Answer:
y(t) = a sin(ωt).
Step-by-step explanation:
The graph of the motion starts at y-0 t = 0 so we use sine in the equation
y(t) = A sin (2π t / T) where A = the amplitude and T = the period so here we can write:
Displacement at t = y(t) = a sin(2π/ 2π/ω)t
y(t) = a sin(ωt)
This is about graph of simple harmonic motion.
y(t) = a sin (ωt)
We are told the condition of the simple harmonic motion we want to model is at y = 0 and t = 0. This condition means the motion starts at the origin. Therefore, we will make use of the solution;y(t) = A sin ωt
Where;
A is amplitude
ω is angular frequency
y(t) is the displacement at time(t)
Now, we know that;ω can also be expressed as;
ω = 2π/T
Where T is period.
Thus;
y(t) = A sin (2π/T)t
We are given that;Period; T = 2π/ω
Thus
y(t) = A sin (2π/(2π/ω))t
2π will cancel out to give;
y(t) = A sin (ωt)
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Consider the following Polynomial.
S(x)= -3x^2 +x-9
Describe the behavior of the graph of S(x) as x ---> +/- ∞
S(x)--> ? as x--> -∞
S(x)-->? as x-->∞
Answer:
S(x) → -∞ as x → -∞S(x) → -∞ as x → ∞Step-by-step explanation:
The leading term tells you what you want to know. It is of even degree, so the value of S(x) is the same regardless of the sign of x as the magnitude of x gets large.
The sign of S(x) matches the sign of the leading coefficient (-3), so is negative as x gets large.
Hence ...
S(x) → -∞ as x → -∞S(x) → -∞ as x → ∞
What is DC ?
Enter the answer in the box.
Answer:
DC = 30
Step-by-step explanation:
Point D is the circumcenter of triangle ABC, so it is equidistant from A, B, and C.
We are given the measures of DE and AE, so we can figure AD using the Pythagorean theorem:
AD² = AE² + DE² . . . . . . . . . . . . . . . E bisects AB, so AE=AB/2=18
AD = √(18² +24²) = √900 = 30
DC = AD
DC = 30
Larry is using an online calculator to calculate the outputs f(n) for different inputs n. The ordered pairs below show Larry's inputs and the corresponding outputs displayed by the calculator:
(1, 5), (2, 9), (3, 13), (4, 17)
Which of the following functions best represents the rule that the calculator uses to display the outputs?
a
f(n) = 5n − 1
b
f(n) = 5n + 1
c
f(n) = 4n + 1
d
f(n) = 4n − 1
Answer:
Option c:
[tex]f(n)=4n+1[/tex]
Step-by-step explanation:
The functional relationship between two variables can be easily found if it's represented as a line.
Larry's online calculator collects these points
(1, 5), (2, 9), (3, 13), (4, 17)
We can see there is a linear relation because every time the first component increases by 1, the second increases by 4.
The equation of a line is given by
[tex]f(n)=m.n+b[/tex]
Where m is the slope of the line and can be computed as
[tex]\displaystyle m=\frac{d-b}{c-a}[/tex]
Where (a,b), (c,d) are two known points of the line. Let's use the first two points (1, 5), (2, 9)
[tex]\displaystyle m=\frac{9-5}{2-1}=4[/tex]
We now know that
[tex]f(n)=4n+b[/tex]
To compute the value of b, we use one of the points again, for example (1,5):
[tex]5=4(1)+b => b=1[/tex]
The relation is
[tex]f(n)=4n+1[/tex]
We can test our results by using other points like (3,13)
[tex]f(3)=4(3)+1=13[/tex]
And also
[tex]f(4)=4(4)+1=17[/tex]
All points belong to the same function or rule
[tex]f(n)=4n+1[/tex]
please help me will mark brainly
Answer:
A
Step-by-step explanation:
Ok. All coordinate are multiplied by factor of 1/3.
So,
(0,0) becomes (0,0).
