The circumference of a sphere was measured to be 86.000 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum error in the calculated surface area and 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 the linear approximation. The formula for the surface area of a sphere is given by:
Surface Area = 4πr^2
where r is the radius of the sphere. Since we have the circumference of the sphere, we can use the formula for circumference to find the radius.
Circumference = 2πr
Given that the circumference is measured to be 86.000 cm with a possible error of 0.5 cm, we can write this as:
86.000 cm = 2πr ± 0.5 cm
Solving this equation, we can find the value of r. Then, we can calculate the surface area for both the maximum and minimum values of r to find the maximum error in the surface area.
Let's solve the equation to find the value of r:
86.000 cm = 2πr ± 0.5 cm
Dividing both sides by 2π, we get:
r ± 0.25/π = 43/π
So, the radius can be written as:
r1 = (43 + 0.25)/π
r2 = (43 - 0.25)/π
Calculating the surface areas for both r1 and r2, we can estimate the maximum error:
Surface Area1 = 4πr1^2
Surface Area2 = 4πr2^2
The maximum error in the surface area can be calculated as:
Maximum Error = |Surface Area1 - Surface Area2|
Next, to estimate the relative error in the calculated surface area, divide the maximum error by the actual surface area:
Relative Error = Maximum Error / Surface Area
This will give you the estimated relative error in the calculated surface area.