the measurement on one side of a right triangle is found to be 9.5 inches inches, and the angle opposite that side is 27 degrees,45' with a possible error of 15'. (round answers to two decimal places)
a) Approximate the percent error in computing the length of the hypotenuse.
b) Estimate the maximum allowable percent error in measuring the angle of the error in computing the length of the hypotenuse cannot exceed 1%
i don't know the answer
a) To approximate the percent error in computing the length of the hypotenuse, we need to find the actual length of the hypotenuse and the length of the hypotenuse including the possible error.
1. Start by finding the actual length of the hypotenuse using the given side length and angle:
We can use the trigonometric ratio sine to solve for the hypotenuse (H):
sin(angle) = side opposite / hypotenuse
sin(27 degrees 45') = 9.5 inches / H
2. Convert the angle to decimal degrees:
27 degrees 45' = 27 + 45/60 = 27.75 degrees
3. Calculate the value of sin(27.75 degrees):
sin(27.75 degrees) ≈ 0.4646
4. Solve for H:
0.4646 = 9.5 / H
H ≈ 9.5 / 0.4646 ≈ 20.47 inches (rounded to two decimal places)
5. Now let's find the length of the hypotenuse including the maximum possible error:
Since the angle has a possible error of 15', the actual angle could vary between 27 degrees 45' - 15' and 27 degrees 45' + 15':
Lower bound: 27 degrees 45' - 15' = 27 degrees 30'
Upper bound: 27 degrees 45' + 15' = 28 degrees
6. Recalculate the hypotenuse using the lower bound angle:
Calculate the value of sin(27.5 degrees):
sin(27.5 degrees) ≈ 0.4597
Solve for H:
0.4597 = 9.5 / H
H_lower ≈ 9.5 / 0.4597 ≈ 20.65 inches (rounded to two decimal places)
7. Recalculate the hypotenuse using the upper bound angle:
Calculate the value of sin(28 degrees):
sin(28 degrees) ≈ 0.4695
Solve for H:
0.4695 = 9.5 / H
H_upper ≈ 9.5 / 0.4695 ≈ 20.25 inches (rounded to two decimal places)
8. Calculate the percent error:
Percent Error = (|H_upper - H| / H) * 100
Percent Error = (|20.25 - 20.47| / 20.47) * 100
Percent Error ≈ (0.22 / 20.47) * 100 ≈ 1.08% (rounded to two decimal places)
Therefore, the approximate percent error in computing the length of the hypotenuse is 1.08%.
b) To estimate the maximum allowable percent error in measuring the angle so that the error in computing the length of the hypotenuse does not exceed 1%, we need to find the maximum angle error.
1. Calculate the maximum length of the hypotenuse when the percent error is 1%:
Max H = H * (1 + Max Percent Error)
Max H = 20.47 * (1 + 0.01) = 20.47 * 1.01 ≈ 20.68 inches (rounded to two decimal places)
2. Find the maximum angle error:
Calculate the difference between the actual length of the hypotenuse and the maximum length:
Max H - H = 20.68 - 20.47 = 0.21 inches
To find the maximum angle error, we need to use the inverse sine function:
Angle Error = sin^(-1) (Side Error / Hypotenuse)
Angle Error = sin^(-1) (0.21 / 20.47)
3. Calculate the maximum angle error in degrees:
Angle Error ≈ 0.618 degrees (rounded to three decimal places)
4. Convert the angle error from decimal degrees to degree-minutes:
Angle Error ≈ 0 degrees 37.08'
Therefore, the maximum allowable percent error in measuring the angle is approximately 0 degrees 37.08'.