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Q. 57

Expert-verified

Calculus

Found in: Page 168

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function $E (t) = 123 {(1.025)}^{t}$ , where expenditures are measured in billions of dollars and time is measured in years since 1990.
(a) Estimate the total yearly expenditures by these colleges and universities in 1995.
(b) Compute the average rate of change in yearly expenditures between 1990 and 2000.
(c) Compute the average rate of change in yearly expenditures between 1995 and 1996.
(d) Estimate the rate at which yearly expenditures of public colleges and universities were increasing in 1995.

Ans:

(a) The expenditure was $$ 139.163$ billion.

(b) The average rate of change in yearly expenditure is: $$ 3.445$ billion per year

(c) The average rate of change in yearly expenditure is: $$ 3.479$ billions per year

(d) The average expenditure throughout these two years is approximately $$ 3.45$ billion.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Part (a) Step 1. Given information.

given expenditure function,

$E (t) = 123 (1.025)^{t}$

Part (a) Step 2. Here the objective is to determine the total yearly expenditure of the colleges and universities in1995.

For the year $1995$ put $t = 5$ in the function:

$\begin{aligned} E (5) = 123 (1.025)^{5} \\ = 123 \times 1.1314 \\ = 139.163 \end{aligned}$

Therefore the expenditure was $$ 139.163$ billion.

Part (b) Step 3. Here the objective is to compute the average rate of change in expenditure for 1990 to 2000.

For $1990$ the value of $t = 0$ and for the year $2000$ the value of $t = 10$ .

Therefore the average rate of change is:

$\begin{aligned} \frac{E (10) - E (0)}{10 - 0} = \frac{123 (1.025)^{10} - 123}{10} \\ = \frac{34.450}{10} \\ = 3.445 \end{aligned}$

Hence the average rate of change in yearly expenditure is: $$ 3.445$ billions per year.

Part (c) Step 1. Here the objective is to compute the average rate of change in expenditure for 1995 to 1996.

For $1995$ the value of $t = 5$ and for the year $1996$ the value of $t = 6$ .

Therefore the average rate of change is:

$\begin{aligned} \frac{E (6) - E (5)}{6 - 5} = \frac{123 (1.025)^{6} - 123 (1.025)^{5}}{1} \\ = \frac{3.479}{1} \\ = 3.479 \end{aligned}$

Hence the average rate of change in yearly expenditure is billion per year.

Part (d) Step 5. Here the objective is to estimate the rate at which the expenditure was increased from 1995 to 1996.

For $1995$ take $t = 5$ and for 1996 take $t = 6$ .

The average expenditure throughout these two years is approximately localid="1650454347759" $$ 3.45$ billion.

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