1.41: Past-paper question 7

Figure 2 shows a shape $\displaystyle ABCDEOFG$

$\displaystyle ABCO$ is a quarter circle with radius $\displaystyle r$ cm

$\displaystyle CDEO$ and $\displaystyle AOFG$ are congruent rectangles of length $\displaystyle r$ cm and width $\displaystyle x$ cm

The total area of the shape is $\displaystyle 100\mathrm{cm}^2$

The perimeter of the shape is $\displaystyle P$ cm

(b) Use calculus to find the value of $\displaystyle r$ for which $\displaystyle P$ is a minimum, justifying that this value of $\displaystyle r$ gives a minimum value of $\displaystyle P$

(c) Find the minimum value of $\displaystyle P$