A spherical balloon is inflated at a rate of 10 cm³/min. At wh...

Question

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Accepted Answer

Welcome back, everyone. A spherical water droplet is growing at a rate of 20 centimeters cubed per minute. Determine the rate at which the diameter of the droplet is increasing. When the diameter is 8 centimeters, we're given the four answer choices A says 5 divided by 85 centimeters per minute, B 5 divided by 4 centimeters per minute, C 10 divided by pi centimeters per minute, and the 20 divided by pi centimeters per minute. So we're given a spherical water droplet and essentially it has a volume of B equals 43. Pi or cubed where R is radius. This is how we define the volume of a sphere, and we know that radius is simply half of the diameter d. So what we're going to do is solve for B in terms of the. So we're going to get 4/3 multiplied by pi, which is then multiplied by D divided by 2 cubed. This is the same thing as radius, right? We simply want to rewrite V as a function of diameter. So let's simplify volume equals 4/3 multiplied by pi, which is then multiplied by the cubed, which is multiplied by 1/8. Because we can distribute the cube, right? And now we can see that volume is equal to. Now we have 4/3 multiplied by 1/8, so that's 4 divided by 24, and this gives us a 1/6 by the cubed. And then we have to understand that volume is a function of time. And also our diameter is a function of time. They are changing relative to time. We know that. The water droplet is growing at a rate of 20 centimeters per minute, so that's the rate of change of volume. We can verify that, looking at the units, 20 centimeters cubed per minute. We can see that in the numerator of the units we have centimeters cubed, right? So this is the rate of change of volume, and now we want to understand what's the rate at which the diameter of the droplet is increasing when the diameter value D is 8 centimeters. What we want to do is perform differentiation on the given volume function. Specifically, we have function one, and we're going to differentiate both sides with respect to time. On the left we get DV divided by DT, the derivative of volume with respect to time. On the right, we're going to get pi divided by 6, which is our constant. And then let's understand that D is cubed, right? So first of all, we have to apply the chain rule, which means that we're going to apply the power rule, bring down the 3, and decrease the original exponent by 1 unit, which gives us the derivative of the cubed, and this is a 3D squared, but then we need to multiply by. DD divided by DT, which is the rate of change of the diameter relative to time because our diameter is a function of time. And now if we simplify, well, we're going to get DV divided by D T equals. We can essentially divide 3 by 6, which gives us 1/2, so we're going to have 2 in the denominator. In the numerator, we're going to have pi multiplied by the squared, and all of that is multiplied by DD divided by D T. And this is the term that we're looking for. We want to solve for the rate of change of the diameter with respect to time. So we have to multiply both sides by 2. And divide both sides by pid squared. So we're going to get DD divided by D T equals 2 divided by pid squared multiplied by DB divided by DT, and now we can substitute the given values. So we get 2 divided by pi, which is multiplied by the square of the diameter, so that's 8 squad, so 8 centimeters squared. And then we multiply by 2. Centimeter cubed per minute. From here all that we have to do is simplify, so we get 2. Multiplied by 20, so we can simply say that we have 40. In the numerator, the units are centimeters cubed per minute. What about the denominator? Well, we're going to have 64 i. Centimeter squared and now considering the units we can see that those would be centimeters per minute because we're canceling out the cube with the square. So DD divided by the T is going to be equal to, now we need to analyze those. Well, the numerator and the denominator, those would be divisible by 8, so we're going to get 5 at the top. So that would be 5. And at the bottom, we're going to have 8, right? Because 64 divided by 8 is 8, so it's 5 divided by 8 by centimeters per minute. Well done, we have solved the problem. We can conclude that the correct answer to this problem corresponds to the answer choice A. Thank you for watching.

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Key Concepts

Related Rates

Volume of a Sphere

Chain Rule

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