \frac{-72}{-0.45}[/tex]
[tex]A = 160\ pints[/tex]
We substitute the value of A into one of the two equations and solve for B.
[tex]160 + B = 240[/tex]
[tex]B = 80\ pints[/tex]
To make 240 pints of a mixture that is 70% pure fruit juice, you will need 160 pints of the first type of fruit drink (55% pure fruit juice) and 80 pints of the second type of fruit drink (100% pure fruit juice).
Explanation:To solve this problem, we can set up a system of equations. Let's say x represents the number of pints of the first type of fruit drink (55% pure fruit juice) and y represents the number of pints of the second type of fruit drink (100% pure fruit juice). We know that the total number of pints of the mixture is 240, so we can write the equation x + y = 240. We also know that the desired percentage of pure fruit juice in the mixture is 70%, so we can write the equation (55% * x + 100% * y) / 240 = 70%. To solve this system of equations, we can use substitution or elimination method. Let's use substitution:
From the first equation, we can solve for x in terms of y: x = 240 - y. Substituting this into the second equation, we get ((55% * (240 - y)) + 100% * y) / 240 = 70%. Simplifying the equation, we have (0.55(240 - y) + y) / 240 = 0.70. Distributing and combining like terms, we get (132 - 0.55y + y) / 240 = 0.70. Simplifying further, we have (132 + 0.45y) / 240 = 0.70. Cross multiplying, we get 132 + 0.45y = 0.70 * 240. Simplifying, we have 132 + 0.45y = 168. Multiplying 0.45 with y, we get 0.45y = 168 - 132. Subtracting 132 from 168, we get 0.45y = 36. Dividing both sides of the equation by 0.45, we get y = 36 / 0.45. Evaluating this expression, we get y = 80. So, the number of pints of the second type of fruit drink (100% pure fruit juice) needed is 80. Substituting this value back into the first equation, we can solve for x: x + 80 = 240. Subtracting 80 from both sides of the equation, we get x = 240 - 80. Evaluating this expression, we get x = 160. Therefore, the number of pints of the first type of fruit drink (55% pure fruit juice) needed is 160.
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Define a function sinc(x) (pronounced "sink of x") by: sinc(x)= {sin(x)/x if x is not 0 {1 if x = 0 Use this list of Basic Taylor Series to find the Taylor Series for f(x) = (sinc(x)) based at 0. a.Give your answer using summation notation. b.Give the interval on which the series converges.
You probably know that
[tex]\sin x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}[/tex]
Then
[tex]\mathrm{sinc}\,x=\displaystyle\frac1x\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)!}=\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}[/tex]
when [tex]x\neq0[/tex], and 1 when [tex]x=0[/tex].
By the ratio test, the series converges if the following limit is less than 1:
[tex]\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+2}}{(2n+3)!}}{\frac{(-1)^nx^{2n}}{(2n+1)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)!}{(2n+3)!}[/tex]
The limit is 0, so the series converges for all [tex]x[/tex].
It is known that for all tests administered last year, the distribution of scores was approximately normal with mean 74 and standard deviation 7.1. a. A particular employer requires job candidates to score at least 80 on the dexterity test. Approximately what percentage of the test scores during the past year exceeded 80?
Answer: 19.77%
Step-by-step explanation:
Given: Mean : [tex]\mu=74[/tex]
Standard deviation : [tex]\sigma = 7.1[/tex]
The formula to calculate z-score is given by :_
[tex]z=\dfrac{x-\mu}{\sigma}[/tex]
For x= 80, we have
[tex]z=\dfrac{80-74}{7.1}\approx0.85[/tex]
The P-value = [tex]P(z>0.85)=1-P(z<0.85)=1-0.8023374=0.1976626[/tex]
In percent , [tex]0.1976626\times100=19.76626\%\approx19.77\%[/tex]
Hence, the approximate percentage of the test scores during the past year exceeded 80 =19.77%
Two friends went to get ice cream sundaes. They each chose a flavor of ice cream from a list of vanilla and chocolate and toppings from a list of hot fudge, strawberries, sprinkles, peanuts, and whipped cream. Use the sets below describing their choices and find B'.
Let A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
Let B = {vanilla, hot fudge, sprinkles, whipped cream}
Let C = {chocolate, hot fudge, peanuts, whipped cream}
{vanilla, hot fudge, sprinkles, whipped cream}
{chocolate, strawberries, peanuts}
{chocolate, strawberries}
{vanilla, strawberries, peanuts}
Answer:
{chocolate, strawberries, peanuts}
Step-by-step explanation:
Given that three sets are
Let A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
Let B = {vanilla, hot fudge, sprinkles, whipped cream}
Let C = {chocolate, hot fudge, peanuts, whipped cream}
Then Universal set U = AUBUC
= {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}
B'=elements in U but not in B
={chocolate, strawberries, peanuts}
The resulting set is B' = {chocolate, strawberries, peanuts}.
To solve for B', we first need to understand that B' (B complement) consists of elements that are in set A but not in set B.
Given the sets:
A = {vanilla, chocolate, hot fudge, strawberries, sprinkles, peanuts, whipped cream}B = {vanilla, hot fudge, sprinkles, whipped cream}C = {chocolate, hot fudge, peanuts, whipped cream}Set B includes: vanilla, hot fudge, sprinkles, and whipped cream. Therefore, B' will be the elements of set A excluding those in B.
