91Ó°ÊÓ

A radioactive element decays exponentially in proportion to its mass. One half of its original amount remains after \(5,750\) years. If \(10,000\) grams of the element are present initially, how much will be left after \(1,000\) years?

The minute hand of a clock is 6 inches long. Starting from noon, how fast is the area of the sector swept out by the minute hand increasing in in \(^{2} / \min\) at any instant?

A container with a square base, vertical sides, and an open top is to be made from 1000 \(\mathrm{ft}^{3}\) of material. Find the dimensions of the container with the greatest volume.

Find the volume of the solid whose base is the region between the semi-circle \(y=\sqrt{16-x^{2}}\) and the \(x\) -axis and whose cross-sections perpendicular to the \(x\) -axis are squares with a side on the base.

Find the volume of the solid whose base is the region between \(y=x^{2}\) and \(y=4\) and whose perpendicular cross-sections are isosceles right triangles with the hypotenuse on the base.

Boats A and B leave the same place at the same time. Boat A heads due north at 12 km/hr. Boat B heads due east at 18 km/hr. After 2.5 hours, how fast is the distance between the boats increasing (in km/hr)? (A) 21.63 (B) 31.20 (C) 75.00 (D) 9.84

The radius of a sphere is increasing at a rate proportional to itself. If the radius is 4 initially, and the radius is 10 after two seconds, what will the radius be after three seconds? (A) 62.50 (B) 15.81 (C) 16.00 (D) 25.00