A spherical balloon is being inflated. Find the rate of increase of the surface area [tex]$S=4 \pi R^2$[/tex] with respect to the radius [tex]$R$[/tex] when [tex]$R=4$[/tex] ft.
Answer (2)
Find the derivative of the surface area formula S = 4 π R 2 with respect to the radius R , which gives d R d S = 8 π R .
Evaluate the derivative at R = 4 to find the rate of increase at that specific radius.
Calculate the rate of increase: 8 π ( 4 ) = 32 π .
The rate of increase of the surface area with respect to the radius when R = 4 ft is 32 π .
Explanation
Problem Analysis We are given the surface area of a sphere as S = 4 π R 2 , and we want to find the rate of increase of the surface area with respect to the radius R when R = 4 ft. This means we need to find the derivative of S with respect to R , denoted as d R d S , and then evaluate it at R = 4 .
Finding the Derivative First, let's find the derivative of S with respect to R :
d R d S = d R d ( 4 π R 2 ) Using the power rule, we get: d R d S = 4 π ( 2 R ) = 8 π R
Evaluating the Derivative Now, we need to evaluate the derivative at R = 4 :
d R d S ∣ R = 4 = 8 π ( 4 ) = 32 π So, the rate of increase of the surface area with respect to the radius when R = 4 ft is 32 π ft 2 /ft.
Final Answer Therefore, the rate of increase of the surface area S with respect to the radius R when R = 4 ft is 32 π .
Examples
Imagine you're designing spherical balloons for a party. Knowing how quickly the surface area increases as you inflate the balloon (increase the radius) helps you determine how much material you need and how the balloon's appearance changes as it grows. This calculation is crucial for ensuring the balloon looks good and doesn't pop!
The surface area of a spherical balloon increases at a rate of 32 π ft²/ft when the radius is 4 ft. This indicates how much the surface area expands for a small increase in radius. Therefore, understanding this rate can help in practical applications like balloon design.
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