Aspherical balloon is inflated with gas at a rate of 900 cubic centimeters per minute.
(a) how fast is the radius of the balloon changing at the instant the radius is 40 centimeters?
/min

(b) how fast is the radius of the balloon changing at the instant the radius is 90 centimeters? cm/min

Use the chain rule to find the relation between the change in Volume, dV / dt and the change in radius, dr / dt

Change in Volume = dV / dt = [dV / dr] * [dr / dt] => dr / dt = [dV / dt] / [dV/ dr]

Also, Volume, V = [4/3]π(r^3) => dV / dr = 4π (r^2)
And dV/dt = 900 cm^3 / min

Then, dr / dt =  [900 cm^ / min] / [dV/dr] = 900 / [4π r^2] = 225 / (π r^2)

a) r = 40 cm

dr / dt = 225 / [π (40^2) ] = 0.14 cm/min

b) r = 90 cm

dr / dt = 225 / (π 90^2) = 0.027 cm/min

Step-by-step explanation:

Equation 1: Sum:

 

x + y = 12

 

Equation 2: Difference:

 

x - y = 4

 

Solution:

 

x + y = 12

x  - y =   4

 

Add the two equations together:

 

2x = 16

 

Divide both sides by 2:

 

x = 8

 

Solve for y:

 

Use equation 1 substituting for x:

 

x + y = 12

 

8 + y = 12

 

Subtract 8 from both sides:

 

y = 4

 

 

one number is 8 and the other number is 4.

 

8 + 4 = 12

 

8 - 4  =  4