(6,9) becomes (2,3).
(15,0) becomes (5,0).
In order to obtain a sample of voters in Pennsylvania, a simple random sample of 10 counties is selected. From each of the selected counties, 10 precincts are chosen at random. Finally, from each of these 100 precincts , a simple random sample of 20 voters students is selected. Thus, the final sample consists of 2,000 voters.
This is an example of which type of sampling strategy?a) Simple random samplingb) Stratified samplingc) Multistage samplingd) Convenience sampling
Answer:
c) Multistage sampling.In statistics, there are several strategies to select a sample. Remember that a sample is a sub group selected from a population. This sample selection must be random to have more reliability in the research, each answer option represents a type of sampling.
In this case, the researcher is using a multistage sampling, because it consists in taking smaller sample units at each stage, which is being done in this example. First, is selected a number of counties, then the precincts, and at the end the voters. By this way, the sample is being reduced to less subjects at each stage.
The sampling strategy described is an example of c) multistage sampling. This technique helps in reducing complexity and cost.
The sampling strategy described in the question is an example of multistage sampling.
In this process:
A simple random sample of 10 counties in Pennsylvania is selected.From each of these counties, 10 precincts are chosen randomly.Finally, from each of these 100 precincts, a simple random sample of 20 voters is selected.This results in a final sample of 2,000 voters.Multistage sampling is used to reduce the complexity and cost of data collection by breaking the population into smaller groups at each stage.
What is the area of the figure?
Answer:
90 in²
Step-by-step explanation:
The figure's area is that of four right triangles, each with legs of 6 in and 7.5 in. The area of each triangle is half the product of the leg lengths, so is ...
triangle area = (1/2)(6 in)(7.5 in)
Then the area of 4 of those triangles is ...
figure area = 4 · triangle area = 2(6 in)(7.5 in) = 90 in²
A textbook store sold a combined total of 257 math and psychology textbooks in a week. The number of math textbooks sold was 87 more than the number of psychology textbooks sold. How many textbooks of each type were sold?
Answer:
85 psychology172 mathStep-by-step explanation:
Let p represent the number of psychology textbooks sold. Then the total number sold was ...
p + (p+87) = 257
2p = 170 . . . . . .subtract 87; next divide by 2
p = 85 . . . . . . . psychology books sold
p+87 = 172 . . . math books sold
85 psychology textbooks and 172 math textbooks were sold.
Explanation:Let's denote the number of psychology textbooks sold as x. According to the problem, the number of math textbooks sold, which is 87 more than the psych books, can be represented as x + 87. The total number of textbooks sold is expressed in the problem as 257, which is the sum of the psych books (x) and the math books (x + 87). So:
x (psych books) + x + 87 (math books) = 257
We can simplify this to 2x + 87 = 257. If we subtract 87 from both sides, we'll get 2x = 170. Dividing both sides by 2 results in x = 85. This tells us that 85 psychology books were sold.
Then to find the number of math books, we can use the original relationship given: math books = psychology books + 87, therefore, 85 + 87 = 172 math books.
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HELP NEEDED, GIVING BRAINLIEST!!
Identify the statement as true or false and justify your answer.
Plane M is perpendicular to line s through point Q. Therefore it is the only plane perpendicular to s through point Q.
A. False; If a line is perpendicular to a plane, any line perpendicular to that line at the point of intersection of the line and the plane is contained by the plane.
B. True; If a line is perpendicular to a plane, any line perpendicular to that line at the point of intersection of the line and the plane is contained by the plane.
C. True; Given a point on a line, there is one and only one plane perpendicular to the line through that point.
D. False; Given a point on a line, there is one and only one plane perpendicular to the line through that point.
The statement is false. Multiple planes can be perpendicular to the same line through a given point.
Explanation:The statement is False. If a line is perpendicular to a plane, it does not mean that it is the only plane perpendicular to the line.
