The measurement of the circumference of a circle is found to be 70 centimeters, with a possible error of 0.9 centimeter.
a) Approximate the percent error in computing the area of the circle. (Round your answer to two decimal places.)
b) Estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 5%. (Round your answer to one decimal place.)
a) To approximate the percent error in computing the area of the circle, we need to first calculate the actual circumference and radius of the circle.
Given:
Circumference = 70 cm
Possible error in circumference = 0.9 cm
We know that the circumference of a circle is given by the formula C = 2πr, where C is the circumference and r is the radius. Rearranging the formula, we have r = C / (2π).
Actual Radius = Circumference / (2π)
Actual Radius = 70 cm / (2π)
Approximating the value of π as 3.14, we can calculate the actual radius:
Actual Radius ≈ 70 cm / (2 * 3.14)
Actual Radius ≈ 11.15 cm
The actual area of the circle can be calculated using the formula A = πr²:
Actual Area = π * (Actual Radius)²
Actual Area ≈ 3.14 * (11.15 cm)²
Actual Area ≈ 3.14 * 124.2225 cm²
Actual Area ≈ 389.4585 cm² (rounded to four decimal places)
Now, we can calculate the percent error in computing the area by comparing the actual area to the computed area:
Percent Error = (Actual Area - Computed Area) / Actual Area * 100
Since the possible error in circumference is 0.9 cm, the possible error in the radius is 0.9 cm / (2π) ≈ 0.143 cm.
Computed Radius = Actual Radius ± Possible Error in Radius
Computed Radius = 11.15 cm ± 0.143 cm
Using the upper bound, the Computed Radius becomes:
Computed Radius = 11.15 cm + 0.143 cm
Computed Radius ≈ 11.293 cm
Using the lower bound, the Computed Radius becomes:
Computed Radius = 11.15 cm - 0.143 cm
Computed Radius ≈ 11.007 cm
Now, we can calculate the computed area using both the upper and lower bounds of the radius:
Upper Computed Area = π * (Computed Radius)²
Upper Computed Area ≈ 3.14 * (11.293 cm)²
Upper Computed Area ≈ 3.14 * 127.846049 cm²
Upper Computed Area ≈ 401.9372814 cm² (rounded to four decimal places)
Lower Computed Area = π * (Computed Radius)²
Lower Computed Area ≈ 3.14 * (11.007 cm)²
Lower Computed Area ≈ 3.14 * 121.154049 cm²
Lower Computed Area ≈ 380.9321234 cm² (rounded to four decimal places)
Now we can calculate the percent error for both upper and lower bounds:
Percent Error (Upper) = (Actual Area - Upper Computed Area) / Actual Area * 100
Percent Error (Upper) = (389.4585 cm² - 401.9372814 cm²) / 389.4585 cm² * 100
Percent Error (Upper) = -3.21% (rounded to two decimal places)
Percent Error (Lower) = (Actual Area - Lower Computed Area) / Actual Area * 100
Percent Error (Lower) = (389.4585 cm² - 380.9321234 cm²) / 389.4585 cm² * 100
Percent Error (Lower) = 2.20% (rounded to two decimal places)
Therefore, the approximate percent error in computing the area of the circle is between -3.21% and 2.20%.
b) The maximum allowable percent error in measuring the circumference can be calculated by using the percent error formula for the area and solving for the percent error in the circumference.
Percent Error (Circumference) = (Actual Area - Computed Area) / Actual Area * 100
Since the error in computing the area cannot exceed 5%, we can set the percent error in the circumference to 5% and solve for the maximum allowable percent error in measuring the circumference.
