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:
What is the simplest form of
Answer:
The simplest form of [tex]\sqrt[3]{27a^{3}b^{7}}[/tex] is
3ab²(∛b)
Step-by-step explanation:
The given term is:
[tex]\sqrt[3]{27a^{3}b^{7}}[/tex]
To convert it into its simplest form, we will apply simple mathematical rules to simplify the power of individual terms.
[tex]\sqrt[3]{27a^{3}b^{7}}\\= \sqrt[3]{3^{3} a^{3}b^{7}}\\= \sqrt[3]{3^{3}a^{3}b^{6}b}\\= 3^{3/3} a^{3/3}b^{6/3}b^{1/3}}\\= 3ab^{2}(\sqrt[3]{b})[/tex]
While simplifying the term, we basically took the cube root of individual terms. The powers cancelled out cube root for some terms. In the end, we were left with the simplest form of the expression.
Last year, Susan had 10,000 to invest. She invested some of it in an account that paid 6%
simple interest per year, and she invested the rest in an account that paid 5% simple interest per year. After one year, she received a total of %560 in interest. How much did she invest in each account?
Answer:
In the account that paid 6% Susan invest [tex]\$6,000[/tex]
In the account that paid 5% Susan invest [tex]\$4,000[/tex]
Step-by-step explanation:
we know that
The simple interest formula is equal to
[tex]I=P(rt)[/tex]
where
I is the Final Interest Value
P is the Principal amount of money to be invested
r is the rate of interest
t is Number of Time Periods
Part a) account that paid 6% simple interest per year
in this problem we have
[tex]t=1\ years\\ P=\$x\\r=0.06[/tex]
substitute in the formula above
[tex]I1=x(0.06*1)[/tex]
[tex]I1=0.06x[/tex]
Part b) account that paid 5% simple interest per year
in this problem we have
[tex]t=1\ years\\ P=\$10,000-\$x\\r=0.05[/tex]
substitute in the formula above
[tex]I2=(10,000-x)(0.05*1)[/tex]
[tex]I2=500-0.05x[/tex]
we know that
[tex]I1+I2=\$560[/tex]
substitute and solve for x
[tex]0.06x+500-0.05x=560[/tex]
[tex]0.01x=560-500[/tex]
[tex]0.01x=60[/tex]
[tex]x=\$6.000[/tex]
therefore
In the account that paid 6% Susan invest [tex]\$6,000[/tex]
In the account that paid 5% Susan invest [tex]\$4,000[/tex]
Susan invested $6,000 at 6% and the remainder, $4,000, at 5% interest.
Susan invested $10,000 in two different accounts, one with a 6% simple interest and the other with a 5% simple interest. After one year, she received a total of $560 in interest. We need to find out how much she invested in each account.
Let's denote x as the amount invested at 6% and (10,000 - x) as the amount invested at 5%. Using the formula for simple interest, interest = principal × rate × time, we can set up two equations based on the given information:
The interest from the account with 6% interest: 0.06 × x
The interest from the account with 5% interest: 0.05 × (10,000 - x)
The sum of these interests is $560, so the equation is:
0.06x + 0.05(10,000 - x) = 560
Now we solve for x:
0.06x + 500 - 0.05x = 560
0.01x = 60
x = 60 / 0.01
x = $6,000
Therefore, Susan invested $6,000 at 6% and the remainder, $4,000, at 5% interest.
x = Temperature (°C) 1100 1200 1300 1100 1500 1200 1300 y = Porosity (%) 30.8 19.2 6 13.5 11.4 7.7 3.6 (a) Fit the simple linear regression model using the method of least squares. Find the least squares estimates of the intercept and slope in the simple linear regression model.
Answer:
intercept: 55.6256slope: -0.0341585Step-by-step explanation:
This sort of problem is best worked by a tool such as a graphing calculator or spreadsheet.
Gianna is going to throw a ball from the top floor of her middle school. When she throws the ball from 48feet above the ground, the function h(t)=−16t2+32t+48 models the height, h, of the ball above the ground as a function of time, t. Find the times the ball will be 48feet above the ground.
[tex]\bf \stackrel{height}{h(t)}=-16t^2+32t+48\implies \stackrel{48~ft}{~~\begin{matrix} 48 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}=-16t^2+32t~~\begin{matrix} +48 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} \\\\\\ 0=-16t^2+32t\implies 16t^2-32t=0\implies 16t(t-2)=0\implies t= \begin{cases} 0\\ 2 \end{cases}[/tex]
t = 0 seconds, when the ball first took off, and t = 2, 2 seconds later.
Answer: [tex]t_1=0\\t_2=2[/tex]
Step-by-step explanation:
We know that the function [tex]h(t)=-16t^2+32t+48[/tex] models the height "h" of the ball above the ground as a function of time "t".
Then, to find the times in which the ball will be 48 feet above the ground, we need to substitute [tex]h=48[/tex] into the function and solve fot "t":
[tex]48=-16t^2+32t+48\\0=-16t^2+32t+48-48\\0=-16t^2+32t[/tex]
Factorizing, we get:
[tex]0=-16t(t-2)\\t_1=0\\t_2=2[/tex]
Identify the parameters p and n in the following binomial distribution scenario. The probability of winning an arcade game is 0.718 and the probability of losing is 0.282. If you play the arcade game 20 times, we want to know the probability of winning more than 15 times. (Consider winning as a success in the binomial distribution.)
