Answer:
The area is growing at a rate of 
Step-by-step explanation:
<em>Notice that this problem requires the use of implicit differentiation in related rates (some some calculus concepts to be understood), and not all middle school students cover such.</em>
We identify that the info given on the increasing rate of the circle's radius is 3 
and we identify such as the following differential rate:
Our unknown is the rate at which the area (A) of the circle is growing under these circumstances,that is, we need to find 
.
So we look into a formula for the area (A) of a circle in terms of its radius (r), so as to have a way of connecting both quantities (A and r):
We now apply the derivative operator with respect to time (
) to this equation, and use chain rule as we find the quadratic form of the radius:
![\frac{d}{dt} [A=\pi\,r^2]\\\frac{dA}{dt} =\pi\,*2*r*\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7Bd%7D%7Bdt%7D%20%5BA%3D%5Cpi%5C%2Cr%5E2%5D%5C%5C%5Cfrac%7BdA%7D%7Bdt%7D%20%3D%5Cpi%5C%2C%2A2%2Ar%2A%5Cfrac%7Bdr%7D%7Bdt%7D)
Now we replace the known values of the rate at which the radius is growing ( ), and also the value of the radius (r = 12 cm) at which we need to find he specific rate of change for the area :
which we can round to one decimal place as:
B; make sure to make the denominators equal before adding
17. 1lb is the better deal
18. about $17
19. 1.5lbs I think
20. 7000lbs
Answer:
x=5, x=-1
Step-by-step explanation:
Since both equations are equal to y, we can set them equal to each other
y = -x + 7 and y = 0.5(x - 3)2
-x + 7 = 0.5(x - 3)^2
Distribute the right hand side by Foiling
-x+7 =.5 ( x^2 -3x-3x+9)
Combine like terms
-x+7 = .5( x^2 -6x+9)
Distribute the .5
-x+7 = .5x^2 -3x+4.5
Add x to each side
-x+x+7 = .5x^2 -3x+x+4.5
7 = .5x^2 -2x+4.5
Subtract 7 from each side
7 -7= .5x^2 -2x+4.5-7
0 = .5x^2 -2x-2.5
Multiply by 2 to clear the decimals
0 = x^2 -4x -5
Factor
0 = (x-5) (x+1)
Using the zero product property
x-5=0 x+1 =0
x=5, x=-1