Thus, B' is:
chocolatestrawberriespeanutsTherefore, the set B' = {chocolate, strawberries, peanuts}.
This method can help you understand combinations without repetition effectively.
A sanitation supervisor is interested in testing to see if the mean amount of garbage per bin is different from 50. In a random sample of 36 bins, the sample mean amount was 48.47 pounds and the sample standard deviation was 3.1 pounds. Conduct the appropriate hypothesis test using a 0.05 level of significance.\
Answer with explanation:
Given : Sample mean =[tex]\overline{x}=\text{48.47 pounds}[/tex]
Standard deviation : [tex]\sigma=\text{ 3.1 pounds.}[/tex]
Sample size : n = 36
Claim : [tex]\mu\neq50[/tex]
∴ [tex]H_0:\mu=50[/tex]
[tex]H_1:\mu\neq50[/tex]
Since the alternative hypothesis is two tail , then the test is two tail test.
By using a z statistic and a 0.05 level of significance. Reject [tex]H_0[/tex] if z < -1.960 or is z> 1.960.
Then , the test static for population mean is given by :-
[tex]z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
[tex]z=\dfrac{48.47-50}{\dfrac{3.1}{\sqrt{36}}}\approx-2.96[/tex]
We reject [tex]H_0[/tex] since [tex]-2.96\leq -1.96[/tex]. We have statistically significant evidence at [tex]\alpha=0.05[/tex] to show that the mean amount of garbage per bin is not different from 50.
To test if the mean amount of garbage per bin is different from 50, we set up null and alternative hypotheses and use a z-test to calculate a z-score. The z-score of -1.49 falls within the acceptance region; hence, we fail to reject the null hypothesis, implying that the mean amount of garbage per bin is not significantly different from 50 pounds.
Explanation:First, we need to set up our null and alternative hypotheses. The null hypothesis (H0) states that the mean amount of garbage is 50, or μ = 50. The alternative hypothesis (H1) is that the mean is different from 50, or μ ≠ 50.
Second, calculate the z-score which is given by the formula Z = (X - μ) / (σ/√n), where X is the sample mean, μ is the population mean, σ is the standard deviation and n is the sample size. Substituting the given values, we get Z = (48.47 - 50) / (3.1 / √36) = -1.49
For a two-tailed test, we are checking if the z-score falls within the rejection region. Since our level of significance is 0.05, the critical values for a two-tailed test are -1.96 and 1.96. Our computed z-score of -1.49 falls within the acceptance region. Therefore, we fail to reject the null hypothesis. This suggests that the mean amount of garbage per bin is not statistically different from 50 pounds at the 0.05 level of significance.
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Round the following decimal to the nearest tenth.
543.67 ◯543.6 ◯543.68 ◯543.7 ◯543.670 ◯544
Answer:
543.7
Step-by-step explanation:
Answer:
543.7, 543.6, 543.7, 543.7, 543., 544.0
Step-by-step explanation:
trains A and B are 240 miles apart. Both start at the same time and travel toward each other. They meet 3 hours later. The speed of train A is 20 miles faster than train B. Find the speed of each train.
Final answer:
Train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.
Explanation:
Let's say the speed of train B is x miles per hour. Since train A is 20 miles per hour faster, the speed of train A is x + 20 miles per hour.
Distance = Speed x Time
For train A, Distance = (x +20) * 3
For train B, Distance = x * 3
Since they meet 240 miles apart, the sum of their distances is 240:
(x + 20) * 3 + x * 3 = 240
3x + 60 + 3x = 240
6x = 180
x = 30
Hence, train B travels at a speed of 30 miles per hour and train A travels at a speed of 50 miles per hour.
VWX and NOP are similar. If mV = 44° and mP = 66°, what is
mW?
Answer: 70°
Step-by-step explanation:
ΔVWX ≅ ΔNOP ⇒ ∠V=∠N, ∠W=∠O, and ∠X=∠P
Since ∠V=44° and ∠P=66°=∠X
then we can use the Triangle Sum Theorem:
∠V + ∠W + ∠X = 180°
44° + ∠W + 66° = 180°
∠W + 110° = 180°
∠W = 70°
Find S for the given geometric series. Round answers to the nearest hundredth, if necessary.
a1=0.2, a5=259.2, r=6
a) 311
b)51
c)222.2
d)624.96
Answer:
A
Step-by-step explanation:
Sum of the first n terms of a geometric series is:
S = a₁ (1 - r^n) / (1 - r)
Here, a₁ = 0.2, r = 6, and n = 5.
S = 0.2 (1 - 6^5) / (1 - 6)
S = 311
Answer:
Option A
Step-by-step explanation:
For the given geometric series
a₁ = 0.2
a₅ = 259.2
r = 6
Then we have to find the sum of initial 5 terms of this series
[tex]S_{n} =\frac{a_1(r^4-1)}{(r-1)}=\frac{0.2(6^3-1)}{(6-1)}[/tex]
[tex]=\frac{0.2(7776-1)}{5}[/tex]
[tex]\frac{0.2\times 7775}{5}[/tex]
= 311
Option A is the answer.