Answer:
p = 0.718 and n = 20
Step-by-step explanation:
p is the probability of success and n is the number of trials.
Here, p = 0.718 and n = 20.
Answer:
There is a 29.50% probability of winning more than 15 times.
Step-by-step explanation:
For each time you play the arcade game, there are only two possible outcomes. Either you win, or you lose. This means that we can solve this problem using the binomial probability distribution.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinatios of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
The probability of winning a game is 0.718. So [tex]p = 0.718[/tex].
The game is going to be played 20 times, so [tex]n = 20[/tex].
If you play the arcade game 20 times, we want to know the probability of winning more than 15 times.
This is
[tex]P(X > 15) = P(X = 16) + P(X = 17) + P(X = 18) + P(X = 19) + P(X = 20) = 0.2950[/tex].
There is a 29.50% probability of winning more than 15 times.
Travel: One Cyclist drives at Six miles per hour faster then another Cyclist. Express the speed of the faster Cyclist in terms of the speed of the lowest Cyclist .....
Answer:
Faster cyclist: 6x
Slower: x
Step-by-step explanation:
x = mph
The speed of the faster cyclist is expressed as v + 6 mph, where v represents the speed of the slower cyclist.
To express the speed of the faster cyclist in terms of the speed of the slower cyclist, let's denote the speed of the slower cyclist as v mph. The problem states that the faster cyclist travels at a speed that is six miles per hour faster than the slower cyclist. Therefore, the speed of the faster cyclist can be expressed as v + 6 mph.
For example, if the slower cyclist is travelling at a speed of 9 mph, the faster cyclist would be traveling at 9 mph + 6 mph = 15 mph.
A market researcher obtains a list of all streets in a town. She randomly samples 10 street names from the list, and then administers survey questions to every family living on those 10 streets. What type of sampling is this?
Answer: Simple random sampling
Step-by-step explanation:
Given: A market researcher obtains a list of all streets in a town. She randomly samples 10 street names from the list, and then administers survey questions to every family living on those 10 streets.
Since she randomly samples street names , therefore the type of sampling is simple random sampling.
A simple random sample is a sample that is a subset of the population the researcher surveyed selected in a way such that all the individuals in the population has an equal chance to be selected.
Write 1.052 as a percent
1.052 is equal to 105.2 percent.
Given that a decimal number 1.052, we need to write 1.052 as a percent,
To express a decimal number as a percent, you need to multiply it by 100.
Let's calculate 1.052 as a percent:
1.052 x 100 = 105.2
To understand this, let's break it down:
The number 1.052 represents 105.2% because it is greater than 1 (100%). By multiplying it by 100, we shift the decimal point two places to the right, resulting in 105.2.
In percentage terms, 105.2% means that 1.052 is 105.2 parts out of 100. This can also be interpreted as 105.2 per hundred or simply 105.2 out of every 100 units.
Therefore, 1.052 can be written as 105.2%.
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Lines a and b are parallel. Line c is perpendicular to both line a and line b. Which statement about lines a, b, and c is NOT true?
Line a and line b have the same slope.
The sum of the slopes of line b and line c is 0.
The product of the slopes of line c and line b is −1.
The product of the slopes of line a and line c is −1.
m + (-1/m) ≠ 0
⇒ The sum of the slopes of line b and line c is 0.
⇒ False ⇒ NOT true
Answer:
The sum of the slopes of line b and line c is 0.
Step-by-step explanation:
Remember that the product of the slopes of two parallel lines is -1, so in order to be -1 you have to multiply M*-1/m=-1 so since to add them up you would do it like this m+(-1/m) taht wouldn´t get as result 0, so that would be the option that is not correct, remember that parallel lines have the same slope, so that also eliminates all of the other options.
Use f’( x ) = lim With h ---> 0 [f( x + h ) - f ( x )]/h to find the derivative at x for the given function. 5-x²
The derivative of the function f(x) is:
[tex]f'(x)=-2x[/tex]
Step-by-step explanation:We are given a function f(x) as:
[tex]f(x)=5-x^2[/tex]
We have:
[tex]f(x+h)=5-(x+h)^2\\\\i.e.\\\\f(x+h)=5-(x^2+h^2+2xh)[/tex]
( Since,
[tex](a+b)^2=a^2+b^2+2ab[/tex] )
Hence, we get:
[tex]f(x+h)=5-x^2-h^2-2xh[/tex]
Also, by using the definition of f'(x) i.e.
[tex]f'(x)= \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}[/tex]
Hence, on putting the value in the formula:
[tex]f'(x)= \lim_{h \to 0} \dfrac{5-x^2-h^2-2xh-(5-x^2)}{h}\\\\\\f'(x)=\lim_{h \to 0} \dfrac{5-x^2-h^2-2xh-5+x^2}{h}\\\\i.e.\\\\f'(x)=\lim_{h \to 0} \dfrac{-h^2-2xh}{h}\\\\f'(x)=\lim_{h \to 0} \dfrac{-h^2}{h}+\dfrac{-2xh}{h}\\\\f'(x)=\lim_{h \to 0} -h-2x\\\\i.e.\ on\ putting\ the\ limit\ we\ obtain:\\\\f'(x)=-2x[/tex]
Hence, the derivative of the function f(x) is:
[tex]f'(x)=-2x[/tex]
Answer:
The derivative of given function is -2x.
