The number of electric scooters e that a factory can produce per day is a function of the number of hours h it operates and is given by e(h) = 290h, 0 <= h <= 10. The daily cost c to manufacture e electric scooters is given by the function c(e) = 0.05e^2 + 65e + 1000. (a) Find (c @ e)(h). (b) Evaluate (c @ e)(13).

Question
Composite functions
asked 2021-03-05
The number of electric scooters e that a factory can produce per day is a function of the number of hours h it operates and is given by \(\displaystyle{e}{\left({h}\right)}={290}{h},{0}\le{h}\le{10}.\)
The daily cost c to manufacture e electric scooters is given by the function \(\displaystyle{c}{\left({e}\right)}={0.05}{e}^{{2}}+{65}{e}+{1000}.\)
(a) Find \(\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}.\)
(b) Evaluate \(\displaystyle{\left({c}\circ{e}\right)}{\left({13}\right)}.\)

Answers (1)

2021-03-06
The composition of the function c(e) and e(h) is denoted by \(\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}\) is given by the formula,
\(\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}={c}{\left({e}{\left({h}\right)}\right)}\)
(a) Compute the composite function as,
\(\displaystyle{e}{\left({h}\right)}={290}{h},{0}\le{h}\le{10}\)
\(\displaystyle{c}{\left({e}\right)}={0.05}{e}^{{2}}+{65}{e}+{1000}\)
\(\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}={c}{\left({e}{\left({h}\right)}\right)}\)
\(\displaystyle={c}{\left({290}{h}\right)}\)
\(\displaystyle={0.05}{\left({90}{h}\right)}^{{2}}+{65}{\left({290}{h}\right)}+{100}\)
\(\displaystyle{3920}{h}^{{2}}+{18850}{h}+{1000},{0}\le{h}\le{10}\)
(b) The composite function \(\displaystyle{\left({c}\circ{e}\right)}{\left({h}\right)}\) is not defined for h = 13.
Hence, the function \(\displaystyle{\left({c}\circ{e}\right)}{\left({13}\right)}\) cannot be evaluated.
0