5% = (389.4585 cm² - Computed Area) / 389.4585 cm² * 100
Now, we rearrange the formula and solve for Computed Area:
(389.4585 cm² - Computed Area) / 389.4585 cm² = 5% / 100
389.4585 cm² - Computed Area = 0.05 * 389.4585 cm²
Computed Area = 389.4585 cm² - 0.05 * 389.4585 cm²
Computed Area = 389.4585 cm² - 19.472925 cm²
Computed Area ≈ 369.985575 cm²
Using the formula for the area of a circle, we can calculate the computed circumference using the computed area:
Computed Circumference = 2π * √(Computed Area / π)
Computed Circumference = 2π * √(369.985575 cm² / π)
Computed Circumference ≈ 2π * √117.778 cm²
Computed Circumference ≈ 2π * 10.856 cm
Computed Circumference ≈ 68.128 cm (rounded to three decimal places)
Now, we can calculate the maximum allowable percent error in measuring the circumference:
Maximum Allowable Percent Error = (Actual Circumference - Computed Circumference) / Actual Circumference * 100
Since the actual circumference is 70 cm, we can calculate the maximum allowable percent error:
Maximum Allowable Percent Error = (70 cm - 68.128 cm) / 70 cm * 100
Maximum Allowable Percent Error ≈ 2.68% (rounded to one decimal place)
Therefore, the maximum allowable percent error in measuring the circumference is approximately 2.68%.
To approximate the percent error in computing the area of a circle, we need to know the formula for the area of a circle. The formula for the area of a circle is A = πr^2, where A represents the area and r represents the radius of the circle.
To find the radius, we need to divide the circumference by 2π. The formula for the circumference of a circle is C = 2πr, where C represents the circumference and r represents the radius.
Since we are given the circumference of the circle as 70 centimeters with a possible error of 0.9 centimeters, we can find the radius as follows:
C = 2πr
70 = 2πr
Divide both sides by 2π:
r = 70 / (2π)
Now we can calculate the exact area of the circle using the formula A = πr^2:
A = π(70 / (2π))^2
Simplify:
A = π(70^2 / (2π)^2)
A = 1225 / 4π
To calculate the approximate percent error in computing the area, we need to calculate the difference between the exact area and the estimated area, and then divide it by the exact area. The estimated area is calculated using the approximate circumference (70 cm + 0.9 cm) and the same formula for the area:
Estimated Area = π( (70 + 0.9) / (2π))^2
Estimated Area = π( 35.9 / (2π))^2
Estimated Area = (35.9 / 2π)^2
Now we can calculate the percent error:
Percent Error = (Estimated Area - Exact Area) / Exact Area x 100%
Percent Error = ((35.9 / 2π)^2 - 1225 / 4π) / (1225 / 4π) x 100%
Using a calculator to get the approximate value, we find that the percent error is approximately 1.61%.
Now, let's move to part b.
To estimate the maximum allowable percent error in measuring the circumference if the error in computing the area cannot exceed 5%, we can set up an inequality using the same formulas for the area and circumference.
Let's assume the maximum allowable percent error in measuring the circumference is x%. This means that the estimated area should not exceed 105% of the exact area.
Estimated Area / Exact Area <= 105% / 100%
Using the formulas for the area and circumference, we can substitute them into the inequality and solve for x:
(π( (C + ΔC) / (2π))^2) / (1225 / 4π) <= 1.05
Simplify the equation:
( (C + ΔC) / (2π))^2 <= 1.05(1225 / 4π)
Take the square root of both sides:
(C + ΔC) / (2π) <= sqrt(1.05(1225 / 4π))
Multiply both sides by 2π:
C + ΔC <= 2πsqrt(1.05(1225 / 4π))
Substitute C = 70 and ΔC = 0.9 into the equation:
70 + 0.9 <= 2πsqrt(1.05(1225 / 4π))
Simplify:
70.9 <= 2πsqrt(1.05(1225 / 4π))
Now divide both sides by 2π and square the result:
(70.9 / (2π))^2 <= sqrt(1.05(1225 / 4π))
Simplify:
(35.45 / π)^2 <= sqrt(1.05(1225 / 4π))
Take the square of both sides:
(35.45 / π) <= sqrt(sqrt(1.05(1225 / 4π)))
Simplify:
35.45 / π <= (1.05(1225 / 4π))^0.25
Multiply both sides by π:
35.45 <= (π^0.25) * (1.05(1225 / 4π))^0.25
Now, calculate the right-hand side of the inequality using a calculator:
35.45 <= 1.078
Therefore, the maximum allowable percent error in measuring the circumference should be approximately 1.1%.
C = 2 pi r so r = 35/pi
so
dC = 2 pi dr
so
dr = dC/ (2 pi) = .9/(2pi)
a = pi r^2
da = 2 pi r dr
da = 70 pi dr
da/a = 70 pi dr/a
multiply by 100 to get percent