Approximating changes

Approximate the change ...

Question

Approximating changes

Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

Accepted Answer

Welcome back, everyone. A cylindrical water tank with a fixed radius of 10 centimeters is being drained. Calculate the change in the water volume when the water level decreases from 30 centimeters to 29.5 centimeters. Use the formula for the volume of a cylinder V of H equals piR2 H. We given for answer choices as negative. 50 B 50 C 100 pi, and the 100 pi all given in centimeters cubed. Now how do we estimate the change in the value of a function? We have to recall that the change in the value of a function delta F can be approximated as the derivative of the function at the initial value X1 multiplied by the change in X, right? So in this case we're given a different function which is V of H, so the change in volume can be approximated as the derivative of the volume function at H1, the initial height multiplied by the change in height. Now we can identify the change in height to begin with, which is the final height H2 minus the initial height H1. Our final height is 29.5 centimeters and the initial height is 30 centimeters. When we subtract the values, we're going to get -0.5 centimeters, right? Height decreases. And that we want to identify the derivative of the volume function with respect to H. This means that pi or squared is a constant, and we are multiplying by the derivative of H, which is simply 1, and we get pi R2. Now let's identify the value of the derivative at the initial. Height, right? So the initial height is 30 centimeters. So we are evaluating the at 30. And this gives us pi are squad right because B is independent of h. We don't have any h, so we're simply multiplying pi by radius square, and our radius is 10 centimeters. When we square that, we're going to get 100 centimeters squared. Well done. And now our change in volume is approximately equal to 100 squared multiplied by the change in height, so -0.5 centimeters. Which gives us negative 50. By centimeters cube and this corresponds to the answer choice a. Thank you for watching.

Approximating changes

Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 to h = 11.9 cm (V(h) = πr²h).

Key Concepts

Differential Calculus

Volume of a Cylinder

Linear Approximation

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