Question

(5) (10 points) The radius and height of a cone are measured as 3m and 4m, respectively, with an error of measurement of 0.1 cm in each dimension. Use differentials to estimate the error in the calculated surface area of the cone. Then, use this to decide on the least amount of cm2 of material one should obtain to be certain that they can build the surface of this cone. (The surface area of a cone is S(r,h)= √r²+h²​ ). (For the last part, just calculate the surface area of a cone with radius 3m and height 4m and add the quantity you found in the first part of the problem).

Answer 1

To estimate the error in the cone's surface area due to measurement errors, apply differentials to the surface area formula and then add the calculated error to the actual surface area to find the least amount of material needed.

Step-by-step explanation:

The question involves estimating the error in the surface area of a cone when there is an error in measuring the radius and height. To solve the problem, we will use differentials. The surface area of a cone is given by the formula A = πr(r + √r²+h²). The differential of A, dA, which represents the error in the surface area, can be found using the derivatives dr and dh (the errors in the radius and height, respectively).

To estimate the error in the surface area, we first differentiate the formula for the surface area with respect to r and h. We get dA = π(dr)(r + √r²+h²) + πr(rac{rdr + hdh}{√r²+h²}). Substituting the given values r = 3m, h = 4m, dr = 0.001m, and dh = 0.001m into this equation will give us the estimated error in the surface area (dA).

After calculating dA, we'll add this error to the actual surface area of the cone to determine the least amount of material required. The actual surface area for a cone with radius 3m and height 4m is A = π × 3m(3m + √(3m)²+(4m)²), rounded to an appropriate number of significant figures based on the measurement error.