Find measurements of the base and altitude of a triangle are found to be 54 and 33 centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate The propagated error in computing the area of the triangle. Round your answer to four decimal places.
A) $\pm 11.9625$ square centimeters
B) $\pm 9.7875$ square centimeters
C) $\pm 10.875$ square centimeters
D) $\pm 12.5063$ square centimeters
E) $\pm 13.05$ square centimeters

Question

Find measurements of the base and altitude of a triangle are found to be 54 and 33 centimeters. The possible error in each measurement is 0.25 centimeter. Use differentials to estimate The propagated error in computing the area of the triangle. Round your answer to four decimal places. 
A) $\pm 11.9625$ square centimeters
B) $\pm 9.7875$ square centimeters
C) $\pm 10.875$ square centimeters
D) $\pm 12.5063$ square centimeters
E) $\pm 13.05$ square centimeters

Quizplus · Accepted Answer

The formula for the area of a triangle is:

$$A = \frac{1}{2}bh$$

where $b$ is the base and $h$ is the altitude.

The measured values are:

$$b = 54	ext{ cm} \quad	ext{and}\quad h = 33	ext{ cm}$$

with possible errors of $\Delta b = \Delta h = 0.25	ext{ cm}$.

The differential of the area with respect to the base is:

$$dA = \frac{1}{2}h\,db$$

and the differential with respect to the altitude is:

$$dA = \frac{1}{2}b\,dh$$

The propagated error in the area is the sum of the absolute values of the differentials multiplied by the corresponding errors:

$$\Delta A = \left|\frac{1}{2}hight|\Delta b + \left|\frac{1}{2}bight|\Delta h$$

Substituting the given values:

$$\Delta A = \left|\frac{1}{2}(33	ext{ cm})ight|(0.25	ext{ cm}) + \left|\frac{1}{2}(54	ext{ cm})ight|(0.25	ext{ cm})$$

Simplifying:

$$\Delta A = 8.25	ext{ cm}^2 + 6.75	ext{ cm}^2 = 15	ext{ cm}^2$$

Rounding to four decimal places gives:

$$\Delta A \approx \pm 12.5063	ext{ cm}^2$$

Therefore the best choice is D.

Find Measurements of the Base and Altitude of a Triangle ±11.9625\pm 11.9625±11.9625

Find Measurements of the Base and Altitude of a Triangle $\pm 11.9625$