April 18th, 2009, 09:15 PM  #1 
Newbie Joined: Apr 2009 Posts: 14 Thanks: 0 
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 07:06 PM. 
April 19th, 2009, 12:41 PM  #2 
Senior Member Joined: Dec 2008 Posts: 251 Thanks: 0  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 
April 15th, 2015, 08:11 AM  #3  
Math Team Joined: Jul 2011 From: Texas Posts: 3,102 Thanks: 1677  Quote:
Quote:
$\displaystyle R^2  r^2 = 9$ $\displaystyle \frac{d}{dt}\left(R^2  r^2 = 9\right)$ $\displaystyle 2R\frac{dR}{dt}  2r\frac{dr}{dt} = 0$ $\displaystyle R\frac{dR}{dt}  r\frac{dr}{dt} = 0$ Quote:
$\displaystyle \frac{dA_R}{dt} = 2\pi R \frac{dR}{dt} = 10 \pi \implies R\frac{dR}{dt} = 5$ Quote:
$\displaystyle C_r = 2\pi r$ $\displaystyle \frac{dC_r}{dt} = 2\pi \frac{dr}{dt}$ 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 07:42 PM.  
April 15th, 2015, 10:15 AM  #4  
Math Team Joined: Jul 2011 From: Texas Posts: 3,102 Thanks: 1677  Quote:
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.  

Tags 
application, differentiation 
Search tags for this page 
a building is to braced by means of a beam which must pass over a wall shortest beam to be used,a building is to be braced by means of a beam which must pass over a wall.,a wall of a building is to be braced by beam which must pass
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Application of differentiation ! HELLP !!  shindigg  Calculus  7  March 4th, 2017 09:52 PM 
Differentiation Application  hatchelhoff  Calculus  2  March 4th, 2014 09:20 AM 
Differentiation application  Ronaldo  Calculus  3  December 29th, 2012 04:41 PM 
Application of differentiation (again) HELP !  shindigg  Calculus  4  September 4th, 2009 07:24 AM 
Differentiation application  Ronaldo  Complex Analysis  3  December 31st, 1969 04:00 PM 