Step-by-step explanation:
The first principle of differentiation is
[tex]f'(x)=lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}[/tex]
The given function is
[tex]f(x)=5-x^2[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{5-(x+h)^2-(5-h^2}{h}[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{5-(x^2+2xh+h^2)-5+h^2}{h}[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{5-x^2-2xh-h^2-5+h^2}{h}[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{-x^2-2xh}{h}[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{-x^2}{h}-\frac{2xh}{h}[/tex]
[tex]f'(x)=lim_{h\rightarrow 0}\frac{-x^2}{h}-2x[/tex]
Apply limit.
[tex]f'(x)=\frac{-x^2}{0}-2x[/tex]
[tex]f'(x)=0-2x[/tex]
[tex]f'(x)=-2x[/tex]
Therefore, the derivative of given function is -2x.
Find the derivative of the function using the definition of derivative. g(x) = 5 − x (1) Find g'(x) (2) State the domain of the function. (Enter your answer using interval notation.) (3) State the domain of its derivative. (Enter your answer using interval notation.)
Answer:
Answer is contained in the explanation
Step-by-step explanation:
[tex]g(x)=5-x\\g'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{[5-(x+h)]-[5-x]}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{5-x-h-5+x}{h}\\g'(x)=\lim_{h \rightarrow 0} \frac{-h}{h}\\g'(x)=\lim_{h \rightarrow 0} -1\\g'(x)=-1[/tex]
g(x)=5-x has domain all real numbers (you can plug an a number and always get a number back)
So in interval notation this is [tex](-\infty, \infty)[/tex]
g'(x)=-1 has domain all real numbers (the original function had domain issues... and no matter the number you plug in you do get a number, that number being -1)
So in interval notation this is [tex](-\infty, \infty)[/tex]
The derivative of given function g(x) is
g'(x)=-1
Domain of function g(x) is (-∞,∞)
Domain of derivative is (-∞,∞)
Given :
[tex]g(x) = 5 - x[/tex]
Lets find derivative using definition of derivative
[tex]\lim_{h \to 0} \frac{g(x+h)-g(x)}{h} \\g(x)=5-x\\g(x+h)=5-(x+h)\\g(x+h)=5-x-h\\\lim_{h \to 0} \frac{5-x-h-(5-x)}{h} \\\\\lim_{h \to 0} \frac{5-x-h-5+x}{h} \\\\\lim_{h \to 0} \frac{-h}{h} \\\\-1[/tex]
Derivative g'(x)=-1
g(x) is a linear function . for all linear function the domain is set of all real numbers
Domain of function g(x) is (-∞,∞)
Derivative function g'(x) =-1. For all values of x the value of y is -1
So domain is set of all real numbers
Domain of derivative is (-∞,∞)
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Which expression is equivalent to
Answer:
The correct answer is second option
4a²b²c²∛b)
Step-by-step explanation:
It is given an expression, ∛(64a⁶b⁷c⁹)
Points to remember
Identities
ⁿ√x = x¹/ⁿ
To find the equivalent expression
We have, ∛(64a⁶b⁷c⁹)
∛(64a⁶b⁷c⁹) = (64a⁶b⁷c⁹)1/3
= (4³/³ a⁶/³ b⁷/³ c⁹/³) [Since 64 = 4³]
= 4a² b² b¹/³ c³
= 4a²b²c³(b¹/³)
= 4a²b²c³ (∛b)
Therefore the correct answer is second option
4a²b²c³(∛b)
Question 1: Factor out the Greatest
Common Factor
3t5s − 15t2s3
Question 1 options:
3(t5s − 5t2s3)
3t2(t3s − 5s3)
3t2s(t3 − 5s2)
-12t3s2
For this case we have that by definition, the GCF or (Greatest Common Factor) is given by the greatest common factor that divides both terms without leaving a residue.
15: 1,3,5,15
3: 1.3
Then we have the GCF of the expression is:
[tex]3t ^ 2s (t ^ 3-5s ^ 2)[/tex]
ANswer:
Option C
How do I simply this radical expression?
Answer:
[tex]1000±100\sqrt{55}[/tex]
Step-by-step explanation:
To simplify that expression, first we need to find the largest common of the expression inside the radical, in this case: 2.200.000.
We know that 2.200.000 = 2 · 2 · 2 · 2 · 2 · 2 · 5 · 5 · 5 · 5 · 5 · 11 = [tex]2^{6}[/tex] ×[tex]5^{5}[/tex]× [tex]11[/tex]
Now, [tex]\sqrt{2^{6}5^{5}11} = 200\sqrt{55}[/tex].
Now we have: [tex]\frac{2000±200\sqrt{55}}{2}[/tex]
Dividing by 2: [tex]1000±100\sqrt{55}[/tex]
So the simplified expression is: [tex]1000±100\sqrt{55}[/tex]
A town's January high temperatures average 36degreesF with a standard deviation of 8degrees, while in July the mean high temperature is 72degrees and the standard deviation is 9degrees. In which month is it more unusual to have a day with a high temperature of 57degrees? Explain.
