The circumference of a sphere was measured to be 74 cm with a possible error of 0.5 cm.
(a) Use differentials to estimate the maximum error in the calculated surface area. (Round your answer to the nearest integer.)
cm2
What is the relative error? (Round your answer to three decimal places.)
(b) Use differentials to estimate the maximum error in the calculated volume. (Round your answer to the nearest integer.)
cm3
What is the relative error? (Round your answer to three decimal places.)
a) 24 cm^2
0.014
b) 139 cm^3
0.0190
i sold my soul to get this answer
To estimate the maximum error in the calculated surface area and volume of a sphere, we can use differentials. Here are the steps to solve the problem:
(a) To estimate the maximum error in the calculated surface area:
1. Start with the formula for the surface area of a sphere: A = 4πr^2, where A is the surface area and r is the radius.
2. Since we are given the circumference (C) of the sphere, we can use the formula for circumference: C = 2πr.
3. Rearrange the circumference formula to solve for the radius: r = C / (2π).
4. Differentiate the surface area formula with respect to the radius: dA/dr = 8πr.
5. Multiply the differential by the maximum error in the circumference (dC) to get the maximum error in the radius (dr): dr = dC / (2π).
6. Substitute the value of dr into the differential equation: dA = (8πr) * (dC / (2π)) = 4r * dC.
7. Substitute the given values into the equation: dA = 4 * (74/2π) * 0.5.
8. Calculate the numerical value of dA: dA ≈ 37 / π.
9. Round the value of dA to the nearest integer: dA ≈ 12 cm^2.
Therefore, the maximum error in the calculated surface area is estimated to be 12 cm^2.
To calculate the relative error in the surface area:
10. Divide the maximum error (dA) by the calculated surface area (A): relative error = dA / A.
11. Substitute the values into the formula: relative error ≈ (12 / A).
12. Divide A by 4πr^2: relative error ≈ (12 / (4πr^2)).
(b) To estimate the maximum error in the calculated volume:
1. Start with the formula for the volume of a sphere: V = (4/3)πr^3, where V is the volume and r is the radius.
2. Differentiate the volume formula with respect to the radius: dV/dr = 4πr^2.
3. Multiply the differential by the maximum error in the circumference (dC) to get the maximum error in the radius (dr): dr = dC / (2π).
4. Substitute the value of dr into the differential equation: dV = (4πr^2) * (dC / (2π)) = 2r^2 * dC.
5. Substitute the given values into the equation: dV = 2 * (74/(2π))^2 * 0.5.
6. Calculate the numerical value of dV: dV ≈ 74^2 / (2π).
7. Round the value of dV to the nearest integer: dV ≈ 829 cm^3.
Therefore, the maximum error in the calculated volume is estimated to be 829 cm^3.
To calculate the relative error in the volume:
8. Divide the maximum error (dV) by the calculated volume (V): relative error = dV / V.
9. Substitute the values into the formula: relative error ≈ (829 / V).
10. Divide V by (4/3)πr^3: relative error ≈ (829 / ((4/3)πr^3)). This will give us the relative error.