The circumference of a sphere was measured to be 88000 cm with a possible error of 050000 cm. Use linear approximation to estimate the maximum error in the calculated surface area.
Estimate the relative error in the calculated surface area.
Oops the 88000 is supposed to be 88 cm and 0.5 cm
To estimate the maximum error in the calculated surface area of a sphere, we can use linear approximation. Linear approximation is based on the assumption that small changes in a function can be approximated by the linear change in the function's tangent line.
Let's start by finding the formula for the surface area of a sphere. The surface area of a sphere is given by the formula:
A = 4πr^2
where A is the surface area and r is the radius of the sphere.
To estimate the maximum error in the surface area, we need to find the derivative of the surface area function with respect to the radius:
dA/dr = 8πr
Next, we'll calculate the maximum error in the circumference measurement. The actual circumference of the sphere is 88000 cm, and it has a possible error of 050000 cm. The maximum error is half of the possible error, so the maximum error in the circumference is equal to 050000 cm / 2 = 025000 cm.
Since the circumference of a sphere is given by the formula:
C = 2πr
We can use linear approximation to estimate the maximum error in the radius. The linear approximation formula is:
Δf ≈ f'(a)Δx
where Δf is the change in the function, f'(a) is the derivative of the function at the point a, and Δx is the change in the input.
In this case, the function is the circumference, f(r) = 2πr, and the input is the radius r. The change in the circumference Δf is the maximum error in the circumference measurement, 025000 cm.
By rearranging the formula, we have:
Δx ≈ Δf / f'(a)
where Δx is the maximum error in the radius.
Substituting the values, we get:
Δx ≈ 025000 cm / (2π(a))
To estimate the maximum error in the surface area, we need to substitute the maximum error in the radius back into the derivative formula for the surface area:
ΔA ≈ (dA/dr)(Δx)
Substituting the values, we get:
ΔA ≈ (8π(a))(025000 cm / (2π(a)))
Simplifying the expression, we have:
ΔA ≈ 4(a)(025000 cm)
Now, to estimate the relative error in the calculated surface area, we divide the maximum error ΔA by the actual surface area A:
Relative error = ΔA / A
Substituting the values, we get:
Relative error ≈ (4(a)(025000 cm)) / (4π(a^2))
Simplifying the expression, we have:
Relative error ≈ (025000 cm) / (π(a))
To have an actual estimate of the relative error, we would need to know the value of the radius a or have additional information about the problem.
Circumference, C=88 cm
C = 2πr
Area, A=πr²
Differentiating:
dA/dr = 2πr
Using linear approximation:
δA = 2πr δr
=2πr δr
=C δr
=88 cm * 0.5 cm
=44 cm²