The Measurment of the circumference of a sphere is 54 cm. find the % error in computing the surface area of?

Question

*continuation of the question: the sphere if the possible error in measurement of circumference is 0.6 cm. Give your answer in two decimal places.
The answer should be 2.22%.

Answer 1

The perimeter, P, of a sphere = 2*pi*r, where r is the radius. The surface area, S, is 4*pi*r^2. Therefore, in terms of the perimeter, the surface area is

S(P) = 4*pi*r^2 = 4*pi*(P/(2*pi))^2 = P^2/pi

Now we are given that the possible error in measurement of the circumference is 0.6cm. This means that the perimeter value is accurate to within +/- 0.3 cm (total possible error = 0.6 cm).

The percent error of the surface area can be determined by looking at the following

(S(P+0.3) - S(P-0.3))/S(P)

The numerator is the difference in surface area at the extremes of the possible perimeter measurements. The denominator is the surface area at the middle of the possible perimeter measurements. We can simplify this as

[(P + 0.3)^2/pi - (P - 0.3)^2/pi]/(P^2/pi)

= [P^2 + 0.6P + 0.09 - (P^2 - 0.6P + 0.09)]/P^2
= 1.2P/P^2
=1.2/P
=1.2/54 = 2.22%

Math Rules!

Answer 2

Circumference is directly proportional to radius. The error on the circumference (and thus, on the radius, too) is:
(0.6 cm) / (54 cm)
= 0.0111
= 1.11%

The surface area of a sphere goes as the radius squared. So, our error is compounded when we multiply the too-large radius factor twice. The error on the surface area is:
(1 + 0.0111)^2
= 1.0111^2
= 1.02234
= 2.23%

2.23% is a more accurate answer than 2.22%.

Answer 3

You know pi reliably, so the diameter is therefor:

pi*Dia = circumference

Diameter is 17.19 cm plus/minus 0.191 cm.

Area is 4*pi*rad^2, so:

Low: 907.68 cm²
Middle: 928.19 cm²
High: 948.93 cm²

(High - Low)/2 /Middle = 2.22%.