The circumference of a sphere was measured to be 82 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum and relative error in the calculated surface area.
To estimate the maximum error in the calculated surface area of a sphere, we can use linear approximation.
First, let's recall the formulas for the circumference and surface area of a sphere:
Circumference of a sphere = 2 * π * r
Surface area of a sphere = 4 * π * r^2
Given that the circumference of the sphere is measured to be 82 cm with a possible error of 0.5 cm, we can express this as:
Circumference = 82 cm ± 0.5 cm
Now, let's use linear approximation to estimate the maximum error in the calculated surface area.
Step 1: Find the corresponding error in the radius.
Using the formula for circumference, we can rearrange it to solve for the radius:
Circumference = 2 * π * r
82 cm ± 0.5 cm = 2 * π * r
Simplifying this equation, we have:
41 cm ± 0.25 cm = π * r
Therefore, the radius can be expressed as:
r = (41 cm ± 0.25 cm) / π
Step 2: Estimate the maximum error in the surface area.
Next, we'll calculate the surface area with the maximum possible radius error. We already know the formula for the surface area of a sphere:
Surface area = 4 * π * r^2
By using the maximum radius error value, we can calculate the maximum surface area:
Maximum surface area = 4 * π * (r + δr)^2
where δr represents the maximum error in radius.
Substituting the expression for r from Step 1, we have:
Maximum surface area = 4 * π * [(41 cm ± 0.25 cm) / π + δr]^2
Expanding and simplifying this equation:
Maximum surface area = 4 * π * [(41 cm ± 0.25 cm) / π]^2
Finally, we can evaluate this equation to estimate the maximum error in the calculated surface area.
Step 3: Calculate the relative error.
To find the relative error, we need to divide the maximum error in surface area by the actual surface area.
Relative error = (Maximum surface area - Actual surface area) / Actual surface area
Remember, the actual surface area can be calculated using the accurate radius measurement:
Actual surface area = 4 * π * r^2
By substituting the measurement for r into the equation above, we can calculate the actual surface area.
By following these steps, you can estimate the maximum and relative error in the calculated surface area of a sphere based on the given measurements.