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%
a) Well, well, well, looks like we've got some calculations to do! To approximate the percent error in computing the length of the hypotenuse, we need to first calculate the actual length, and then take into account the possible error. Exciting stuff, right?
Now, let's find the actual length of the hypotenuse using the side length and the angle opposite. We know that the hypotenuse is the side opposite the right angle, so it's the longest side in a right triangle.
Using some trigonometry magic, we can use the formula: hypotenuse = (side length) / (sin(angle)). Plugging in the values, we get:
hypotenuse = 9.5 / sin(27°45')
Now, to calculate the percent error, we'll need to consider the maximum allowable error. We were given that the possible error in the angle measurement is 15', so we'll calculate the hypotenuse with the maximum angle value (27°45' + 15') and then calculate the percent difference between that value and the actual value.
b) Oh, now we're estimating the maximum allowable percent error in measuring the angle? This is like playing "Guess the Error!" Here's how we can do it:
To estimate the maximum allowable percent error, we'll calculate the change in hypotenuse when the angle measurement has a 1% error. We know that the hypotenuse changes when there's a change in angle, so we'll use the formula: change in angle = hypotenuse * (percent error / 100).
We'll set up an equation to solve for the percent error when the change in hypotenuse is 1%, which will look something like this: 1% = (change in angle) / (hypotenuse).
Solve that equation, and voila! We've estimated the maximum allowable percent error in measuring the angle.
Hey, hope I could bring a little bit of laughter to your math queries. Remember, math may not always be funny, but a little humor can go a long way!
a) To approximate the percent error in computing the length of the hypotenuse, we need to use trigonometry. The formula to calculate the length of the hypotenuse is:
c = a / sin(A)
Where:
c = length of the hypotenuse
a = length of the side opposite the given angle A
Using the given values, we have:
a = 9.5 inches
A = 27 degrees 45' = 27.75 degrees
Convert the angle to radians:
A_radians = A * (π/180) = 27.75 * (π/180) ≈ 0.4848 radians
Now we can calculate the length of the hypotenuse:
c = 9.5 / sin(0.4848) ≈ 19.55 inches
To find the percent error, we need to compare the approximate length of the hypotenuse (19.55 inches) with the actual length. Let's assume the actual length is "x".
Percent error = (|x - 19.55| / x) * 100
b) To estimate the maximum allowable percent error in measuring the angle, we can use the formula for the percent error in trigonometric functions. The formula is:
Percent error = (|Δx / x|) * 100
Where:
Δx = maximum allowable error in the measurement of the angle
x = value of the angle
Since we are given that the percent error should not exceed 1% when measuring the angle, we can set up the equation:
(ΔA / A) * 100 ≤ 1
Where:
ΔA = maximum allowable error in the measurement of the angle
A = value of the angle in degrees
Solving for ΔA, we have:
(ΔA / 27.75) * 100 ≤ 1
Dividing both sides by 100 and multiplying by 27.75:
ΔA ≤ (1/100) * 27.75
ΔA ≤ 0.2775
Therefore, the maximum allowable error in measuring the angle cannot exceed 0.2775 degrees.
To find the answers to these questions, we need to use trigonometry and formulas for the right triangle.
Let's start with part (a) - approximate the percent error in computing the length of the hypotenuse.
Step 1: Use trigonometry (specifically, the sine function) to find the length of the hypotenuse.
The formula for finding the length of the hypotenuse (c) in a right triangle is:
c = a / sin(A)
where:
a = length of one side of the triangle
A = angle opposite to the given side
Given:
a = 9.5 inches
A = 27 degrees, 45' (with a possible error of 15')
Step 2: Convert the angle from degrees to radians.
To use the trigonometric functions in most calculators, we need to convert the angle from degrees to radians. Since the given angle is already in degrees, we can skip this step.
Step 3: Compute the sine of the angle.
Using a calculator, find the sine of 27 degrees, which is approximately 0.45587.
Step 4: Calculate the length of the hypotenuse.
Using the formula c = a / sin(A), we can substitute the values:
c = 9.5 / 0.45587 ≈ 20.84 inches
Step 5: Determine the error in the length of the hypotenuse.
The given error for the angle is 15'. To convert this to radians, divide by 60:
Error = 15' / 60 = 0.25 degrees
Step 6: Find the new length of the hypotenuse with the maximum error.
To calculate the maximum length of the hypotenuse, we need to add the error to the given angle.
New hypotenuse length = a / sin(A + error)
New hypotenuse length ≈ 9.5 / sin(27.25) ≈ 20.80 inches
Step 7: Calculate the difference in length due to the estimated error.
Difference = Original hypotenuse length - New hypotenuse length
Difference ≈ 20.84 - 20.80 = 0.04 inches
Step 8: Calculate the percent error.
Percent Error = (Difference / Original hypotenuse length) x 100
Percent Error ≈ (0.04 / 20.84) x 100 ≈ 0.19%
Therefore, the approximate percent error in computing the length of the hypotenuse is 0.19%.
Moving on to part (b) - estimate the maximum allowable percent error in measuring the angle.
We need to find the maximum error in measuring the angle that will result in a percent error of 1% or less.
Step 1: Define the maximum allowable percent error.
Let's call the maximum allowable percent error in measuring the angle E.
Step 2: Calculate the maximum error in the length of the hypotenuse.
Given the maximum allowable percent error of 1%, we know that:
Percent Error = (Maximum error / Original hypotenuse length) x 100
1% = (Maximum error / 20.84) x 100
Step 3: Solve for the maximum error.
Rearranging the equation, we have:
Maximum error = (1% / 100) x 20.84
Maximum error ≈ 0.01 x 20.84 ≈ 0.2084 inches
Step 4: Convert the maximum error to degrees.
To convert the maximum error from inches to degrees, we need to use the formula:
Error in degrees = (Maximum error / hypotenuse length) x 180 / π
Error in degrees ≈ (0.2084 / 20.84) x 180 / π ≈ 1.00 degrees
Therefore, the maximum allowable percent error in measuring the angle is approximately 1.00 degrees.
Remember to round your answers to two decimal places as requested.
27°45' = 27.75° = 0.484 radians
15' = .25° = 0.00436 radians
the hypotenuse h is
h = 9.5 cscθ
h(.484) = 20.416
dh = -9.5 cscθ cotθ dθ
for dh <= 1% at θ=.484
.00484 >= 9.5(2.149)(1.902) dθ
dθ <= 0.012%