There can be multiple planes perpendicular to the same line through a given point. For example, consider a line s passing through point Q and a plane M perpendicular to s at point Q. Now, we can also have another plane N perpendicular to line s at point Q, which is different from plane M. Therefore, the statement is false.
Triangle XYZ and EFG are given. ΔXYZ≅ΔEFG by SAS. If m∠EFG = 5p-2, YZ=2n-5 and GF=n+5 then which of the following statements are true.
A.ZY=15
B.m∠XZY=52
C.p=8
D.p is 2 more than n.
E.m∠EFG=38
F.GF=8
Answer:
A. ZY=15
Step-by-step explanation:
Insufficient information is given about angles to make any statement about the value of p or the measures of any angles. (Eliminates B,C,D,E)
Side YZ corresponds to side FG. Since they are congruent, their measures are the same. This means ...
2n -5 = n +5
n = 10 . . . . . . . . add 5-n
YZ = ZY = 2·10 -5 = 15 . . . . . . matches choice A
I am thinking of a number. When I double my number and then subtract the result from five, I get negative one. What is my number? Write and solve an equation
Answer:
3
Step-by-step explanation:
To turn the word problem into an equation, when we read:
"I am thinking of a number" we write "x"
"when I double my number" we write "2x"
"and then subtract the result from 5" we write "5 - 2x"
"I get negative one" we write "-1 = 5 - 2x"
Now we solve for the number, which is x.
Equation: -1 = 5 - 2x
-1 = 5 - 2x
subtract 5 from both sides
-6 = -2x
divide both sides by -2
3 = x
There we go! The number is 3
The mathematical expression of the given phrase is 5 - 2x = -1 thus the number will be 3.
What is an expression?A mixture of variables, numbers, addition, subtraction, multiplication, and division are called expressions.
An expression is a mathematical proof of the equality of two mathematical expressions.
Let's say that number is x.
Double 2x
Subtract from 5
5 - 2x = -1
-2x = - 1 - 5
-2x = -6
x = 3
Hence "The mathematical expression of the given phrase is 5 - 2x = -1 thus the number will be 3".
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If A is the initial amount put into an account, P is the annual percentage rate of interest, which remains fixed, and the account compounds quarterly, which of the following is an expression, in terms of A and P, for the amount in the Account after 5 years?
A 4A(p100)5
B A(p100)20
C 4A(1+p100)5
D A(1+p25)20
E A(1+p400)20
Answer: The amount in the account is A = A(1 + P/400)^20
Step-by-step explanation:
Initial amount deposited into the account is A. This means that the principal is A, so
P = A
It was compounded quarterly. This means that it was compounded once in four months. So
n = 4
The rate at which the principal was compounded is P%. So
r = P/100
It was compounded for a total of 5 years. So
n = 5
The formula for compound interest is
A = P(1+r/n)^nt
A = total amount in the account at the end of n years.
Let A = B
B = A(1 + (P/100)/4)^4×5
A = A(1 + P/400)^20
Final answer:
The correct expression for the amount in an account after 5 years, where interest is compounded quarterly, is D. [tex]A(1 + \frac{P}{25})^{20}[/tex].
Explanation:
When calculating the future value of an investment with compound interest, it's important to consider the initial amount A, the annual interest rate P, the number of times the interest is compounded per year, and the total number of years the money is invested. In this question, where the account compounds quarterly for 5 years, the formula for compound interest applies, which is:
P(t) = [tex]A(1 + \frac{r}{n})^{nt}[/tex]
Here, r represents the annual nominal interest rate in decimal form [tex](\frac{P}{100})[/tex], n is the number of times interest is compounded per year, and t is the number of years the money is invested. Given that the account compounds quarterly, n equals 4. So after 5 years, the formula would be:
P(5) = [tex]A(1 + \frac{P}{400})^{(4 \times 5)}[/tex]
Based on the options provided in the question, the correct expression for the amount in the account after 5 years, in terms of A and P, when compounded quarterly is:
D. [tex]A(1 + \frac{P}{25})^{20}[/tex]
Slader records show that the average life expectancy of a pair of shoes is 2.2 years with astandard deviation of 1.7 years. A manufacturer gaurantees that shoes lasting less than a year are replaced for free. For every 1000 pairs sold how many pairs should the manufacturer expect to replaces free? Assume a normal distributiom.
