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November 2nd, 2016, 10:17 AM   #1
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Question Optimization

Hello!

I am having problems solving the following exercise:

An athletics track has two ‘straights’ of length l m and two semicircular ends
of radius x m. The perimeter of the track is 400 m.
a. Show that l = 200 - π(pi)x and hence write down the possible values that
x may have.
b. What values of l and x maximise the shaded rectangle inside the track? What is this maximum area?


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Last edited by musiclover; November 2nd, 2016 at 10:21 AM.
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November 2nd, 2016, 10:42 AM   #2
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Quote:
Originally Posted by musiclover View Post
Hello!

I am having problems solving the following exercise:

An athletics track has two ‘straights’ of length l m and two semicircular ends
of radius x m. The perimeter of the track is 400 m.
a. Show that l = 200 - π(pi)x and hence write down the possible values that
x may have.
b. What values of l and x maximise the shaded rectangle inside the track? What is this maximum area?


$P = 2 \ell + 2 \pi x = 400$

$\ell = \dfrac 1 2 (400 - 2 \pi x) = 200 - \pi x$

$\ell \geq 0 \Rightarrow 0 \leq x \leq \dfrac{200}{\pi}$

You want to maximize

$A = x \ell = x (200 - \pi x) = 200x - \pi x^2$

we do this is usual way, solve $\dfrac{dA}{dx} = 0$ for $x$

$\dfrac{dA}{dx} = 200 - 2\pi x = 0$

$x = \dfrac{100}{\pi}$

$\ell = 200 - \pi\left(\dfrac{100}{\pi}\right) = 100$

$A = \dfrac{10000}{\pi}$
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