A race track 460 m long is to be built around a field that has the shape of a rectangle with semicircles attached at opposite ends. Let $r$ be the radius of each semicircular part of the track, $x$ be the length of each straight section of the track, and $L$ be the length of the field. Move the slider below to see the possible shapes of the race track.
Answer the following. Enter $\pi$ as "pi". Do not round any number. Use interval notation to express domains.
a) Find $r$ as a function of $x$. What is the domain of this function?
$r(x) = \frac{(230-x)}{\pi}$
Domain: $x \in (0, 230)$
[m]
[m]
b) Find $L$ as a function of $x$. What is the domain of this function?
$L(x) = x(1 - \frac{2}{\pi}) + \frac{460}{\pi}$
Domain: $x \in (0, 230)$
[m]
[m]
c) Find $x$ as a function of $L$. What is the domain of this function?
$x(L) = \frac{L - \frac{460}{\pi}}{1 - \frac{2}{\pi}}$
Domain: $L \in$
[m]
[m]
d) Find the area $A$ (in m²) of the entire field as a function of $r$. What is the domain of this function?
$A(r) = 460r - \pi r^2$
Domain: $r \in (0, \frac{230}{\pi})$
[m]
[m²]
e) Find the area $A$ (in m²) of the entire field as a function of $x$. What is the domain of this function?
$A(x) = $
Domain: $x \in (0, 230)$
[m]
[m²]
