The measurement of one side of a right triangle is found to be 9.5 inches, and the angle opposite that side is 26°45' with a possible error of 1.5'.
Approximate the percent error in computing the area of the triangle.
=0.46688 rad ± 0.000002315 rad.
should read 0.46688 rad ±0.000436 rad.
The calculated error should therefore read:
"I get ±0.097 sq.in. using both methods."
Area of right triangle
=(1/2)L*(L)cot(θ)
=(1/2)L²cot(θ)
L=9.5,
θ=26°45' ± 1.5'
=0.46688 rad ± 0.000002315 rad.
If you have not done calculus yet, calculate the area based on the given θ, then calculate the largest and smallest possible value of θ to give the higher and lower limits.
If you have done calculus, set
A(θ)=L²cot(θ) and differentiate with respect to θ to get e=A'(θ). Multiply e by the error in θ to give the error in area.
I get ±0.000516 sq.in. using both methods.
To approximate the percent error in computing the area of the triangle, we need to first calculate the actual area of the triangle and then calculate the maximum and minimum possible areas considering the given measurement and angle errors.
The area of a right triangle can be calculated using the formula:
Area = (1/2) * base * height
Given that one side of the right triangle is 9.5 inches, we can assume it to be the base. To find the height, we can use trigonometry. The trigonometric function relating the angle and the sides of a right triangle is given by:
tan(angle) = opposite/adjacent
In this case, the angle is 26°45' with a possible error of 1.5'. Considering the error, the actual angle could vary between 25°15' and 28°15'. Let's calculate the actual height for both minimum and maximum possible angles:
For the minimum angle:
tan(25°15') = height / 9.5 inches
Rearranging the formula, we can find the height:
height_min = tan(25°15') * 9.5 inches
Similarly, for the maximum angle:
tan(28°15') = height / 9.5 inches
Again, rearrange to find the height:
height_max = tan(28°15') * 9.5 inches
Now, we can calculate the actual area of the triangle using the actual base and calculated heights:
area_actual = (1/2) * 9.5 inches * (height_min + height_max) / 2
Next, we need to find the maximum and minimum possible areas considering the measurement and angle errors.
For the maximum possible area, we use the maximum possible base length and maximum possible height:
area_max = (1/2) * (9.5 inches + 0.5 inches) * (height_max + 0.025 inches)
For the minimum possible area, we use the minimum possible base length and minimum possible height:
area_min = (1/2) * (9.5 inches - 0.5 inches) * (height_min - 0.025 inches)
Now, we can calculate the percent error using the formula:
percent_error = ((area_max - area_min) / area_actual) * 100
Replace the values in the formula to approximate the percent error in computing the area of the triangle.