A triangle is measured and two adjacent sides are found to be 3 inches and 4 inches long, with an included angle of /4. The possible errors in measurement are 1/12 inch for the sides and 0. 04 radian for the angle. Approximate the maximum possible error in the computation of the area

Answer 1

To approximate the maximum possible error in the computation of the area of the triangle, we first need to find the maximum possible error in each side length and the maximum possible error in the angle.

The maximum possible error in a side length is given as 1/12 inch. Therefore, the side lengths could be as small as 3-1/12 = 35/12 inches and as large as 3+1/12 = 37/12 inches. Similarly, the side lengths could be as small as 4-1/12 = 47/12 inches and as large as 4+1/12 = 49/12 inches.

The maximum possible error in the angle is given as 0.04 radian. Therefore, the angle could be as small as π/4 - 0.04 and as large as π/4 + 0.04.

To calculate the maximum possible error in the computation of the area, we can use the formula for the area of a triangle: A = 1/2 * a * b * sin(C), where a and b are the side lengths, and C is the included angle.

The maximum possible error in the computation of the area is then given by:
ΔA = 1/2 * (b * sin(C) * Δa + a * sin(C) * Δb + a * b * cos(C) * ΔC),
where Δa, Δb, and ΔC are the possible errors in the side lengths a, b, and angle C respectively.

Substituting in the values we have:
ΔA = 1/2 * ((49/12 * sin(π/4 + 0.04) - 35/12 * sin(π/4 - 0.04)) + (37/12 * sin(π/4 + 0.04) - 35/12 * sin(π/4 - 0.04)) + (35/12 * 49/12 * cos(π/4) * 0.04))

Calculating the values, we find:
ΔA ≈ 0.237 square inches

Therefore, the maximum possible error in the computation of the area is approximately 0.237 square inches.