A spherical balloon is inflated and its volume increases at a ...

Question

A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Determine the rate of change of the radius of a soap bubble if its volume increases at a rate of 20 cubic centimeters per minute. The radius of the bubble is 5 centimeters. Awesome. So it appears for this particular problem, we're ultimately trying to figure out what the rate of change is for this radius of the specific soap bubble, if its volume is increasing at a rate of 20 cubic centimeters per minute, provided that the radius of this soap bubble is 5. 5 centimeters. So now that we know that we're ultimately trying to figure out what the rate of change is for the radius, let us read off our multiple choice answers to see what our final answer might be, noting that they all for all of our multiple choice answers, they state that DR by DT is equal to some value, and they're all in the same units of centimeters per minute. So A is -1 divided by 10 pi, B is 1 divided by 10, C is -1 divided by 5 pi. And D is 1 divided by 5 pi. Awesome. So, first off, in order to help us solve for this particular problem, let us write down all of our known variables and what we're ultimately trying to solve for. So what we're ultimately trying to solve for is the unknown variable that we're trying to solve for. So let us state that capital V, so let's say V will be the volume of the spiritual spherical bubble, and the volume. So that's gonna be the volume V once again is the volume of our spherical, which means it's spear-shaped soap bubble, and let us say that R is going to be the radius of our bubble, and let us note and recall that the volume of sphere, so we need to recall and use the equation for a volume of a sphere, and the volume of a sphere V is equal to 4 divided by 3, 4/3 multiplied by pi, multiplied by R to the power of 3. And we also need to recall and note that the rate of change of the volume with respect to time, which we could write this as DV by DT, is equal to 20. Cubic centimeters per minute. Fantastic. And what we're ultimately trying to solve for is we're trying to find the rate of change of the radius with respect to time. So we're ultimately trying to solve for DR by DT. This value is unknown to us, and this is when When the radius is equal to 5 centimeters. Awesome. So now that's our first step was defining all of our variables and determine what we're trying to solve for. And now our second step is we need to recall and use the chain rule to to relate DV by DT and DR by DT. So we need to differentiate the volume equation with respect to T. So we need take DV by DT and note that this is equal to D by DT where D by DT is just a fancy way of saying we're taking the derivative of T, we're taking the derivative with respect to T. So that's what D by DT means. We're just differentiating with respect to T. So we're saying that DV by DT is equal to D by DT of 4/3, 4 divided by 3, multiplied by pi, multiplied by R to the power of 3. Fantastic. So now, we need to apply the chain rule, and when we apply the chain rule, we will find that DV by DT will be equal to 4 multiplied by pi, multiplied by R to the power up to DR and this will be multiplied by DR by DT. Fantastic. And once again, that is what our result will be when we take the chain rule for our volume equation. So when we take our volume equation once again, and we apply the chain rule, we should get 4 multiplied by pi, multiplied by R square, multiplied by DR by DT. So now, our 3rd step now is we need to substitute in our known variables into our differentiated equation to solve for DR by DT. So once again, noting that we're told that DV by DT is equal to 20 cubic centimeters per minute and R is equal to 5 centimeters, we will find that 20 cubic centimeters per minute is equal to 4 multiplied by pi, multiplied by plugging in our value for R, which is 5 centimeters squared, multiplied by DR by DT. And when we take this expression, and we simplify the equation to isolate and solve for DR by DT, we will find that 20. Cubic centimeters, so we need to take 20. Cubic centimeters per minute is equal to 100 multiplied by pi, multiplied by DR by DT, and it's going to be in units of centimeters square. So when we get DR by DT all by itself, you will find that DR by DT is equal to 20, divided by 100 pi. centimeters. Per minute will be our new units. So when we simplify one last time, you will find that this is equal to 1 divided by 5 centimeters per minute. So it's 1 divided by 5 centimeters per minute, and that is our final answer for DR by DT. Hooray, we did it. We found our rate of change, and this corresponds with multiple choice answer letter D. Which once again states that DR by DT is equal to 1 divided by 5 centimeters per minute. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A spherical balloon is inflated and its volume increases at a rate of 15 in³/min. What is the rate of change of its radius when the radius is 10 in?

Key Concepts

Volume of a Sphere

Related Rates

Chain Rule

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