Problem 36 - Calculus Early Transcendentals

Question

Changes in cone volume The volume of a right circular cone with radius \(r\) and height \(h\) is \(V=\pi r^{2} h / 3\) a. Approximate the change in the volume of the cone when the radius changes from \(r=6.5\) to \(r=6.6\) and the height changes from \(h=4.20\) to \(h=4.15\) b. Approximate the change in the volume of the cone when the radius changes from \(r=5.40\) to \(r=5.37\) and the height changes from \(h=12.0\) to \(h=11.96\)

Step-by-Step Solution

Answer

Answer: In order to find the approximate changes in volume for cases a and b, calculate \(\Delta V_a \approx V_2 - V_1\) and \(\Delta V_b \approx V_2 - V_1\) respectively using a calculator. Make sure to use the given radius and height values mentioned in the solution for each case.

Step 1: (Case a: Initial volume calculation)

(Calculate the initial volume of the cone using the given radius and height values: \(r_1 = 6.5\) and \(h_1 = 4.20\). Plug these values into the volume formula, \(V = \pi r^2 h / 3\):) \(V_1 = \frac{\pi (6.5)^2(4.20)}{3}\)

Step 2: (Case a: New volume calculation)

(Calculate the new volume of the cone using the updated radius and height values: \(r_2 = 6.6\) and \(h_2 = 4.15\). Plug these values into the volume formula:) \(V_2 = \frac{\pi (6.6)^2(4.15)}{3}\)

Step 3: (Case a: Change in volume calculation)

(Find the approximate change in volume by subtracting the initial volume from the new volume:) \(\Delta V_a \approx V_2 - V_1\)

Step 4: (Case b: Initial volume calculation)

(Calculate the initial volume of the cone using the given radius and height values: \(r_1 = 5.40\) and \(h_1 = 12.0\). Plug these values into the volume formula:) \(V_1 = \frac{\pi (5.40)^2(12.0)}{3}\)

Step 5: (Case b: New volume calculation)

(Calculate the new volume of the cone using the updated radius and height values: \(r_2 = 5.37\) and \(h_2 = 11.96\). Plug these values into the volume formula:) \(V_2 = \frac{\pi (5.37)^2(11.96)}{3}\)

Step 6: (Case b: Change in volume calculation)

(Find the approximate change in volume by subtracting the initial volume from the new volume:) \(\Delta V_b \approx V_2 - V_1\) Now you can use a calculator to find the approximate value of the change in volume for both cases, \(\Delta V_a\) and \(\Delta V_b\).