The circumference of a sphere was measured to be 77 cm with a possible error of .8 cm.
Use differentials to estimate the maximum error in the calculated surface area.
Estimate the relative error in the calculated surface area.
please help me..thank you!
Area= 4 PI radius^2
dA= 8 PI radius dR
What is your difficulty with this?
well to be quite honest i am still not sure how to solve the two questions. i tried taking the derivative of the surface area and multiplying by .8 but this is not the correct answer.i know that the radius is is 77/2pi but could u elaborate more? thanks
To estimate the maximum error in the calculated surface area of a sphere, we can use differentials. Here's how to solve the problem step by step:
1. Given information:
- Circumference of the sphere = 77 cm
- Possible error in the circumference = 0.8 cm
2. Let's start by finding the radius of the sphere using the formula for circumference:
Circumference = 2πr, where r is the radius.
Since the circumference is given as 77 cm, we have:
77 = 2πr
Dividing both sides by 2π, we get:
r = 77 / (2π) ≈ 12.27 cm (approximated to two decimal places)
3. Now, let's calculate the maximum error in the radius. Since the possible error in the circumference is 0.8 cm, the maximum error in the radius will be half of that, as it affects both ends:
Maximum error in the radius = 0.8 / 2 = 0.4 cm
4. To estimate the maximum error in the calculated surface area (dA), we can use the differential formula you mentioned: dA = 8πr * dr, where dr represents the change in radius.
Substituting the values, we have:
dA = 8π(12.27 cm) * (0.4 cm)
Simplifying this, we find:
dA ≈ 39.11π cm^2
So, the estimated maximum error in the calculated surface area of the sphere is approximately 39.11π cm^2.
To find the relative error in the calculated surface area, we divide the maximum error in the surface area by the actual surface area and express it as a ratio or percentage.
5. Calculating the actual surface area (A):
The formula for the surface area of a sphere is A = 4πr^2.
Substituting the radius we calculated earlier, we have:
A = 4π(12.27 cm)^2
A ≈ 1899.73π cm^2
6. Now, let's calculate the relative error in the surface area:
Relative error = (dA / A) * 100%
Relative error = (39.11π cm^2 / 1899.73π cm^2) * 100%
Simplifying this, we find:
Relative error ≈ 2.06%
Therefore, the estimated relative error in the calculated surface area of the sphere is approximately 2.06%.