The circumference of the equator of a sphere is measured as 10...

Question

The circumference of the equator of a sphere is measured as 10 cm with a possible error of 0.4 cm. This measurement is used to calculate the radius. The radius is then used to calculate the surface area and volume of the sphere. Estimate the percentage errors in the calculated values of

a. the radius.
b. the surface area.
c. the volume.

a. the radius.

b. the surface area.

c. the volume.

Accepted Answer

Welcome back, everyone. A scientist measures the diameter of a circular petri dish as 10 centimeters with a possible error of 0.2 centimeters. The scientists used this diameter to calculate the radius. Then the radius is used to calculate the area of the petri dish. Approximate the percentage errors in the calculated values of the radius and the area. A says the percentage area in the radius is 2% while the percentage error in the area is 6%. B says they are 2 and 4% respectively. C 1 and 6%, and D 1 and 4%. Now what do we know here? Let's first make note of the information we have. So far we know that the diameter of the Petri dish, let's call that D is equal to 10 centimeters. And that diameter has a possible error delta D of 0.2 centimeters. This must mean then that the radius of the Petri dish is going to be equal to half of the, which is 5 centimeters, OK? And the error in the radius is going to be delta R. No, we're going to solve for the percentage area in the calculated value of the radius. In other words, we would like to figure out the ratio of delta R to R as a percentage, so we're multiplying it by 100, and we'll also want to find a percentage error in the calculated value of the area. In other words, Delta A divided by a multiplied by 100%. So how can we do that? Well, let's first start with the percentage error in the calculated value of the radius. We already know the radius is 5 centimeters. So what is going to be a delta R or a possible error in the radius? Well, Since R equals 5 centimeters, which is half the error of the diameter, then the error in the radius is going to be a half of the error in the diameter. So that's a half of delta 2, that's 0.2 centimeters divided by 2, which equals 0.1 centimeters. Since we have Delta R and R. Then this means that our percentage error in the radius. Is going to be equal to 0.1 centimeters divided by 5 centimeters, OK, multiplied by 100%, which is going to be equal to 2%. OK, so the percentage error in the radius is 2%. Next, what about the er in the area? Let's move on to that part of our problem. Now what do we know? Well, the relationship between the radius and the area is not linear in the same way it is between the radius and the, the, the diameter. But what do we know about the area of a circle? Well, recall that the area. Of a circle A is equal to pi R2. OK. So thus we can use the differentials to help us figure out the error in the area because if we differentiate A with respect to R, OK, then it's going to be equal to 2 R. Thus this means then. That or change or or error in the area, OK, is going to be equal to the derivative of A with respect to R multiplied by the change in R. Which is gonna be 2 pi R multiplied by delta R, OK. I know if we substitute our known values. We know, let me put this down here. We know that R is 5 centimeters, so that's 2 multiplied by 5 centimeters, multiplied by 0.1 centimeters, delta R, and thus, this is going to be equal to pi, OK? So this is pi square centimeters. Now what about our area itself? Well, we also know that since the areas pi are squared, that's pi multiplied by 5 squared, which is gonna be equal to 25 square centimeters. So now, this means then. OK? That or a percentage error in the area, Delta A multiplied by A, sorry, divided by a multiplied by 100% is gonna be equal to pi divided by 25 pi, multiplied by 100%. Which is 1/25 of 100, which is 4%. Thus, the percentage area in the the percentage error in the area is 4%, and the percentage error in the area in the radius, sorry, is 2%, which means then that B is our correct answer. Thanks for watching everyone. I hope this video helped.

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