The circumference of a sphere was measured to be 74 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum error in the calculated surface area.
The (small) relative error in a linear dimension, which in this case is 0.676%, gets doubled when you are calculating the relative error in an area dimension. That is the "linear approximation".
Here's why:
Area = C * r^2 * (1 + e)^2
= C r^2 (1 + 2e + e^2)
where e is the relative error in radius (or circumference). For e <<1, you can neglect e^2. 2e becomes the area realtive error. C is a constant. The same rule can be applied to any shape or figure. The relative error in a volume dimension is 3e.
To estimate the maximum error in the calculated surface area, we will use linear approximation.
The circumference of a sphere is given by the formula C = 2πr, where r is the radius of the sphere. We are given that the circumference C was measured to be 74 cm, with a possible error of 0.5 cm.
Let's denote the measured circumference as C_measure and the actual circumference as C_actual, and the radius as r.
C_measure = C_actual ± 0.5 cm
We can write the linear approximation of C_actual as:
C_actual ≈ C_measure ± δC
Where δC is the error in the measured circumference and can be calculated as:
δC = C_actual - C_measure
Since C_measure = 74 cm and the possible error is 0.5 cm, we have:
δC = C_actual - C_measure
= C_actual - 74 cm
Now, let's express the actual circumference C_actual in terms of the radius r:
C_actual = 2πr
Substituting this into the equation for δC:
δC = 2πr - 74 cm
To estimate the maximum error in the surface area, we will use the derivative of the surface area equation with respect to the radius. The surface area of a sphere is given by the formula A = 4πr^2.
Taking the derivative of the surface area with respect to the radius:
dA/dr = 8πr
Now, let's calculate the maximum error in the surface area using linear approximation:
Maximum error in surface area = |(dA/dr) * (δC)|
Substituting the values we found earlier:
Maximum error in surface area = |(8πr) * (2πr - 74 cm)|
So, the maximum error in the calculated surface area is |(8πr) * (2πr - 74 cm)|.
To estimate the maximum error in the calculated surface area of the sphere, we can use linear approximation.
The surface area of a sphere can be calculated using the formula A = 4πr^2, where A is the surface area and r is the radius of the sphere.
First, let's find the measured radius of the sphere using the circumference. The formula for circumference is C = 2πr, where C is the circumference and r is the radius.
Given that the circumference is 74 cm with a possible error of 0.5 cm, we can write this as:
C = 74 cm ± 0.5 cm
Now, solve for the radius:
C = 2πr
74 cm ± 0.5 cm = 2πr
To estimate the maximum error in the calculation of the surface area, we need to find the maximum and minimum values for the radius.
For the maximum value of the radius, add the maximum error to the measured value of the circumference:
74 cm + 0.5 cm = 74.5 cm
For the minimum value of the radius, subtract the maximum error from the measured value of the circumference:
74 cm - 0.5 cm = 73.5 cm
Now we will calculate the surface areas for both the maximum and minimum values of the radius and determine the difference:
Maximum surface area:
A_max = 4π(74.5 cm)^2
Minimum surface area:
A_min = 4π(73.5 cm)^2
Now, find the difference between the maximum surface area and the minimum surface area:
ΔA = A_max - A_min
This difference will give us the maximum error in the calculated surface area.
I will now calculate the values for you.