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
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.
To find the maximum possible error in the computation of the area, we need to calculate the partial derivatives with respect to the variables involved (sides and angle).
Let's denote the sides as a and b, and the angle as θ.
The formula to calculate the area of a triangle is:
A = (1/2) * a * b * sin(θ)
Taking the partial derivative of A with respect to each variable, we have:
∂A/∂a = (1/2) * b * sin(θ)
∂A/∂b = (1/2) * a * sin(θ)
∂A/∂θ = (1/2) * a * b * cos(θ)
To find the maximum possible error in the computation of the area, we can use Taylor's theorem:
ΔA ≈ (∂A/∂a) * Δa + (∂A/∂b) * Δb + (∂A/∂θ) * Δθ
Where Δa, Δb, and Δθ represent the errors in the measurement of sides a, b, and angle θ respectively.
Given the maximum errors in measurement as:
Δa = 1/12 inch
Δb = 1/12 inch
Δθ = 0.04 radian
Let's substitute these values into the formula:
ΔA ≈ ((1/2) * b * sin(θ)) * (1/12) + ((1/2) * a * sin(θ)) * (1/12) + ((1/2) * a * b * cos(θ)) * (0.04)
Simplifying the equation:
ΔA ≈ (1/24) * ((b * sin(θ)) + (a * sin(θ))) + (0.02) * (a * b * cos(θ))
ΔA ≈ (1/24) * ( (3 * sin(π/4)) + (4 * sin(π/4)) ) + (0.02) * (3 * 4 * cos(π/4))
ΔA ≈ (1/24) * ( (3 * √2/2) + (4 * √2/2) ) + (0.02) * (12 * √2/2)
ΔA ≈ (1/24) * ( (3√2 + 4√2)/2 ) + (0.02) * (12√2/2)
ΔA ≈ (1/24) * ( (7√2)/2 ) + (0.02) * (6√2)
ΔA ≈ (1/24) * ( (7√2)/2 ) + (0.02) * (6√2)
ΔA ≈ (7√2/48) + (0.12√2)
ΔA ≈ (7√2/48) + (0.12√2)
ΔA ≈ 0.146 inch
Therefore, the maximum possible error in the computation of the area is approximately 0.146 inch.