The legs of a right triangle are measured to be and inches with a maximum error of inches in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the hypotenuse.
A) is the maximum error in the hypotenuse.
B) is the maximum error in the hypotenuse.
C) is the maximum error in the hypotenuse.
D) is the maximum error in the hypotenuse.
E) is the maximum error in the hypotenuse.

Question

The legs of a right triangle are measured to be  and  inches with a maximum error of  inches in each measurement. Use differentials to estimate the maximum possible error in the calculated value of the hypotenuse.
A) is the maximum error in the hypotenuse. 
B) is the maximum error in the hypotenuse. 
C) is the maximum error in the hypotenuse. 
D) is the maximum error in the hypotenuse. 
E) is the maximum error in the hypotenuse.

Quizplus · Accepted Answer

Let the legs of the right triangle be a = 11.04 inches and b = 11.01 inches, each with a maximum error of 0.01 inches. The hypotenuse c of the right triangle is calculated using the Pythagorean theorem:

c² = a² + b²
c² = (11.04)² + (11.01)²
c² = 243.0585

Taking the square root gives:

c ≈ 15.5875

To estimate the maximum possible error in c using differentials, we find the partial derivatives of c with respect to a and b:

∂c/∂a = 2a/2c = a/c
∂c/∂b = 2b/2c = b/c

Using the formula for the differential, we have:

Δc ≈ (∂c/∂a)Δa + (∂c/∂b)Δb
Δc ≈ (a/c)Δa + (b/c)Δb
Δc ≈ (11.04/15.5875)(0.01) + (11.01/15.5875)(0.01)
Δc ≈ 0.0070798

Therefore, the maximum possible error in c is approximately 0.007080 inches, or 11ead0bc_8dd1_d700_99a0_351ad0650b10_TB6988_11 inches.

The Legs of a Right Triangle Are Measured to Be