Question

The radius of a puddle is claimed to be 12.0 inches, correct to within 0.01 inch. Use linear approximation to estimate the resulting error, measured in square inches, in the area of the puddle.

(a) ±0.24π.
(b) ±0.024π.
(c) ±0.06π.
(d) ±0.0004166.

Answer 1

Option (a) is correct.

Explanation:

We know that area of the circle is
`\pi r^2` where r is the radius of the circle.

Let
`f(r)=\pi r^2`

Differentiate with respect to r

`f'(r)=\pi 2r^(2-1)=2\pi r`

As
`r= 12 \,inches`,

`f'(12)=2\pi (12)=24\pi`

Also, its given that
`\Delta r=0.01\,\,inches`

We know that as per linear approximation to estimate the resulting error,

`\Delta f(r)\approx f'(r)\,\Delta r`

Put
`f'(r)=24\pi\,,\,\Delta r=0.01 \,\,inches`

`\Delta f(r)\approx f'(r)\,\Delta r\\=(24\pi)(0.01)\\=0.24\pi\,\,square \,\,inches`

Therefore,

the error is
`\pm 0.24\pi`

So, option (a) is correct.