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%
To solve this problem, we need to use trigonometric ratios and understand the concept of percent error.
a) To approximate the percent error in computing the length of the hypotenuse, we need to calculate the actual length of the hypotenuse and the estimated length using the given measurements.
1. Calculate the length of the hypotenuse using the given measurements:
To find the length of the hypotenuse, we can use the trigonometric ratio for the sine function. In a right triangle, the ratio is defined as: sin(angle) = opposite/hypotenuse.
Given: angle = 27 degrees, 45' (in decimal form, 27.75 degrees) and the side opposite the angle = 9.5 inches.
Using the sine ratio: sin(angle) = opposite/hypotenuse
Substituting the values we found: sin(27.75 degrees) = 9.5/hypotenuse
2. Solve the equation for the hypotenuse:
Multiply both sides of the equation by the hypotenuse: hypotenuse * sin(27.75 degrees) = 9.5
Divide both sides of the equation by sin(27.75 degrees): hypotenuse = 9.5 / sin(27.75 degrees)
Using a calculator, evaluate the value: hypotenuse = 20.66 inches (rounded to two decimal places).
3. Calculate the estimated length of the hypotenuse:
To estimate the length of the hypotenuse, we'll consider the maximum possible error in the angle measurement.
The given error in the angle measurement is 15'. To convert this to degrees, divide 15 by 60: 15/60 = 0.25 degrees.
To find the maximum estimated length of the hypotenuse, we need to calculate the hypotenuse using the maximum possible angle measurement:
angle = 27.75 degrees + 0.25 degrees = 28 degrees.
Using the same trigonometric ratio as before: sin(28 degrees) = estimated hypotenuse / 9.5
We can now solve for the estimated hypotenuse: estimated hypotenuse = 9.5 * sin(28 degrees)
Using a calculator, evaluate the value: estimated hypotenuse = 21.14 inches (rounded to two decimal places).
4. Calculate the percent error:
To calculate the percent error, we'll use the formula: percent error = (|estimated - actual| / actual) * 100.
Using the values we found: percent error = (|21.14 - 20.66| / 20.66) * 100
Evaluate this expression: percent error = (0.48 / 20.66) * 100 ≈ 2.32%
Therefore, the approximate percent error in computing the length of the hypotenuse is approximately 2.32%.
b) To estimate the maximum allowable percent error in measuring the angle, we need to consider the percent error in the computed length of the hypotenuse.
Given: maximum allowable error in the computed length of the hypotenuse = 1%
To find the maximum allowable error in the angle measurement, we'll consider the formula for percent error.
Using the formula: percent error = (|estimated - actual| / actual) * 100
Substituting the given values: 1 = (|21.14 - 20.66| / 20.66) * 100
To solve this equation, isolate the absolute value term:
1/100 = |21.14 - 20.66| / 20.66
1/100 * 20.66 = |21.14 - 20.66|
20.66/100 = |0.48|
Now solve for the absolute value:
0.2066 = 0.48
Since 0.2066 is not equal to 0.48, we can conclude that the maximum allowable percent error in measuring the angle cannot exceed 1%.
In summary:
a) The approximate percent error in computing the length of the hypotenuse is approximately 2.32%.
b) The maximum allowable percent error in measuring the angle is 1%.