A right circular cylinder with a height of 10 cm and a surface...

Question

A right circular cylinder with a height of 10 cm and a surface area of S cm² has a radius given by r(S)=1/2(√100+2S/π −10).

Find lim S→0^+ r(S) and interpret your result.

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. The radius of a right cylinder having a height of 15 centimeters and a surface area of U2 centimeters square is given as R of U is equal to 15th multiplied by the square root of 225 plus 5 U divided by pi minus 15. Calculate the limit. As you approaches 0 from the right of our of you and provide an interpretation. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. Firstly, we're asked to calculate the value of this limit, and our second answer we're trying to figure out is we're trying to provide an interpretation for the specific limit. So with that in mind, let's read off our multiple choice answers, and in an effort to save time, I won't read off each individual interpretation, but just note that each interpretation will be as you approaches zero. From the positive side, it will tell you if the radius is approaching, like will approach the specific value that we get for the calculated limit. And it will indicate for a very small surface area that it will be that specific evaluation, that calculated value for the limit. So as you approach a zero from the positive side, the radius will approach whatever value we calculated for the limit, and this will indicate that for a very small surface area, the radius of the cylinder will be approximately the value that we calculated for the limit. So with that said, let's read them off to see what our final answer might be. So A is 1/2 multiplied by the square root of 105, B is the B is 1/5 multiplied by the square root of 21, C is 1/5 multiplied by the square root of 210. And D is 12 multiplied by the square root of 210. OK. So first off, what we need to do in order to solve this particular problem is we need to identify the target variable and all of our known variables, and the target variable is the variable we're ultimately trying to solve for. So the target variable. We're ultimately trying to solve for is we're trying to figure out the limit as you approaches 0 from the right. That's what the little plus sign and the power means. So we're trying to figure out the limit as you approaches 0 from the right of our of you. Awesome. And our known variables are, and let's write these in blue, our known variables are you, the surface area. And R of you, where R of you is the radius as a function of you. Awesome. So now our next step is we need to write our given function for the radius. So we're told that our of U is equal to 1/5 multiplied by the square root of 225 plus, and we need to take this and add 5 multiplied by you divided by pi. And then we need to subtract 15 and note that our subtraction or the 15 is outside of the square root. Or the square root symbol or the square root bar. Awesome. So, With that said, we now need to substitute U equals 0 into the function of R of U. So when we do that, we will find that R of 0 is equal to 15th multiplied by the square root of 225 plus 5 multiplied by 0 divided by pi minus 15. And now we need to simplify the expression inside of the square root, and when we do that, we will get that R of 0 is equal to 1/5 multiplied by the square root of 225 minus 15. Awesome. So now, we need to calculate the value inside of the square root, and when we do that, we will find that R of 0 is equal to 1/5. Multiplied by the square root of 210. So now we need to simplify the expression, which right now R of 0 is equal to 1/5 multiplied by the square root of 210. That is our most simplified form, but we could just get rid of the Parenthesis to be more precise. So that will mean that our final answer is R of 0 is equal to 15th multiplied by the square root of 210. So that is our final answer. Hooray, we did it. And as you can see, this corresponds to multiple choice answer letter C. So when we go to look up at our multiple choice answers, the correct answer has to be letter C. Which, as for our interpretation, We need to note that as U approaches 0 from the positive side, the radius of the cylinder approaches 1/5 multiplied by the square root of 210, and this will indicate that for a very small surface area, the radius of the cylinder will be approximately 1/5 multiplied by the square root of 210. Awesome. So that means once again for C our final answer is 1/5 multiplied by the square root of 210, and our interpretation once again is as you approaches zero from the positive side, the radius of the cylinder will approach 1/5 multiplied by the square to 210, and this indicates that for a very small surface area, the radius of the cylinder will be approximately 1/5 multiplied by the square root of 210. Awesome, and that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

A right circular cylinder with a height of 10 cm and a surface area of S cm² has a radius given by r(S)=1/2(√100+2S/π −10).
Find lim S→0^+ r(S) and interpret your result.

Key Concepts

Limits

Surface Area of a Cylinder

Function Interpretation

Watch next