Chapter 3, Problem 18TP

Solution

The area of a 3 inch by 5 inch index card.

The inverse function is, $y=\frac{1}{0.03937}x$

Area of index card is: $9677.4\text{ square millimeters}$

Given:

The function for converting x millimeters to y inches is $y=0.03937x$ . Find the inverse of the function.

Concept used:

• The graph of the function $y=f\left(x\right)$ and its inverse $f^{-1}\left(x\right)$ are symmetric to the line $y=x$ .
• For any point $\left(a,b\right)$ on the graph of $f\left(x\right)$ , the corresponding point on the graph of inverse function $f^{-1}\left(x\right)$ is $\left(b,a\right)$ .
• If a function is one-to-one, then it has an inverse.

Calculation:

In order to find inverse of the function $y=0.03937x$ , first switch x and y and then solve further for y to get the required inverse function as shown below

  $\begin{array}{c}y=0.03937x\\ x=0.03937y\left(\text{Switch }x\text{and }y\right)\\ y=\frac{1}{0.03937}x\end{array}$

Thus, the invers of the function is $y=\frac{1}{0.03937}x$ .

The inverse function converts the x inches into y millimeters.

Since the index card has dimension 3 inch by 5 inch, in millimeters the dimension is

  $\begin{array}{l}\frac{1}{0.03937}\times 3\text{ millimeters by }\frac{1}{0.03937}\times 5\text{millimeters}\\ \text{=}76.2\text{ millimeters by }127\text{millimeters}\end{array}$

Thus, the area of the index card is

  $\begin{array}{c}\text{Area of index card}=76.2\text{ millimeters }\times 127\text{millimeters}\\ =9677.4\text{ square millimeters}\end{array}$

Thus, the area of the index card is $9677.4\text{ square millimeters}$ .