A circle has an initial radius of 50 ft when the radius begins...

Question

A circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

Accepted Answer

Below 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. A circular ice rink is melting, causing its radius to shrink. The initial radius of the ice rink is 40 m, and it decreases at a rate of 1 m per minute. Find the rate at which the area is changing when the radius is 15 m. Awesome. So it appears for this particular problem, we're trying to figure out the rate at which the area of this ice rink is changing when the radius is 15 m, noting that the initial radius of this ice rink is 40 m. So now that we know that we're ultimately trying to figure out what this rate of change is for this specific radius of the ice rink. Let us read off our multiple choice answers to see what our final answer might be, noting that for all of our multiple choice answers, it states that DA by DT is equal to some value, and they're all in the same units of meters squad per minute. So A is 30 pi, B is negative 30 pi, C is 15 pi, and D is negative 15 pi. Awesome. So first off, let us define all of our known variables. So we're told that our initial radius, which we'll call R0, our initial radius is going to be equal to 40 m, and we're also told that the rate of change of our radius, which we'll call DR by DT. And our rate of change of the radius is equal to -1 m per minute, and this value is negative, as we should recall, due to the fact that the radius is decreasing. So, as we should recall, because our radius is decreasing, this rate of change value will be negative. Let us also note that the radius at the moment of interest will be R is equal to 15 m. And what we're ultimately trying to figure out, or what we're trying to determine is we're trying to find the rate of change of the area. So we're trying to solve for DA by DT. This value is unknown to the unknown to us, and this is where it's at once again R equals 15. meters. So we're trying to figure out DA by DT when it's at R equals 15 m. So once again, putting what we have written as an expression into words, we're trying to figure out the rate of change of the area when R is equal to 15 m. So now, our next step is we need to recall and use the equation for an area of a circle, which an area of a circle states that the area A. is equal to pi multiplied by the radius squared. And now we need to take our area by circle, and we need to differentiate our area with respect to time T. So we need to take DA by DT. And DA by DT is equal to D by DT, where D by DT is just a fancy way of saying we're taking the derivative with respect to T. So we're taking D by DT of pi multiplied by R2, and when we do take that derivative, we will find that it's equal to 2 multiplied by pi, multiplied by R, multiplied by DR by DT. Fantastic. So now our next step is we need to substitute in all of our known variables back into our differentiated equation. So we will find that DA by DT is equal to 2 or 2 multiplied by pi. So 2 pi multiplied by 15 m. Multiplied by -1 m per minute. Fantastic. And when we simplify our equation, so when we simplify this expression to find the rate of change of the area, so once again we're simplifying our expression for DA by DT. You will find that DA by DT is equal to -30 pi, and once again it's units of measurement. are going to be, and don't forget, it's important um that don't forget that when we took the derivative. That we have 2 pi, so we're taking 2 multiplied by 15, which will be 30, so it'll be 30 pi. And don't forget we have a -1, so our final answer when we simplify will be negative 30 pi, and it's units of Measurement are going to be 1 m squad now, means you're taking 15 m multiplied by -1 m per minute. So you're gonna have 1 m squared per minute will be your new units, and that will be our final answer. Hooray, we did it. We found the rate of change of the area, and this corresponds with multiple choice answer letter B, which once again states that D A I D T is equal to -30 pi and its units of measurement are meters squad 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 circle has an initial radius of 50 ft when the radius begins decreasing at a rate of 2 ft/min. What is the rate of change of the area at the instant the radius is 10 ft?

Key Concepts

Related Rates

Area of a Circle

Differentiation

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