Answer: July
Step-by-step explanation:
Formula of z score :
[tex]z=\dfrac{X-\mu}{\sigma}[/tex]
Given: The mean high temperature in January = [tex]\mu_1=36^{\circ} F[/tex]
Standard deviation : [tex]\sigma_1=8^{\circ}F[/tex]
For X = [tex]57^{\circ}F[/tex]
[tex]z=\dfrac{57-36}{8}=2.625[/tex]
The mean high temperature in July = [tex]\mu_1=72^{\circ} F[/tex]
Standard deviation : [tex]\sigma_1=9^{\circ}F[/tex]
[tex]z=\dfrac{57-72}{8}=-1.875[/tex]
⇒ 57° F is about 2.6 standard deviations above the mean of January high temperatures, and 57° F is about 1.9 standard deviations below the mean of July’s high temperatures.
A general rule says that z-scores lower than -1.96 or higher than 1.96 are considered unusual .
Hence, the 57˚F is more unusual in January.
Final answer:
A high temperature of 57 degrees is more unusual in January than in July, as it is 2.625 standard deviations above the January mean, compared to 1.667 standard deviations below the July mean.
Explanation:
To determine in which month it is more unusual to have a high temperature of 57 degrees Fahrenheit, we can calculate the z-score for each month. The z-score tells us how many standard deviations away from the mean a particular value is.
For January, the z-score is calculated as follows:
Z = (57 - 36) / 8 = 21 / 8 = 2.625
This means that a temperature of 57 degrees in January is 2.625 standard deviations above the January mean.
For July, the z-score is calculated as follows:
Z = (57 - 72) / 9 = -15 / 9 = -1.667
This means that a temperature of 57 degrees in July is 1.667 standard deviations below the July mean.
Since the absolute value of the January z-score (2.625) is higher than the absolute value of the July z-score (-1.667), a high temperature of 57 degrees is more unusual in January than in July.
Suppose that you believe that the probability you will get a grade of B or better in Introduction to Finance is .6 and the probability that you will get a grade of B or better in Introduction to Accounting is .5. If these events are independent, what is the probability that you will receive a grade of B or better in both courses?
Answer: Probability that he will receive a grade of B or better in both courses is 0.30.
Step-by-step explanation:
Since we have given that
Probability that he will get a grade of B or better in introduction to Finance say P(A) = 0.6
Probability that he will get a grade of B or better in introduction to Accounting say P(B) = 0.5
Since A and B are independent events.
We need to find the probability that he will receive a grade of B or better in both the courses.
So, it becomes,
[tex]P(A\cap B)=P(A).P(B)\\\\P(A\cap B)=0.6\times 0.5\\\\P(A\cap B)=0.30[/tex]
Hence, Probability that he will receive a grade of B or better in both courses is 0.30.
The probability that you will receive a grade of B or better in both courses is 0.3.
The probability that you will receive a grade of B or better in both Introduction to Finance and Introduction to Accounting, given that the events are independent, is calculated by multiplying the individual probabilities of each event.
[tex]\( P(F) = 0.6 \) \( P(A) = 0.5 \)[/tex]
Since the events are independent, the probability of both events occurring is given by the product of their individual probabilities:
[tex]\( P(F \text{ and } A) = P(F) \times P(A) \)[/tex]
Substituting the given probabilities:
[tex]\( P(F \text{ and } A) = 0.6 \times 0.5 \) \( P(F \text{ and } A) = 0.3 \)[/tex]
The final answer is [tex]\(\boxed{0.3}\).[/tex]
If y = e2x is a solution to y''- 5y' + ky = 0, what is the value of k?
Answer:
The value of k is 6
Step-by-step explanation:
we need to find the value of k
Given : - [tex]y=e^{2x}[/tex] is the solution [tex]y''-5y'+ky=0[/tex]
[tex]y=e^{2x}[/tex] ........(1)
differentiate [tex]y=e^{2x}[/tex] with respect to 'x'
[tex]\frac{dy}{dx}=\frac{d}{dx}e^{2x}[/tex]
Since, [tex]\frac{d}{dx}e^{x} =e^{x}\frac{d}{dx}(x)[/tex]
[tex]\frac{dy}{dx}=e^{2x}\frac{d}{dx}(2x)[/tex]
[tex]\frac{dy}{dx}=e^{2x}\times 2[/tex]
[tex]\frac{dy}{dx}=2e^{2x}[/tex]
so, [tex]y'=2e^{2x}[/tex] ..........(2)
Again differentiation above with respect to 'x'
[tex]\frac{d}{dx}\frac{dy}{dx}=\frac{d}{dx}2e^{2x}[/tex]
[tex]\frac{d^{2}y}{dx^{2}}=2e^{2x}\frac{d}{dx}(2x)[/tex]
[tex]\frac{d^{2}y}{dx^{2}}=2e^{2x}\times 2[/tex]
[tex]\frac{d^{2}y}{dx^{2}}=4e^{2x}[/tex]
so, [tex]y''=4e^{2x}[/tex] ........(3)
Now, put the value of [tex]y\ ,y' \ \text{and} \ y''[/tex] in [tex]y''-5y'+ky=0[/tex]
[tex]4e^{2x}-5(2e^{2x})+(e^{2x})k=0[/tex]
[tex]4e^{2x}-10e^{2x}+e^{2x}k=0[/tex]
[tex]-6e^{2x}+e^{2x}k=0[/tex]
add both the sides by [tex]6e^{2x}[/tex]
[tex]e^{2x}k=6e^{2x}[/tex]
Cancel out the same terms from left and right sides
[tex]k=6[/tex]
Hence, the value of k is 6
A house was valued at $100,000 in the year 1987. The value appreciated to $165,000 by the year 2002.