Answer:
For every 1000 pairs sold, the manufacturer expect to replace 239 pairs for free.
Step-by-step explanation:
Given:
Mean (μ) = 2.2, Standard deviation(S.D) (σ) = 1.7 years and x = 1 (1 year)
Let's find the Z score.
Z = [tex]\frac{x - mean}{S.D}[/tex]
Now plug in the given values in the above formula, we get
Z = [tex]\frac{1 - 2.2}{1.7} = -0.71[/tex]
Now we have to use the z-score table.
The z-score for 0.71 is 0.2611
Since it z is negative, so we subtract 0.2611 from 0.5000
0.5000 - 0.2611 = 0.2389
Percentage = 0.2389 × 100 = 23.89%
To find replaces for 1000 pairs, we need to multiply 23.89% by 1000
= [tex]\frac{23.89}{100} .1000 = 238.9[/tex]
= 239
The cannot be in decimal, when we round off to the nearest whole, we get
239
Select the three ratios that are equivalent to 2 adults5 children. CLEAR CHECK 8 adults20 children 5 adults8 children 20 adults50 children 4 adults10 children
Answer:
Step-by-step explanation:
The ratio of adult to children is determined by number of adult / number of children.
We want to determine the three ratios that are equivalent to 2 adults 5 children. So we will divide each of the given number of adults and children.
1) 8 adults 20 children = 8/20 = 2/5
2) 5 adults 8 children = 5/8
3) 20 adults 50 children = 20/50 = 2/5
4) 4 adults 10 children = 4/10 = 2/5
So the three ratios that are equivalent to 2 adults 5 children are
8 adults 20 children,
20 adults 50 children and
4 adults 10 children
Find the solution u(x, y) of Laplace's equation in the rectangle 0 < x < a, 0 < y < b, that satisfies the boundary conditions u(0, y) = 0, u(a, y) = 0, 0 < y < b, u(x, 0) = 0, u(x, b) = g(x), 0 ≤ x ≤ a.
Answer:
The solution has been given in the attachment.
Step-by-step explanation:
20 POINTS AND BRAINIEST FOR THOSE WHO ANSWER CORRECTLY
~
What is the simplest radical form of the expression?
(x^4y^7)^3/4
~
Thank you!
Answer:
x^2 y ^4 ∛ [x^2 y ^2 ] is the answer that I got
Step-by-step explanation:
Answer:
PLEASE MARK BRAINLIEST!Step-by-step explanation:
[tex](x^{4}y^{7})^{\frac{3}{4}}[/tex]
Answer 1:
[tex]= x^{3}y^{5} \sqrt[4]{y}[/tex]
Answer 2:
[tex]x^{3}y^{\frac{21}{4}}[/tex]
Answer 3:
[tex]\sqrt[4](x^{4}y^{7})^{3}[/tex]
I didn't know which one was correct, so I included all of them. I hope this helps!
A bank wishes to estimate the mean credit card balance owed by its customers. The population standard deviation is estimated to be $300. If a 98 percent confidence interval is used and an interval of $78 is desired, how many customers should be sampled?A. 725B. 80C. 57D. 320
Answer: B. 80
Step-by-step explanation:
We know that the formula to find the sample size is given by :-
[tex]n=(\dfrac{z^*\cdot\sigma}{E})^2[/tex]
, where [tex]\sigma[/tex] = population standard deviation.
