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August 17th, 2014, 06:39 AM   #1
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Question is attached:

don't understand part c.

100m by 112.5m

Solutions would be much appreciated!
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Last edited by skipjack; August 17th, 2014 at 07:23 AM.

 August 17th, 2014, 07:28 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,096 Thanks: 1905 You need to assume that all of the available fencing is used. For part (c), they want you to calculate x and y.
August 17th, 2014, 09:57 PM   #3
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Hello, jessjans11!

Quote:
 5. 1800 m of fencing is available to enclose 6 identical pig pens. Code:  * - - * - - * - - * | | | | y * - - * - - * - - * | | | | y * - - * - - * - - * x x x (a) Explain why $9x+8y \:=\:1800.$

There are 9 lengths of fencing which are $x$ m each.
There are 8 lengths of fencing which are $y$ m each.

The total fencing is 1800 m: $\:9x + 8y \:=\:1800\;\;[1]$

Quote:
 (b) Show that the area of each pen $\quad$is given by: $\,A \:=\:-\tfrac{9}{8}x^2 + 225x$

Solve equation [1] for $y$.
$\quad y \;=\;\dfrac{1800-9x}{8} \quad\Rightarrow\quad y \;=\;225 - \tfrac{9}{8}x\;\;[2]$

The area of a pen is: $\:A \;=\;xy \;=\;x\left(225-\frac{9}{8}x\right)$

Therefore: $\:A \;=\;-\frac{9}{8}x^2 + 225$

Quote:
 (c) If the enclosed area is to be a maximum, $\quad$what are the dimensions of each pen?

We want to maximize $A$.

The equation is a down-opening parabola.
Its maximum is at its vertex.

The vertex is: $\:x \:=\:\dfrac{\text{-}b}{2a} \:=\:\dfrac{\text{-}225}{2(\text{-}\frac{9}{8})} \:=\:100$

Substitute into [2]: $\:y \:=\:225 - \frac{9}{8}(100) \:=\: \dfrac{225}{2}$

Therefore, the dimensions are: $\:100\text{m} \times 112.5\text{m}$.

August 18th, 2014, 07:19 AM   #4
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Quote:
 Originally Posted by soroban $\quad y \;=\;\dfrac{1800-9x}{8} \quad\Rightarrow\quad y \;=\;225 - \tfrac{9}{8}x\;\;[2]$ The area of a pen is: $\:A \;=\;xy \;=\;x\left(225-\frac{9}{8}x\right)$ Therefore: $\:A \;=\;-\frac{9}{8}x^2 + 225$ Substitute into [2]: $\:y \:=\:225 - \frac{9}{8}(100) \:=\: \dfrac{225}{2}$ Therefore, the dimensions are: $\:100\text{m} \times 112.5\text{m}$.
Your A expression is incorrect. You left the x off of "225x."

And then when you substituted, you substituted into the wrong expression.

You should have substituted the x-value into

A = 225x - (9/8)x^2.

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