Question

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft) is

• A. $(0.088)(\sqrt{2}+1)$
• B. $(0.616)(\sqrt{2}+1)$
• C. $(0.616)(\sqrt{2})$
• D. $(0.088)(\sqrt{2})$

Hint:

Error propagation: Use differentials to approximate the error in the calculated quantity.

Answer 1

The Correct Option is B

The total surface area of a right circular cone is given by the formula:

$$A = \pi r(r + l)$$

where \(r\) is the radius, and \(l\) is the slant height of the cone. The slant height \(l\) is determined using the Pythagorean theorem as:

$$l = \sqrt{r^2 + h^2}$$

Given \(r = 7\) feet and \(h = 7\) feet, the slant height \(l\) is:

$$l = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2}$$

Thus, the surface area \(A\) is:

$$A = \pi \times 7 \times (7 + 7\sqrt{2}) = 7\pi (7 + 7\sqrt{2})$$

Now, we compute the differential to estimate the error in the surface area due to the errors in measurements of \(r\) and \(h\). The differential for the area is:

$$dA = \frac{\partial A}{\partial r}dr + \frac{\partial A}{\partial l}dl$$

The partial derivatives are:

$$\frac{\partial A}{\partial r} = \pi(2r + l)$$

$$\frac{\partial A}{\partial l} = \pi r$$

Errors for both \(r\) and \(h\) are \(\pm 0.002 \times 7\) feet, thus:

$$dr = dl = 0.014$$

Substitute into the total differential:

$$dA = \pi(14 + 7\sqrt{2}) \times 0.014 + \pi \times 7 \times 0.014$$

Combining, we have:

$$dA = \pi \times 0.014 \times [(14 + 7\sqrt{2}) + 7]$$

$$= \pi \times 0.014 \times (21 + 7\sqrt{2})$$

Thus:

$$dA = 0.014 \pi \times 7(\sqrt{2} + 1)$$

Simplifying:

$$dA \approx 0.616(\sqrt{2} + 1)$$

The error in the total surface area is \((0.616)(\sqrt{2} + 1)\) square feet, matching the provided correct answer.