Suggested languages for you:

Americas

Europe

Q. 57

Expert-verifiedFound in: Page 168

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

The total yearly expenditures by public colleges and universities from 1990 to 2000 can be modeled by the function $E\left(t\right)=123{(1.025)}^{t}$ , 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.

given expenditure function,

$E\left(t\right)=123(1.025{)}^{t}$

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

$\begin{array}{r}E\left(5\right)=123(1.025{)}^{5}\\ =123\times 1.1314\\ =139.163\end{array}$

Therefore the expenditure was $\$139.163$ billion.

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

Therefore the average rate of change is:

$\begin{array}{r}\frac{E\left(10\right)-E\left(0\right)}{10-0}=\frac{123(1.025{)}^{10}-123}{10}\\ =\frac{34.450}{10}\\ =3.445\end{array}$

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

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

Therefore the average rate of change is:

$\begin{array}{r}\frac{E\left(6\right)-E\left(5\right)}{6-5}=\frac{123(1.025{)}^{6}-123(1.025{)}^{5}}{1}\\ =\frac{3.479}{1}\\ =3.479\end{array}$

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

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.

94% of StudySmarter users get better grades.

Sign up for free