E= margin of error
z*= Two -tailed critical z-value
Given : Confidence level = 98% =0.98
[tex]\alpha=1-0.98=0.02[/tex]
Population standard deviation : [tex]\sigma=300[/tex]
Also, from z-table for [tex]\alpha/2=0.01[/tex] (two tailed ), the critical will be = [tex]z^*=2.326[/tex]
Then, the required sample size must be :
[tex]n=(\dfrac{2.326\cdot300}{78})^2\\\\ n=(8.94615)^2\\\\ n=80.0336686391\approx80[/tex] [To the nearest option]
Hence, the required sample size = 80
Hence, the correct option is option B. 80
Final answer:
To estimate the mean credit card balance owed by the bank's customers using a 98 percent confidence interval and a desired interval of $78, the sample size should be 725 customers.
Explanation:
To estimate the mean credit card balance owed by the bank's customers, we need to determine the sample size. We can use the formula for sample size calculation for a mean with a desired margin of error: n = (Z * σ / E)².
Here, Z is the Z-score for the desired confidence level, σ is the population standard deviation, and E is the desired margin of error. In this case, the Z-score for a 98 percent confidence level is approximately 2.33. Plugging in the values, we get: n = (2.33 * 300 / 78)² = 724.9.
Since we can't have a fractional sample size, we round up to the nearest whole number. Therefore, the bank should sample 725 customers.
Terrell Trucking Company is in the process of setting its target capital structure. The CFO believes that the optimal debt-to-capital ratio is somewhere between 20% and 50%, and her staff has compiled the following projections for EPS and the stock price at various debt levels: Debt/Capital Ratio Projected EPS Projected Stock Price 20% $3.00 $34.75 30 3.65 36.50 40 3.80 37.75 50 3.55 32.25 Assuming that the firm uses only debt and common equity, what is Terrell's optimal capital structure? Round your answers to two decimal places. % debt % equity At what debt-to-capital ratio is the company's WACC minimized? Round your answer to two decimal places. %
Answer:
40% or 0.4
Step-by-step explanation:
The optimal capital structure (OCS) of a firm is defined as "the proportion of debt and equity that results in the lowest weighted average cost of capital (WACC) for the firm"
The brief explanation of this is that OCS is the factor used by a company in maximising their stock price, and this generally calls for a Debt-to-capital or "Debit-to-equity" ratio.
From the table above, the company's stock ratio is highest or maximised at 37.75 (under Projected Stock Price Column)
This can be traced to 40% under Debt/Capital ratio column
Hence, the Debt/Capital Ratio of 40%,
Because it must equate to 100%, we say that the firm's optimal capital structure is 40% debt and 60% equity.
This is also the debt to capital ratio, where the firms WACC is minimized.
The optimal capital structure for Terrell Trucking Company is a 40% debt-to-capital ratio, indicating a mix of 40% debt and 60% equity. Assuming the WACC is minimized at the optimal capital structure, the company's WACC would also be minimized at a 40% debt-to-capital ratio.
Explanation:In order to determine Terrell's optimal capital structure, we need to identify at what debt-to-capital ratio both the Earnings Per Share (EPS) and the stock price are highest. Based on the provided projections, the EPS and stock price are highest at a 40% debt-to-capital ratio. Therefore, the optimal debt-to-capital ratio for the company is 40%. This would indicate that Terrell's optimal capital structure is 40% debt and 60% equity.
Typically, the Weighted Average Cost of Capital (WACC) is minimized at the optimal capital structure. Assuming this holds true for Terrell Trucking Company, the company's WACC would also be minimized at a 40% debt-to-capital ratio.
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Simplify the following polynomial expression??
Answer:
B
Step-by-step explanation:
just multiply and add
Quiz: The Discriminant and Modeling Data 7:Solving Quadratic Equations
esday
Find the number of real number solutions for the equation. x2 - 10x + 25 = 0
oo
O
1
2
cannot be determined
100%
For this case we must find the solution of the following quadratic equation:
[tex]x ^ 2-10x + 25 = 0[/tex]
Where:
[tex]a = 1\\b = -10\\c = 25[/tex]
Then, the solution is given by:
[tex]x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}[/tex]
Substituting the values:
[tex]x = \frac {- (- 10) \pm \sqrt {(- 10) ^ 2-4 (1) (25)}} {2 (1)}\\x = \frac {10 \pm \sqrt {100-100}} {2}\\x = \frac {10 \pm \sqrt {0}} {2}\\x = \frac {10} {2} = 5[/tex]
Thus, we have two equal real roots.