Use the compound interest form S=P(1+r)^t to answer the following questions.
A) What was the annual growth rate between 1987 and 2002? (Round to 4 decimal places.)
B) What is the correct answer to part A written in percentage form?
C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2005? (Round to the nearest thousand dollars.)
Please help ASAP I need an answer by today!!! :(
Answer:
A) The annual multiplier was 1.0339; the annual increase was 0.0339 of the value.
B) 3.39% per year
C) $182,000
Step-by-step explanation:
A) Let's let t represent years since 1987. Then we can fill in the numbers and solve for r.
165000 = 100000(1 +r)^15
1.65^(1/15) = 1 +r . . . . . divide by 100,000; take the 15th root
1.03394855265 -1 = r ≈ 0.0339
The value was multiplied by about 1.0339 each year.
__
B) The value increased by about 3.39% per year.
__
C) S = $100,000(1.03394855265)^18 ≈ $182,000
Find the dimensions of the box described. The length is twice as long as the width. The height is 4 inches greater than the width. The volume is 48 cubic inches. Find the length, width, height
Answer:
Width of box= 2inches
Length of box= 4inches
Height of box= 6inches.
Step-by-step explanation:
Let width of box=x inches
Length of box = twice of width=[tex]2\times x[/tex]=[tex]2x[/tex]
Height of box= 4 inches greater than width= [tex]x+4[/tex]
Volume of box= 48 cubic inches
We know that the formula of volume of cuboid= [tex] length\times breadth\times height[/tex]
Apply the formula
Volume of box= [tex]x\times 2x\times (x+4)[/tex]
Volume of cube = [tex]2x^2(x+4)[/tex]
[tex]2x^2(x+4)=48[/tex]
[tex]x^2(x+4)=24[/tex]
[tex]x^3+x^2-24[/tex]
Apply inspection method to solve the equation
Put [tex]x=0[/tex]
Then we get [tex]-24\neq0 [/tex]
Hence, x=0 is not the solution of x
Put x=1 in the equation then we get
[tex]-22\neq 0[/tex]
Hence x=1 is not the solution of equation.
Put x=2 then we get
[tex](2)^3+(4)^2-24[/tex]
8+16-24=0
Hence, x=2 is the solution of equation .
[tex] (x-2)(x^2+6x+12)[/tex]=0
Now substitute equation [tex]x^2+6x+12[/tex]=0
Sum roots =6
Product of roots=12
When sum of roots is greater than zero and product of roots is greater than zero then value of roots of equation is imaginary.
Hence, the roots of equation [tex]x^2+6x+12=0[/tex] are imaginary.
Lenght , widht and height are dimensions of box therefore, imaginary value are not possible.
Hence,[tex] x=2 [/tex] is the only real values of root of equation .Therefore, it is possible and other two imaginary value of roots are not possible .
Widht of box=2 inches
Length of box = [tex]2\times2[/tex]=4inches
Height of box=[tex]x+4[/tex]=2+4=6 inches
The dimensions of the box are: length = 4 inches, width = 2 inches, and height = 6 inches.
Explanation:
Let's use the given information to solve for the dimensions of the box:
Let the width of the box be represented by x inches.
The length of the box is twice as long as the width, so the length is 2x inches.
The height is 4 inches greater than the width, so the height is (x + 4) inches.
The volume of a box can be calculated by multiplying the length, width, and height. Since the volume of the box is given as 48 cubic inches, we can set up the equation: 2x * x * (x + 4) = 48.
Simplifying the equation, we get 2x^3 + 8x^2 - 48 = 0.
Factoring the equation, we find that (x - 2)(x + 4)(x + 6) = 0.
The possible solutions are x = 2, x = -4, or x = -6.
Since we are dealing with dimensions, the width cannot be negative, so we can disregard the negative solutions. The width, therefore, is 2 inches.
The length is twice as long as the width, so the length is 2 * 2 = 4 inches.
The height is 4 inches greater than the width, so the height is 2 + 4 = 6 inches.
Therefore, the dimensions of the box are: length = 4 inches, width = 2 inches, and height = 6 inches.
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The number of bricks in the bottom row of a brick wall is 49. The next row up from the bottom contains 47 bricks, and each subsequent row contains 2 fewer bricks than the row immediately below it. The number of bricks in the top row is 3. If the wall is one brick thick, what is the total number of bricks in the wall?
Answer:
624
Step-by-step explanation:
The sequence is 49, 47, 45,...., 7, 5, 3. This is an arithmetic sequence, because the difference between terms is the same.
The sum of the first n terms of an arithmetic sequence is:
S = n/2 (a₁ + an)
where a₁ is the first term and an is the nth term.
Here, we know that a₁ = 49 and an = 3. But we need to find what n is. To do that, we use definition of an arithmetic sequence:
an = a₁ + (n-1) d
where d is the common difference (in this case, -2)
3 = 49 + (n-1) (-2)
2(n-1) = 46
n - 1 = 23
n = 24
So there are 24 terms in the sequence.
The sum is:
S = 24/2 (49 + 3)
S = 12 (52)
S = 624
There are 624 bricks in the wall.
The total number of bricks in the wall is 624. This is a math problem that involves arithmetic sequence, where each term is obtained from the previous one by subtracting a fixed number (2, in this case), and concepts from algebra (equations).
