The circumference of a sphere was measured to be 74.000 cm with a possible error of 0.50000 cm. Use linear approximation to estimate the maximum error in the calculated surface area?
Also Estimate the 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. The formula for the surface area of a sphere is given by:
Surface Area = 4πr²
Where r is the radius of the sphere.
We are given the circumference of the sphere, which is related to the radius by the formula:
Circumference = 2πr
To use linear approximation, we need to find the derivative of the surface area formula with respect to the radius.
d(Surface Area)/dr = 8πr
Now, we can use the percentage error formula to estimate the maximum error in the surface area:
Maximum Error = (d(Surface Area)/dr) * Maximum error in radius
The maximum error in the radius can be calculated by taking half of the possible error in the circumference because we are assuming the error evenly affects the radius. Therefore,
Maximum error in radius = 0.5 * (0.50000 cm) = 0.25000 cm
Plugging in the values, we have:
Maximum Error = (8πr) * 0.25000 cm
As for estimating the relative error in the calculated surface area, we can use the formula:
Relative Error = (Maximum Error in Surface Area) / Surface Area
To find the relative error, we need to calculate the value of the surface area. Given the circumference is 74.000 cm, we can use the equation:
Circumference = 2πr
Solving for the radius, we get:
r = Circumference / (2π) = 74.000 cm / (2π) cm ≈ 11.7790 cm
Now, we can calculate the surface area using the formula:
Surface Area = 4πr² = 4π(11.7790 cm)² ≈ 1739.06 cm²
Finally, plugging in the values, we have:
Relative Error = (Maximum Error) / (Surface Area)
During your calculations, substitute the value of π as needed.
To estimate the maximum error in the calculated surface area of the sphere, we can use linear approximation. The linear approximation formula is given by:
Δf ≈ f'(x) * Δx
In this case, the circumference of the sphere is given as 74.000 cm, with a possible error of 0.50000 cm. The surface area of a sphere is given by the formula:
A = 4πr²
To calculate the maximum error in the surface area, we need to first find the derivative of the surface area formula with respect to the radius:
dA/dr = 8πr
Now, let's calculate the maximum error in the surface area using the formula for linear approximation:
ΔA ≈ (8πr) * Δr
To find the maximum error, we need to substitute the values given. We know that the circumference of the sphere is equal to 2πr, so by rearranging the formula, we can solve for the radius:
2πr = 74.000 cm
r = 74.000 cm / (2π) ≈ 11.772 cm
Now, we can substitute this value into the linear approximation formula:
ΔA ≈ (8π * 11.772 cm) * 0.50000 cm
Simplifying this expression gives:
ΔA ≈ 235.44 cm² * 0.50000 cm
ΔA ≈ 117.72 cm²
Therefore, the maximum error in the calculated surface area of the sphere is approximately 117.72 cm².
To estimate the relative error in the calculated surface area, we need to divide the maximum error by the actual surface area of the sphere. So, we can use the formula:
Relative error = (ΔA) / A
Substituting the values we have:
Relative error ≈ (117.72 cm²) / (4π * (11.772 cm)²)
Calculating this expression gives:
Relative error ≈ 0.249
Therefore, the relative error in the calculated surface area of the sphere is approximately 0.249, or 24.9%.