Problem 2 - Calculus Early Transcendentals

Question

A frustum of a cone is generated by revolving the graph of \(y=4 x\) on the interval [2,6] about the \(x\) -axis. What is the area of the surface of the frustum?

Step-by-Step Solution

Answer

Based on the step-by-step solution provided, to find the area of the surface of the frustum generated by revolving the graph of the function \(y = 4x\) on the interval \([2, 6]\) about the x-axis, we first find the large and small radii, \(R\) and \(r\), which are 24 and 8 respectively. Then, we compute the slant height, which is \(\sqrt{544}\). Finally, we use the formula for the surface area of a frustum, \(A = \pi(R + r)L\), to find the area, which is approximately \(932.56\) square units.

Step 1: Find the large and small radii

To find the large and small radii, plug the endpoints of the interval \([2, 6]\) into the function \(y = 4x\). For the small radius \(r\), plug in \(x = 2\): \(r = 4(2) = 8\) For the large radius \(R\), plug in \(x = 6\): \(R = 4(6) = 24\) Now, we have the large and small radii, \(R = 24\) and \(r = 8\).

Step 2: Find the heights of the two cones

We need to find the heights of the large cone (the entire cone) and the small cone truncated, measured along the x-axis. The height of the large cone is the distance from \(x = 0\) to \(x = 6\): \(H_{large} = 6 - 0 = 6\) The height of the small cone is the distance from \(x = 0\) to \(x = 2\): \(H_{small} = 2 - 0 = 2\)

Step 3: Compute the slant height

We can use the Pythagorean theorem to find the slant height \(L\). Consider the large and small right triangles formed by the height and radii of the cones. They have side lengths \((L, R, H_{large})\) and \((L, r, H_{small})\), respectively. For the large triangle: \(L^2 = R^2 + H_{large}^2\), where \(R = 24\) and \(H_{large} = 6\). So, \(L^2 = 24^2 + 6^2 = 576 + 36 = 612\) For the small triangle: \(L^2 = r^2 + H_{small}^2\), where \(r = 8\) and \(H_{small} = 2\). So, \(L^2 = 8^2 + 2^2 = 64 + 4 = 68\) Taking the difference of the above two equations, \(612 - 68 = L^2 - L^2\) therefore, \(544 = 24^2 - 8^2\) Thus, the slant height of the frustum is \(\sqrt{544}\).

Step 4: Calculate the surface area of the frustum

Now we will use the formula for the surface area of a frustum, \(A = \pi(R + r)L\), with \(R = 24\), \(r = 8\), and \(L = \sqrt{544}\). \(A = \pi(24 + 8)\sqrt{544} = \pi(32)\sqrt{544} \approx 932.56\) square units. The area of the surface of the frustum is approximately \(932.56\) square units.