Relevant Questions

asked 2021-02-11
Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.\) Solve (8.86) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{e}}}\) can be estimated from \(\displaystyle{r}_{{{e}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ {\left({8.87}\right)}\)
(b) (Logistic Growth) This model is described by \(\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}\) where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that \(\displaystyle{r}_{{{l}}}\) can be estimated from \(\displaystyle{r}_{{{l}}}={\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{K}\ -\ {N}{\left({0}\right)}}}{{{N}{\left({0}\right)}}}}\right]}\ +\ {\frac{{{1}}}{{{t}}}}\ {\ln{\ }}{\left[{\frac{{{N}{\left({t}\right)}}}{{{K}\ -\ {N}{\left({t}\right)}}}}\right]}\ {\left({8.89}\right)}\) for \(\displaystyle{N}{\left({t}\right)}\ {<}\ {K}.\)
(c) Assume that \(\displaystyle{N}{\left({0}\right)}={1}\) and \(\displaystyle{N}{\left({10}\right)}={1000}.\) Estimate \(\displaystyle{r}_{{{e}}}\) and \(\displaystyle{r}_{{{l}}}\) for both \(\displaystyle{K}={1001}\) and \(\displaystyle{K}={10000}.\)
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of \(\displaystyle{\left[{r}\right]}\) to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when \(\displaystyle\frac{{N}}{{K}}\) is small compared with 1.
asked 2020-11-08
The regular price of a computer is x dollars. Let f(x) = x - 400 and g(x) = 0.75x. Solve,
a. Describe what the functions f and g model in terms of the price of the computer.
b. Find \(\displaystyle{\left({f}\circ{g}\right)}{\left({x}\right)}\) and describe what this models in terms of the price of the computer.
c. Repeat part (b) for \(\displaystyle{\left({g}\circ{f}\right)}{\left({x}\right)}\).
d. Which composite function models the greater discount on the computer, \(\displaystyle{f}\circ{g}\) or \(\displaystyle{g}\circ{f}\)?
asked 2021-03-07
Mathematical modeling is about constructing one or two equations that represent real life situations. What are these math models used for? Provide at least two equations that can be used in the real world. For example: The equation \(s = 30\ h\ +\ 1000\) can be used to find your salary given the fact you earn a fixed salary of $1000 per month, plus $30 per hours. Here s represents the total salary and h is the number of hours you worked.
asked 2020-10-18
For Questions 1 — 2, use the following. Scooters are often used in European and Asian cities because of their ability to negotiate crowded city streets. The number of scooters (in thousands) sold each year in India can be approximated by the function \(N = 61.86t^2 — 237.43t + 943.51\) where f is the number of years since 1990. 1. Find the vertical intercept. What is the practical meaning of the vertical intercept in this situation? 2. Use a numerical method to find the year when the number of scooters sold reaches 1 million. (Note that 1 million is 1,000 thousand, so N = 1000) Show three rows of the table you used to support your answer and write a clear answer to the problem.
asked 2020-12-02
Gastroenterology
We present data relating protein concentration to pancreatic function as measured by trypsin secretion among patients with cystic fibrosis.
If we do not want to assume normality for these distributions, then what statistical procedure can be used to compare the three groups?
Perform the test mentioned in Problem 12.42 and report a p-value. How do your results compare with a parametric analysis of the data?
Relationship between protein concentration \((mg/mL)\) of duodenal secretions to pancreatic function as measured by trypsin secretion:
\(\left[U/\left(k\ \frac{g}{h}r\right)\right]\)
Tapsin secreton [UGA]
\(\leq\ 50\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.7 \\ \hline 2 & 2.0 \\ \hline 3 & 2.0 \\ \hline 4 & 2.2 \\ \hline 5 & 4.0 \\ \hline 6 & 4.0 \\ \hline 7 & 5.0 \\ \hline 8 & 6.7 \\ \hline 9 & 7.8 \\ \hline \end{array}\)
\(51\ -\ 1000\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.4 \\ \hline 2 & 2.4 \\ \hline 3 & 2.4 \\ \hline 4 & 3.3 \\ \hline 5 & 4.4 \\ \hline 6 & 4.7 \\ \hline 7 & 6.7 \\ \hline 8 & 7.9 \\ \hline 9 & 9.5 \\ \hline 10 & 11.7 \\ \hline \end{array}\)
\(>\ 1000\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 2.9 \\ \hline 2 & 3.8 \\ \hline 3 & 4.4 \\ \hline 4 & 4.7 \\ \hline 5 & 5.5 \\ \hline 6 & 5.6 \\ \hline 7 & 7.4 \\ \hline 8 & 9.4 \\ \hline 9 & 10.3 \\ \hline \end{array}\)
asked 2021-03-05
1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”
asked 2020-10-28
Determine the algebraic modeling a.
One type of Iodine disintegrates continuously at a constant rate of \(\displaystyle{8.6}\%\) per day.
Suppose the original amount, \(\displaystyle{P}_{{0}}\), is 10 grams, and let t be measured in days.
Because the Iodine is decaying continuously at a constant rate, we use the model \(\displaystyle{P}={P}_{{0}}{e}^{k}{t}\) for the decay equation, where k is the rate of continuous decay.
Using the given information, write the decay equation for this type of Iodine.
b.
Use your equation to determine the half-life ofthis type of Fodine, That is, find ‘out how many days it will take for half of the original amount to be left. Show an algebraic solution using logs.
asked 2020-10-23
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P \((A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).\)
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
asked 2020-11-26
Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes.
(a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}.\) 12 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Taxol as a function of time after the infusion is completed.
(b) As soon as the infusion of Abraxane is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}\). 24 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Abraxane as a function of time after the infusion is completed.
(c) Find the long-term behavior of the function from part (b). Is this behavior meaningful in the context of the model?
asked 2020-11-01
Given
\(\displaystyle{f{{\left({x}\right)}}}={x}^{{2}}+{x}+{1}\)
h(x) = 3x + 2,
evaluate the composite function.
...