Explanation:The problem describes a scenario where each row of a brick wall has two fewer bricks than the row below it, which characterizes a sequence in mathematics. More specifically, this is an arithmetic sequence, which is characterized by a common difference between terms, in this case, the difference is -2.
To solve the problem, we need to find the sum of an arithmetic sequence. The formula of the sum is given by:
S = n/2 * (a1 + an)
Where S is the sum, n the number of terms, a1 the first term, and an the last term. Here, a1 is 49 and an is 3. To find n, we use the formula n = (a1 - an) / d + 1, with d being the common difference which is -2. Solving the equation we find that n = 24.
We now plug these values into the sum formula and find that the sum S, which represents the total number of bricks in the wall is
S = 24/2 * (49 + 3) = 12 * 52 = 624.
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Consider the vector field F=(x2+y2,4xy). Compute the line integrals ∫c1F⋅ds and ∫c2F⋅ds, where c1(t)=(t,t2) and c2(t)=(t,t) for 0≤t≤1. Can you decide from your answers whether or not F is a gradient vector field? Why or why not?
[tex]\displaystyle\int_{C_1}\vec F\cdot\mathrm d\vec s=\int_0^1(t^2+t^4,4t^3)\cdot(1,2t)\,\mathrm dt=\int_0^1(t^2+9t^4)\,\mathrm dt=\boxed{\frac{32}{15}}[/tex]
Integral over [tex]C_2[/tex]:[tex]\displaystyle\int_{C_2}\vec F\cdot\mathrm d\vec s=\int_0^1(2t^2,4t^2)\cdot(1,1)\,\mathrm dt=\int_0^16t^2\,\mathrm dt=\boxed{2}[/tex]
The value of the line integral depends on the path, so [tex]\vec F[/tex] is not a gradient vector field.
The line integrals ∫c1F⋅ds and ∫c2F⋅ds are computed by replacing x and y with the parametric representations, calculating ds, completing the dot product, and conducting the integration. If the results are identical, F is a gradient vector field.
Explanation:To compute the line integrals, ∫c1F⋅ds and ∫c2F⋅ds, where c1(t)=(t, t2) and c2(t)=(t,t) for 0≤t≤1 of the vector field F=(x2+y2,4xy), we can reduce each of them to an integral over t, the parameter of the path. In the case of c1(t), replace x and y by t and t² correspondingly, for calculation. Similarly, in the case of c2(t), replace x and y by t in calculations.
Let's consider ∫c1F⋅ds. Here, F = (t²+t⁴,4t³) and ds can be calculated using the Pythagorean theorem leading to sqr(1+4t²). The dot product F.ds is then calculated and integrated from 0 to 1. Repeat the process for ∫c2F⋅ds.
A vector field F is said to be a gradient vector field if integral from one point to another remains the same regardless of the path chosen to get from one point to the other. Comparing the obtained results will determine the truth of this statement.
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Explain why vertical lines are a special case in the definition of parallel lines.
Answer:
A vertical line has an infinite or undefined slope since the denominator is zero.
Step-by-step explanation:
Parallel lines by definition refers to lines that never intersect or meet since they have identical slopes. The slope of line is defined as;
(change in y)/(change in x)
For a vertical line, the y values are changing while the x values remain constant. The slope of this line will thus have a zero value in the denominator implying that its slope will not defined or will be infinity.
Answer:
A vertical line has an infinite or undefined slope since the denominator is zero.
Step-by-step explanation:
Solve the system y = -x + 7 and y= 0.5(x - 3)2
Answer:
The solutions of the system of equations are (-1,8) and (5,2)
Step-by-step explanation:
[tex]y=-x+7[/tex] -------> equation A (equation of a line)
[tex]y=0.5(x-3)^{2}[/tex] ----> equation B (vertical parabola open upward)
Solve the system of equations by graphing
Remember that the solution is the intersection points both graphs
using a graphing tool
The intersection points are (-1,8) and (5,2)
see the attached figure
therefore
The solutions of the system of equations are (-1,8) and (5,2)
Answer: (-1,8) and (5,2)
Step-by-step explanation: The person above me is correct. Give him five stars and a thanks!
Federal Rent-a-Car is putting together a new fleet. It is considering package offers from three car manufacturers. Fred Motors is offering 5 small cars, 5 medium cars, and 10 large cars for $500,000. Admiral Motors is offering 5 small, 10 medium, and 5 large cars for $400,000. Chrysalis is offering 10 small, 5 medium, and 5 large cars for $300,000. Federal would like to buy at least 700 small cars, at least 600 medium cars, and at least 700 large cars. How many packages should it buy from each car maker to keep the total cost as small as possible?
Answer:
40 packages from Fred Motors20 packages from Admiral Motors40 packages from ChrysalisStep-by-step explanation:
I would formulate the problem like this. Let f, a, c represent the numbers of packages bought from Fred Motors, Admiral Motors, and Chrysalis, respectively. Then the function to minimize (in thousands) is …
objective = 500f +400a +300c
The constraints on the numbers of cars purchased are …
5f +5a +10c >= 700
5f +10a +5c >= 600
10f +5a +5c >= 700
Along with the usual f >=0, a>=0, c>=0. Of course, we want all these variables to be integers.
Any number of solvers are available in the Internet for systems like this. Shown in the attachments are the input and output of one of them.
