You are blowing air into a spherical balloon at a rate of 11 cubic inches per se

Question

You are blowing air into a spherical balloon at a rate of 11 cubic inches per second. The goal of this problem is to answer the following question: What is the rate of change of the surface area of the balloon at time t = 1 second, given that the balloon has a radius of 3 inches at that instant? Let r(t), S(t), and V(t) denote the radius, the surface area, and the volume of your balloon at time t, respectively. (a) Rewrite the first sentence above as a statement about the known rate of change of the volume of the balloon for all t: V'(t) = cubic inches per second. (b) Next write a formula relating the changing volume V(t) of the sphere to the changing radius r(t). and differentiate that formula with respect to t. Using what you know about V'(t) and r(1), find the rato of change of the radius at t = 1 sec: r'(1) = inches per second. (c) Finally, write a formula relating the changing surface area S(t) of the sphere to the changing radius r(t), and differentiate that formula with respect to t. Use what you know about r(1) and r'(1) to determine the rate of change of the surface area at t = 1 sec: S'(1) = square inches per second. (d) Note that although you're told that the time was t = 1 seconds when the radius was 3 inches, the solution never actually uses the explicit value of t = 1 (only the value of the radius it that time). So the problem could have alternatively asked the following question: What is the rate of change of the surface area of the balloon at the time that it has a radius of 3 inches? In fact, most related rates problems will be stated this way.

Explanation / Answer

a)v'(t)=11cubic inch per second

b)volume of sphere =v =(4/3)pi r3

differentiate with respect to t

v'(t)=(4/3)pi *3r2 (r'(t))

v'(t)=4pi r2 (r'(t))

at t=1, r=3

v'(1)=4pi *32 (r'(1))

11=36pi (r'(1))

r '(1)= 11/(36pi) in/sec

r '(1)= 0.09726 in/sec

c) surface area s =4pi r2

differentiate with respect to t

s'(t)=4pi *2r*r'(t)

at t=1

s'(1)=4pi *2*3* r'(1)

s'(1)=4pi *2*3*11/(36pi)

s'(1)= 2*11/3

s'(1)=22/3

s'(1)=7.33 sq in/sec

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