91Ó°ÊÓ

The equation $$\frac{1}{s_{1}}+\frac{1}{s_{2}}=\frac{1}{f}$$ relates the distance \(s_{1}\) of an object from a thin lens of focal length \(f\) to the distance \(s_{2}\) of the image from the lens. If an object is moving away from a lens of focal length \(15 \mathrm{cm}\) at the rate of \(5 \mathrm{cm} / \mathrm{min}\), how fast is its image moving toward the lens when the object is \(40 \mathrm{cm}\) from the lens?

Find two numbers whose sum is 60 and whose product is a maximum.

A piece of wire \(50 \mathrm{cm}\) long is to be cut in two pieces, with one part bent into a circle and the other into a square. Where should the wire be cut so that the sum of the two enclosed areas is minimal?

A circular oil slick is spreading from an offshore well. Its radius increases at the rate of \(5 \mathrm{km} /\) day. At what rate is its area increasing when its radius is \(30 \mathrm{km} ?\)

Exercises \(11-16\) : Solve by using differentials. A wooden corner brace is made in the shape of a right triangle. The legs of the triangle are measured to be \(5 \mathrm{cm} \pm 0.02 \mathrm{cm}\) and \(12 \mathrm{cm} \pm 0.2\) \(\mathrm{cm} .\) Estimate the maximum error in computing the length of the hypotenuse.

A rectangular poster is to contain \(900 \mathrm{cm}^{2}\) of printed matter with margins of \(10 \mathrm{cm}\) each at top and bottom and \(5 \mathrm{cm}\) at each side. Find the overall dimensions to make the poster in order to minimize the material needed.

A rectangular block of stone is measured to have dimensions \(0.5 \mathrm{m}, 0.8\) \(\mathrm{m},\) and \(1.2 \mathrm{m},\) with a maximum error in any dimension of \(0.03 \mathrm{m} .\) Use differentials to estimate the maximum error in computing the volume.