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April 18th, 2009, 08:15 PM   #1
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1. The area between two varying concentric circle is at all times 9phi in^2. the rate of change of the area of the larger circle is 10phi in^2/sec. How fast is the circumference of the smaller circle changing when it has area 16phi in^2

2. A wall of a building is to be braced by a beam that must rest on the ground and pass over a vertical wall 10ft high that is 8 ft from the building. find the length L of the shortest beam that can be used.

Last edited by skipjack; April 16th, 2015 at 06:06 PM.
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April 19th, 2009, 11:41 AM   #2
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Re: Application of differentiation

I'll work out the answer to the first problem for you.

First, we give variable names to all the quantities under consideration: and for the area of the larger and smaller circle respectively, and for the radii, and for the circumferences. As it turns out, we will not need to consider , but we have the formulas



Now, we find the circumference of the smaller circle in terms of the area of the larger:



and differentiate by :



At and , we have

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April 15th, 2015, 07:11 AM   #3
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Quote:
Originally Posted by Tear_Grant View Post
1. The area between two varying concentric circle is at all times 9phi in^2. the rate of change of the area of the larger circle is 10phi in^2/sec. How fast is the circumference of the smaller circle changing when it has area 16phi in^2
I assume you mean pi ($\pi$) , and not phi ($\phi$)

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The area between two varying concentric circle is at all times 9phi in^2.
$\displaystyle \pi(R^2-r^2) = 9\pi$

$\displaystyle R^2 - r^2 = 9$

$\displaystyle \frac{d}{dt}\left(R^2 - r^2 = 9\right)$

$\displaystyle 2R\frac{dR}{dt} - 2r\frac{dr}{dt} = 0$

$\displaystyle R\frac{dR}{dt} - r\frac{dr}{dt} = 0$

Quote:
the rate of change of the area of the larger circle is 10phi in^2/sec.
$\displaystyle A_R = \pi R^2$

$\displaystyle \frac{dA_R}{dt} = 2\pi R \frac{dR}{dt} = 10 \pi \implies R\frac{dR}{dt} = 5$

Quote:
How fast is the circumference of the smaller circle changing when it has area 16phi in^2
$\displaystyle 16\pi = \pi r^2 \implies r = 4$

$\displaystyle C_r = 2\pi r$

$\displaystyle \frac{dC_r}{dt} = 2\pi \frac{dr}{dt}$

looks like you need to find the value of $\displaystyle \frac{dr}{dt}$ when $\displaystyle r = 4$ ... you have enough information above to determine that value.

Last edited by skipjack; April 16th, 2015 at 06:42 PM.
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April 15th, 2015, 09:15 AM   #4
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2. A wall of a building is to be braced by a beam that must rest on the ground and pass over a vertical wall 10ft high that is 8 ft from the building. find the length L of the shortest beam that can be used.
Note the sketch for the assignment of variables.

Pythagoras ...

$\displaystyle L^2 = (x+8 )^2 + (y+10)^2$

similar triangles ...

$\displaystyle \frac{y}{8} = \frac{10}{x}$

use the similar triangles proportion to solve for y in terms of x (or x in terms of y, your choice) and substitute into the Pythagoras equation to get $L^2$ in terms of a single variable.

let $\displaystyle L^2 = Z$ ...

$\displaystyle Z = (x+8 )^2 + (y+10)^2$

note that minimizing $Z$ will also minimize $L$.

find $\displaystyle \frac{dZ}{dx}$ or $\displaystyle \frac{dZ}{dy}$ and minimize.
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