[tex]x_ {1} = 5\\x_ {2} = 5[/tex]
Answer:
We have two equal real roots.
Answer:
cannot be determined
Step-by-step explanation:
i tried 1,0,2 dont work
A student randomly draws a card from a standard deck and checks to see if it is his favorite suit. He then returns the card to the deck, shuffles, and repeats the experiment. He performs the experiments 30 times. Can the probability of drawing his favorite suit be found by using the binomial probability formula? Why or why not?
Yes. The events are dependent; however, the 5% guideline can be applied to this situation.
No. The trials are fixed, but the probability of success changes for every trial.
No. The probability of success remains the same for every trial, but the trials are not fixed.
Yes. The outcomes can be classified into two categories, the trials are fixed, and the events are independent.
Answer:
Yes. The outcomes can be classified into two categories, the trials are fixed, and the events are independent.
Step-by-step explanation:
Hope this helps!!
write the slope-intercept form of an equation that passes through (4,4) and is perpendicular to y=2x-4
Answer:
Step-by-step explanation:
The equation of a straight line can be represented in the slope-intercept form, y = mx + c
Where c = y intercept
Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis
If two lines are perpendicular, it means that the slope of one line is the negative reciprocal of the slope of the other line. The equation of the given line is
y=2x-4
Slope = 2
Therefore, the slope of the perpendicular line is -1/2
It passes through point (4,4)
We would determine the intercept, by substituting m = -1/2 , y = 4 and x = 4 into the slope intercept equation
y = mx + c
4 = -1/2 ×4 + c
4 = -2 + c
c = 4 + 2 = 6
The equation becomes
y = -x/2 + 6
Use the list of five irrational below to answer the questions
Square root of 2, Square Root of 6, Square Root of 12, Square Root of 18, and Square Root of 24.
Part A. Choose two numbers whose product is RATIONAL. Explain.
Part B. Choose two numbers whose product is IRRATIONAL. Explain.
Answer:
[tex]\text{A.}\ \sqrt{2}\times\sqrt{18}\\\\\text{B.}\ \sqrt{2}\times\sqrt{6}[/tex]
Step-by-step explanation:
A: The root will be rational if the product of the numbers under the radicals is a perfect square. For this part, there are a couple of choices.
[tex]\text{1.}\ \sqrt{2}\times\sqrt{18}=\sqrt{36}=6\\\\\text{2.}\ \sqrt{6}\times\sqrt{24}=\sqrt{144}=12[/tex]
__
B: The root will be irrational if the product of the numbers under the radicals is not a perfect square. For this part, there are many choices.
[tex]\text{1.}\ \sqrt{2}\times\sqrt{6}=\sqrt{12}\\\\\text{2.}\ \sqrt{2}\times\sqrt{12}=\sqrt{24}\\\\\text{3.}\ \sqrt{2}\times\sqrt{24}=\sqrt{48}\\\\\text{4.}\ \sqrt{6}\times\sqrt{12}=\sqrt{72}\\\\\text{5.}\ \sqrt{6}\times\sqrt{18}=\sqrt{108}\\\\\text{6.}\ \sqrt{12}\times\sqrt{18}=\sqrt{216}\\\\\text{7.}\ \sqrt{12}\times\sqrt{24}=\sqrt{288}\\\\\text{8.}\ \sqrt{18}\times\sqrt{24}=\sqrt{432}[/tex]
To choose two numbers whose product is rational, we can use the square roots of 2. For two numbers whose product is irrational, we can use the square roots of 2 and 6.
