Question

The radius of a circular disk is given as 25 cm with a maximal error in measurement of 0.3 cm. Use differentials to estimate the following. (a) The maximum error in the calculated area of the disk. (b) The relative maximum error. (c) The percentage error in that case.

Answer 1

Answer with Step-by-step explanation:

We are given  that

Radius of circular disk=r=25 cm

Maximal error in measurement of radius =$\Delta r=$0.3

We know that

Surface area of circle =A=$\pi r^2$

Differentiate w.r.t r

a.Maximum error=$dA=2\pi rdr$

Substitute the values then we get

$dA=2\times 3.14\times 25\times 0.3=47.1 cm^2$

Hence, the maximum error in area of disk=$47.1 cm^2$

b.

Relative error in A: $\frac{\Delta A}{A}=\frac{2\pi r\Delta r}{\pi r^2}$

Substitute the value in the formula

Then ,we get

Relative maximal error in area of disk=$\frac{2\cdot 25}{(25)^2}(0.3)=0.024$

Hence, the relative maximum error in area  of disk=0.024

c.Relative error  percentage in area of disk=$\frac{\Delta A}{A}\times 100$

Substitute the values then we get

Relative error  percentage in area of disk=$0.024\times 100=2.4$%

Hence, the percentage error in area of disk=2.4%