Q. 11 - Precalculus Enhanced with Graphing Utilities

Question

The variable interest rate on a student loan changes each

July $1$ based on the bank prime loan rate. For the years

$1992-2007$, this rate can be approximated by the model

$r\left(x\right)=-0.115x^2+1.183x+5.623$, where x is the number of

years since $1992$ and r is the interest rate as a percent.

(a) Use a graphing utility to estimate the highest rate during

this time period. During which year was the interest rate

the highest?

(b) Use the model to estimate the rate in $2010$. Does this

value seem reasonable?

Step-by-Step Solution

Answer

If the function for interest rate is$r\left(x\right)=-0.115x^2+1.183x+5.623$ then

(a) the highest interest rate will be $8.67$ % in the year $1997$

(b) the function cannot predict the interest rate in the year $2010$

Step 1. Given data

Function for interest rate is

$r\left(x\right)=-0.115x^2+1.183x+5.623$

Step 2. Part (a)

Plot the graph of the function $r\left(x\right)=-0.115x^2+1.183x+5.623$

the graph is maximum at $(5.14,8.67)$

where$x=5.14$ so the year is$1992+5=1997$

the interest rate was highest in the year $1997$ at the rate of $8.67$ %

Step 3. Part (b)

For year $2010$, x value is

$$\begin{gathered} x=2010-1992 \\ x=18 \end{gathered}$$

substitute$18$ for x in the function

$$\begin{gathered} r\left(x\right)=-0.115x^2+1.183x+5.623 \\ r\left(18\right)=-0.115{\left(18\right)}^2+1.183\left(18\right)+5.623 \\ r\left(18\right)=-10.343 \end{gathered}$$

the interest rate cannot be negative so the function cannot predict interest rate for $2010$