Question

A right circular closed cone is measured to have base radius 4 cm and perpendicular height 2 cm, and it is formed by rotating a line segment given by y = 2x, 0 5x52, about the x-axis. a) The curved surface area of the cone is given by and integral of the form / 21 f(x) dx. Find f (x) and enter it in the box below. f (x) = b) If the rotation about the x-axis is only between the angles of 0 and what is the surface area of this portion of the cone? 3 Area = c) For variable radius r and height h, the total surface area of the closed cone is given in some Maple code below, along with some other Maple results that may be useful. 2ar + V >S := Pi*r^2 + Pi*r*sqrt (r^2+h^2); S := ar2 +arvh2 + x2 > diff(s,r); ar2 + + Vh2 +r2 > diff(sh); arh Vh2 + r2 Suppose that the values of r and h given in part (a) are measured with errors with magnitudes at most 0.03 cm and 0.3 cm respectively. Using the total differential approximation of S, calculate the maximum absolute error M in the measured value of the total surface area S and enter the value in the box below. Enter the numerical part of your answer correct to 2 decimal places in the box below. For example, if your answer is 3.17873, enter 3.18 M = 4

Answer 1

The function f(x) for the curved surface area of a cone from rotating the line segment y = 2x around the x-axis is 4πx^2. The surface area for a partial rotation can be determined proportionally, while the maximum absolute error in the measurement of the total surface area is found using the total differential approximation.

Step-by-step explanation:

To find the function f(x) for the curved surface area of a cone generated by rotating the line y = 2x around the x-axis, we use the formula for the lateral surface area of a cone: A = π r l, where r is the radius and l is the slant height.

Since the problem provides a function y(x) that represents the radius of the cone at a given x, and the line segment rotates around the x-axis, the formula adapts to an integral form to accumulate the infinitesimal surface areas along the x-axis. Thus, f(x) equals 2πxy. Given that y = 2x, f(x) would be 4πx^2.

The partial area of the cone between angles 0 and θ can be found using a proportional segment of the total lateral surface area. For the total surface area of a closed cone, one adds the area of the base, πr^2, to the lateral surface area.

Regarding the error in measurement for the total surface area S with radius r and height h, we can use the total differential approximation, dS = ∂S/∂r dr + ∂S/∂h dh, and the largest possible errors dr and dh to find the maximum absolute error M.