The optimal purchase appears to be …
40 packages from Fred Motors20 packages from Admiral Motors40 packages from ChrysalisThe total cost of these is $40 million.
This is a linear programming problem that requires to minimize a cost function subject to several constraints about the total number of small, medium, and large cars in the fleet. It can be set up using the system of inequalities and then solved using methods like the Simplex one.
Explanation:This problem can be solved through linear programming, which involves creating a system of inequalities to represent the constraints of the problem, and then optimizing a linear function. To start, let's define the variables: x is the number of packages bought from Fred Motors, y is the number from Admiral Motors, and z is the number from Chrysalis.
The fleet requirements translate to the following constraints: 5x + 5y + 10z ≥ 700 (small cars), 5x + 10y + 5z ≥ 600 (medium cars), and 10x + 5y + 5z ≥ 700 (large cars).
Then, the cost to minimize is: $500,000x + $400,000y + $300,000z.
This is a linear programming problem and can be solved using various methods, such as the Simplex method or graphically. Exact solutions would require a more detailed analysis.
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For Mexican American infants born in Arizona in 1986 and 1987, the probability that a child's gestational age is less than 37 weeks is 0.142 and the probability that his or her birth weight is less than 2500 grams is 0.051. Furthermore, the probability that these two events occur simultaneously is 0.031. Please show work!a. are A and B independent?b. For a randomly selected Mexican American newborn, what is the probability that A or B or both occur?c.What is the probability that event A occurs given that event B occurs?
Answer: Hence, a) No, they are not independent
b) 0.193
c) 0.60
Step-by-step explanation:
Since we have given that
Probability that a child's gestational age is less than 37 weeks say P(A)= 0.142
Probability that his or her birth weight is less than 2500 grams say P(B) = 0.051
P(A∩B) = 0.031
We need to check whether it is independent or not.
Since ,
[tex]P(A).P(B)=0.142\times 0.051=0.0072[/tex]
and
[tex]P(A\cap B)=0.051[/tex]
So, we can see that
[tex]P(A).P(B)\neq P(A\cap B)[/tex]
So, it is not independent.
a) Hence, A and B are not independent.
b) P(A∪B) is given by
[tex]P(A\ or B\ or\ both)=P(A)+P(B)\\\\P(A\ or\ B\ or\ both)=0.142+0.051=0.193[/tex]
c) P(A|B) is given by
[tex]P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{0.031}{0.051}=0.60[/tex]
Hence, a) No,
b) 0.193
c) 0.60
Bismuth-210 is an isotope that radioactively decays by about 13% each day, meaning 13% of the remaining Bismuth-210 transforms into another atom (polonium-210 in this case) each day. If you begin with 233 mg of Bismuth-210, how much remains after 8 days?
Approximately 85.87 mg of Bismuth-210 would remain after 8 days.
Explanation:To calculate the amount of Bismuth-210 remaining after 8 days, we need to apply the concept of radioactive decay. Bismuth-210 decays by about 13% each day, meaning that 13% of the remaining Bismuth-210 transforms into another atom (Polonium-210) each day.
Let's calculate the amount remaining:
Start with 233 mg of Bismuth-210.After the first day, 13% of the remaining Bismuth-210 will decay, leaving 87% of the original amount: 0.87 * 233 mg = 202.71 mg.Repeat this process for each subsequent day.After 8 days, the amount remaining would be: (0.87)^8 * 233 mg = 85.87 mg.Therefore, after 8 days, approximately 85.87 mg of Bismuth-210 would remain.
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After 8 days, approximately 76.69 mg of Bismuth-210 remains out of the initial 233 mg.
To calculate the amount of Bismuth-210 remaining after 8 days of radioactive decay, follow these steps:
Step 1:
Understand the decay rate.
Each day, 13% of the remaining Bismuth-210 decays into Polonium-210. This means that 87% of the Bismuth-210 remains each day.
Step 2:
Calculate the remaining amount each day.
Start with the initial amount of 233 mg of Bismuth-210.
After the first day: [tex]\( 233 \text{ mg} \times 0.87 = 202.71 \text{ mg} \)[/tex]
After the second day: [tex]\( 202.71 \text{ mg} \times 0.87 = 176.43 \text{ mg} \)[/tex]
Continue this process for 8 days.
Step 3:
Perform the calculations for 8 days.
[tex]\[ \text{Day 1: } 233 \text{ mg} \times 0.87 = 202.71 \text{ mg} \][/tex]
[tex]\[ \text{Day 2: } 202.71 \text{ mg} \times 0.87 = 176.43 \text{ mg} \][/tex]
[tex]\[ \text{Day 3: } 176.43 \text{ mg} \times 0.87 = 153.62 \text{ mg} \][/tex]
[tex]\[ \text{Day 4: } 153.62 \text{ mg} \times 0.87 = 133.67 \text{ mg} \][/tex]
[tex]\[ \text{Day 5: } 133.67 \text{ mg} \times 0.87 = 116.33 \text{ mg} \][/tex]
[tex]\[ \text{Day 6: } 116.33 \text{ mg} \times 0.87 = 101.28 \text{ mg} \][/tex]
[tex]\[ \text{Day 7: } 101.28 \text{ mg} \times 0.87 = 88.10 \text{ mg} \][/tex]
[tex]\[ \text{Day 8: } 88.10 \text{ mg} \times 0.87 = 76.69 \text{ mg} \][/tex]
Step 4:
Interpret the result.
