Question

A tuition at Rutgers University was $38,148 in 1995. In 2002, the tuition was $42,727.


a)Assuming tuition is increasing at an exponential rate, write a function f(y) representing the cost of tuition y years since 1995.

b)What is the growth rate per year?

c) What is the growth rate per decade?

d) write function f(d) representing the cost of tuition d decades since 1995.

e) if this trend were to continue, what should the tuition be in the year 2020?

Answer 1

a) Let $y$ be the number of years since 1995 and let $f(y)$ be the cost of tuition in dollars. 

Exponential growth function

$$f(y)=f(0)e^{ky}$$

Given that $f(0)=\$38148.$ Then

$$f(y)=38148e^{ky}$$

b) A tuition at Rutgers University was $42,727 in 2002.

$$2002-1995=7$$

$$f(7)=38148e^{7k}=42727$$

Solve for $k$

$$e^{7k}={42727 \over 38148}$$

$$\ln(e^{7k})=\ln\big({42727 \over 38148}\big)$$

$$7k=\ln\big({42727 \over 38148}\big)$$

$$k={1 \over 7}\ln\big({42727 \over 38148}\big)$$

$$k\approx0.016194$$

$$f(y)=38148e^{0.016194y}=38148(1.016326)^y=38148(1+0.016326)^y$$

The growth rate per year between 1995 and 2002

$$r=0.016326$$

c) Let $d$ be the number of decades since 1995 and let $f(d)$ be the cost of tuition in dollars. 

$$f(d)=38148e^{md}$$

A tuition at Rutgers University was $42,727 in 2002

$$f({7 \over 10})=38148e^{m({7 \over 10})}=42727$$

Solve for $m$

$$e^{0.7m}={42727 \over 38148}$$

$$\ln(e^{0.7m})=\ln\big({42727 \over 38148}\big)$$

$$0.7m=\ln\big({42727 \over 38148}\big)$$

$$m={10 \over 7}\ln\big({42727 \over 38148}\big)$$

$$m\approx0.161940$$

$$f(d)=38148e^{0.16194d}=38148(1.175789)^d=38148(1+0.175789)^d$$

The growth rate per decade between 1995 and 2002

$$q=0.175789$$

d)

$$f(d)=38148e^{0.16194d}=38148(1.175789)^d=38148(1+0.175789)^d$$

e)

$$2020-1995=25$$

$$f(25)=38148e^{0.016194(25)}=\$57186.81$$

A tuition at Rutgers University should be $\$57186.81$ in 2020.