Expert: Paul Klarreich - 11/10/2011

Question
A rectangle has constant area of 200 square meters and its lenght L is increasing at the rate of 4meters per second.

A)Find the width W at the instant that the width is decreasing at the rate of .5 meters per second.

B)At what rate is the diagonal D of the rectangle changing at the instant when the width is 10 meters?

Answer
Questioner: Demetria
Country: Pennsylvania, United States
Category: Calculus
Private: No
Subject: Caculus
Question: A rectangle has constant area of 200 square meters and its lenght L is increasing at the rate of 4meters per second.

A)Find the width W at the instant that the width is decreasing at the rate of .5 meters per second.

B)At what rate is the diagonal D of the rectangle changing at the instant when the width is 10 meters?
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Variables:

L = length of the rect
W = width.
D = length of a diagonal

Rates:

dL/dt = 4
dW/dt TO BE FOUND
dD/dt to be found

Relations:

LW = 200

L^2 + W^2 = D^2

Differentiate:

W dL/dt + L dW/dt = 0

Find the width W at the instant that the width is decreasing at the rate of .5 meters per second.

means:

Find the width W when dW/dt = - 0.5

W (4) + L(-0.5) = 0

4W - L/2 = 0

4W = L/2

W = L/8

But L = 200/W:

W = (200/W)/8

W = 25/W

W^2 = 25;   W = 5

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B)At what rate is the diagonal D of the rectangle changing at the instant when the width is 10 meters?

L^2 + W^2 = D^2

2L dL/dt + 2W dW/dt = 2D dD/dt

L dL/dt + W dW/dt = D dD/dt

Use W = 10, L = 20 (why?),  D = 10 sqrt(5)

Now go back to:

W dL/dt + L dW/dt = 0

Use dL/dt = 4, W = 10, L = 20, to find dW/dt  and you are practically there.

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See :

en.allexperts.com/q/Calculus-2063/2009/11/Related-Rates-87.htm

for other R-R examples and how to go about them.