To choose two numbers whose product is rational, we need to find two numbers whose square roots are rational.
Let's consider the numbers Square Root of 2 and Square Root of 2. Their product is 2, which is a rational number.
For two numbers whose product is irrational, we need to find two numbers whose square roots are irrational.
Let's consider the numbers Square Root of 2 and Square Root of 6. Their product is 2 x Square Root of 6, which is an irrational number.
What is the equation of the function?
Answer:
y = x + 1
Step-by-step explanation:
This line passes through (-1, 0) and (0, 1) As we move from the first point to the second, x increases by 1 and y also increases by 1. Therefore, the slope of this line is m = rise / run = 1 / 1, or m = 1.
Start with the general equation of a line y = mx + b. Substitute 1 for m and 1 for b. Then the equation of the line shown is:
y = x + 1
The equation of the graphed function is y = x + 1 .
The equation of the line can be written in the form ;
y = bx + c b = slope; c = interceptThe slope of the line ; is the ratio of the rise to the run of the line ;
b = (3 - 1) / (2 - 0) = 1The intercept which is the value of y when x = 0 from.the graph is 1 .
Hence, the equation is :
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A vendor has 30 umbrellas to sell. He sells them for $20 each. The function m(u) = 20u models the total amount of money the vendor makes from selling u umbrellas. What are the practical domain and range of the function?
Answer:
Step-by-step explanation:
Domain={0,30},Range={0,600
Total number of umbrellas=3
Price for each umbrella is =$20
m(u)=20u
as we know
u = umber of umbrellas
m(u) total amount vendor makes by selling u number of umbrellas
So Practically,The extreme cases are
when the vendor sells 0 umbrellas/ no umbrellas
when he sells all the umbrellas
So Domain is {0,total number of umbrellas}={0,30}
Putting this extreme value of domains as u in function m(u),
we get the range of m(u)={0$20,30$20}
⇒ Range={0,$600}
Answer:
domain 0,30 range 0,600
Step-by-step explanation:
Which composition of transformations will create a pair of similar
Answer:
A rotation, then a dilation
Step-by-step explanation:
When two triangles are congruent, the three sides and angles will be the same.
A dilation is a type of transformation that works with scale factors and enlarges or reduces a figure, to create a new figure.
Now, the composition of transformations that will create a pair of similar but not congruent triangles are - a rotation, then a dilation.
A composition of a rotation followed by a dilation will create a pair of similar, but not congruent, triangles, option D.
The question asks which composition of transformations will create a pair of similar, but not congruent, triangles. In the realm of Euclidean geometry, certain transformations maintain the shape and size of geometric figures, while others maintain only the shape but not the size.
Among the choices given, a rotation followed by a reflection, a translation followed by a rotation, and a reflection followed by a translation will all produce congruent triangles, as they are types of isometries which preserve shape and size.
However, a rotation followed by a dilation is the correct composition that will result in triangles that are similar but not congruent. This is because rotation preserves the shape and size, but when followed by dilation, the size is changed while the shape remains the same, satisfying the condition of the question. Option D is correct.
The top and bottom margins of a poster are each 6 cm and the side margins are each 4 cm. If the area of printed material on the poster is fixed at 384 cm , find the dimensions of the poster with the smallest area.
Answer:
24 cm wide by 36 cm high
Step-by-step explanation:
The poster with the smallest area will have an aspect ratio that makes the margin dimensions the same percentage of overall dimension in each direction.
Since the ratio of margin widths is 6:4 = 3:2, the poster and printed area will have an aspect ratio of 3:2. That is, the width is ...
width of printed area = √(2/3·384 cm²) = 16 cm
Then the width of the poster is ...
width = left margin + printed width + right margin = 4cm + 16 cm + 4 cm
width = 24 cm
The height is 3/2 times that, or 36 cm.
The smallest poster with the required dimensions is 24 cm wide by 36 cm tall.
