The volume V of a sphere of radius r changes over time t.

Question

b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?

Accepted Answer

Welcome back, everyone. The volume V of a balloon is given by the formula for the volume of a sphere where the radius R changes over time T. If the radius of the balloon is increasing at the rate of 3 centimeters per minute, what is the rate at which the volume is changing when the radius is 5 centimeters? We are given 4 answer choices. A says 225 pi, B 75 pi, C 150 pi, and D 300 pi all given in centimeters cubed per minute. So first of all, let's define the equation that gives us the volume of a sphere. So V equals 4/3 pi are cubed. Now, what we want to do is simply differentiate this equation relative to time knowing that volume is a function of time and radius is also a function of time. So first of all, we take the derivative of the left hand side, that would be DB divided by DT indicating that we're taking the derivative of volume with respect to time. We can then factor out 4/3 by, and then for our cubed, well, we can apply the power rule, right? So we can bring down the exponent. And then R is going to have an exponent of 23 minus 1, right? So we're applying the power rule, and from here we are applying the chain rule because we have a composite function. So we have to multiply by the R divided by the T. And if we simplify, well, we're going to say that. DV divided by DT is going to be equal to 4 pi. R squared dr divided by D T Now what are we given? Well, we can essentially see that the radius of the balloon is increasing at a rate of 3 centimeters per minute. So that's our DR divided by DT. That's 3 centimeters per minute. And we also know that the radius is 5 at the point of interest, so R is 5 centimeters, and we can use those in evaluating the rate of change of volume. DV divided by DC is going to be equal to 4 pi multiplied by a radius square, so that's 5. Centimeter squared multiplied by dr. Divided by the T, so that's 3 centimeters per minute. So we have our equation and now we can see that this is going to be for pi multiplied by 25, multiplied by 3. So 300 pi. centimeters cubed per minute which corresponds to the choice the. Thank you for watching.

The volume V of a sphere of radius r changes over time t.
b. At what rate is the volume changing if the radius increases at 2 in/min when when the radius is 4 inches?

Key Concepts

Related Rates

Volume of a Sphere

Chain Rule

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