After 8 days, approximately 76.69 mg of Bismuth-210 remains.
So, after 8 days, approximately 76.69 mg of Bismuth-210 remains.
The tread life of tires mounted on light-duty trucks follows the normal probability distribution with a population mean of 60,000 miles and a population standard deviation of 4,000 miles. Suppose we select a sample of 40 tires and use a simulator to determine the tread life. What is the likelihood of finding that the sample mean is between 59,050 and 60,950?
Answer: 0.8664
Step-by-step explanation:
Given : Mean : [tex]\mu = 60,000\text{ miles}[/tex]
Standard deviation : [tex]\sigma = 4,000\text{ miles}[/tex]
Sample size : [tex]n=40[/tex]
The formula to calculate the z-score :-
[tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]
For x= 59,050
[tex]z=\dfrac{59050-60000}{\dfrac{4000}{\sqrt{40}}}\approx-1.50[/tex]
For x= 60,950
[tex]z=\dfrac{60950-60000}{\dfrac{4000}{\sqrt{40}}}\approx1.50[/tex]
The P-value : [tex]P(-1.5<z<1.5)=P(z<1.5)-P(z<-1.5)[/tex]
[tex]=0.9331927-0.0668072=0.8663855\approx0.8664[/tex]
Hence, the likelihood of finding that the sample mean is between 59,050 and 60,950=0.8664
The likelihood of finding that the sample mean is between 59,050 and 60,950 miles, according to the given normal distribution, is approximately 86.64%.
Explanation:To solve this problem, we consider that the population mean is 60,000 and the standard deviation is 4,000. If we choose a sample of 40 tires, the standard deviation of the sample mean (standard error) is the standard deviation divided by the square root of the sample size (σ/√n).
This gives us 4,000/√40 = 633. The z-scores for the lower and upper bounds of our interval (59,050 and 60,950) are calculated by subtracting the population mean from these values, and dividing by the standard error. For 59,050: (59,050 - 60,000)/633 = -1.5 and for 60,950: (60,950 - 60,000)/633 = 1.5.
Using standard normal distribution tables, we know that the probability associated with a z-value of 1.5 is 0.9332. Since the normal distribution is symmetric, the probability associated with -1.5 is also 0.9332. Therefore, the probability that the sample mean lies between 59,050 and 60,950 is 0.9332 - (1 - 0.9332) = 0.8664 or approximately 86.64%.
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Last year, Scott had 10,000 to invest. He invested some of it in an account that paid 7%
simple interest per year, and he invested the rest in an account that paid 9% simple interest per year. After one year, he received a total of $740 in interest. How much did he invest in each account?
Answer:
$8000 is invested for 7% interest and $2000 is invested for 9% interest
Step-by-step explanation:
Points to remember
Simple interest formula
I = PNR/100
P - Principle amount
N - Number of years
R - Rate of interest
To find the amount of investment
It is given that total amount = 10,000 and total interest = $740
Let 'x' be the amount invested at the rate of 7%
10,000 - x be the amount invested at the rate of 9%
I = PNR/100
740 = (x*1*7)/100 + (10000 - x)*1*9/100
740 = 7x/100 + 90000/100 - 9x/100
740 = 7x/100 + 900 - 9x/100
740-900 = -2x/100
-160 = -2x/100
x = 16000/2 = 8000
10000-8000 = 2000
Therefore $8000 is invested for 7% interest and $2000 is invested for 9% interest
Answer:
$8000 is invested for 7% interest and $2000 is invested for 9% interest
Step-by-step explanation:
Points to remember
Simple interest formula
I = PNR/100
P - Principle amount
N - Number of years
R - Rate of interest
To find the amount of investment
It is given that total amount = 10,000 and total interest = $740
Let 'x' be the amount invested at the rate of 7%
10,000 - x be the amount invested at the rate of 9%
I = PNR/100
740 = (x*1*7)/100 + (10000 - x)*1*9/100
740 = 7x/100 + 90000/100 - 9x/100
740 = 7x/100 + 900 - 9x/100
740-900 = -2x/100
-160 = -2x/100
x = 16000/2 = 8000
10000-8000 = 2000
Therefore $8000 is invested for 7% interest and $2000 is invested for 9% interest
I really hope it helped idiot noob die
for the following right triangle find the side length x
Since there is a right angle, you can use Pythagoras' Theorem:
So x = √(24² + 7²) = 25
---------------------------------------------------------
Answer:
25
College algebra homework review... Having issues calculating this by hand and on TI-84 receiving errors like "8e12" when trying to calculate the actual quadratic equation it calls for in question B.... Please help
checking the vertex of this upside-down parabola, it has a vertex at (1000, 2000000), so that's the U-turn, when as the price "p" increases the revenue goes down.
[tex]\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{a}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)[/tex]
now, if we solve the quadratic using the value of 500000
[tex]\bf \stackrel{R(p)}{500000}=-2p^2+4000p\implies 250000=-p^2+2000p \\\\\\ p^2-2000p+250000=0[/tex]
and we run the quadratic formula on it, we get the values of x = 133.97 and x = 1866.03, one value is obviously when going upwards, the first one, and the other is when going downwards.
so we know that the R(p) is 500,000 at x = 133.97, and it keeps on going up, up to the vertex above at x = 1000, so we can say from x = [134, 1000] R(p) > 500000.