_____
If you need to see the calculus problem, consider the printed area width to be x. Then the printed height is 384/x and the overall dimensions are ...
(x + 8) by (384/x + 12)
We want to minimize the area, which is the product of these dimensions:
a = (x +8)(384/x +12) = 384 +12x +3072/x +96
a = 12x + 3072/x +480
This is a minimum where its derivative is zero.
a' = 12 -3072/x^2 = 0
a' = 1 -256/x^2 = 0 . . . . . . divide by 12; true when x^2 = 256
This has solutions x=±16, of which the only useful solution is x=16.
Prehistoric cave paintings were discovered in a cave in France. The paint contained 10 %10% of the original carbon-14. Use the exponential decay model for carbon-14, Upper A equals Upper A 0 e Superscript negative 0.000121 t=A0e−0.000121t, to estimate the age of the paintings.
Answer:
t=19188.2 y
Step-by-step explanation:
The exponential decay equation is:
[tex] A=A_{0}e^{-0.00012t}[/tex] (1)
But, A is 10% of A₀, it means that A=0.10A₀.
If we put it into equation (1), we will have:
[tex] 0.10A_{0}=A_{0}e^{-0.00012t}[/tex]
[tex] 0.10=e^{-0.00012t}[/tex] (2)
Now, we just need to solve (2) for t.
[tex] t=\frac{ln(0.10)}{-0.000121} = 19188.2 y [/tex]
I hope it helps you!
A construction crew has just built a new road. It took them 8 weeks to build 20.48 kilometers of road. At what rate did they build the road?
Answer:The rate per week =
20.48/8 = 2.56 kilometers per week
Step-by-step explanation:
A construction crew has just built a new road. It took them 8 weeks to build 20.48 kilometers of road. To determine the rate at which the road was built, we would divide the total length of road that was built by the construction company by the number of weeks or days or even hours used in the construction.
The rate per week =
20.48/8 = 2.56 kilometers per week
If we want to find the rate per day,
1 week = 7 days
8 weeks will be 8×7 = 56 days
So the rate per day =
20.48/56 = 0.366 kilometers per day.
The expected costs to make replacements, alterations, or improvements to a building that materially prolong its life and increase its value is referred to as vacancy losses. collection losses. capital expenditures. operating expenses.
Answer:
Capital expenditures
Step-by-step explanation:
The major difference between capital and revenue expenditures are usually seen by certain variables such as; the amount spent, frequency of the spend and whether the spend expands or improves the earning capacity, functionality or operating efficiency of the asset under consideration.
For example, if the money spent on this building was just for painting and it is something that occurs every other year, then the amount spent would be referred to as operating expense.
In the question above, the money spent on the building does the following; materially prolong its life,increase its value. It is evident from these that such expense can be classified as capital expenditure.
Furthermore, this kind of expenditure cannot be carried out every year.
I hope this answer clears your doubt and improves your understanding of what is required.
Two partners agree to invest equal amounts in their business. One will contribute $10,000 immediately. The other plans to contribute an equivalent amount in 2 years. How much should she contribute at that time to match her partner's investment now, assuming an interest rate of 9% compounded quarterly?
Answer:
She should contribute $ 8369.38 ( approx )
Step-by-step explanation:
Let P be the amount invested by the other partner,
∵ The amount formula in compound interest,
[tex]A=P(1+\frac{r}{n})^{nt}[/tex]
Where,
r = annual rate,
n = number of compounding periods in a year,
t = number of years,
Here, r = 9% = 0.09, n = 4 ( quarters in a year ), t = 2 years,
Then the amount after 2 years,
[tex]A = P(1+\frac{0.09}{4})^{8}[/tex]
According to the question,
A = $ 10,000,
[tex]P(1+\frac{0.09}{4})^{8}= 10000[/tex]
[tex]P(1+0.0225)^8 = 10000[/tex]
[tex]\implies P = \frac{10000}{1.0225^8}\approx \$ 8369.38[/tex]