Question

Let $R$ be the region in the first quadrant bounded by the $x$ and $y$ axis and the graphs of $f( x )=\frac{9}{25} x + b$ and $y =f^{-1}( x )$. If the area of $R$ is 49 , then the value of $b$, is

• A. $\frac{18}{5}$
• B. $\frac{22}{5}$
• C. $\frac{28}{5}$
• D. none

Answer 1

Answer: (C)

If $f(x)=m x+b$, then $f^{-1}(x)=\frac{x-b}{m}$ and their point of intersection can be found by setting $x=m y+b$ since they intersect on $y=x$.
Thus $x =\frac{ b }{1- m }$ and the point of intersection is $\left(\frac{ b }{1- m }, \frac{ b }{1- m }\right)$.
Region $R$ can be broken up into congruent triangles $PAB$ and $PCB$ which both have a base of $b$ and a height of $\frac{ b }{1- m }$.
The area of $R$ is $2\left(\frac{ b }{2}\right)\left(\frac{ b }{1- m }\right)=\frac{ b ^2}{1- m }=49$. For $m =\frac{9}{25}, b ^2=\frac{16}{25} \cdot 49 \Rightarrow b =\frac{28}{5}$