91Ó°ÊÓ

A spring has a natural length of 10 in. , and a 30 -lb force stretches it to \(11 \frac{1}{2}\) in. Find the work done in stretching the spring from 10 in. to 12 in. Then find the work done in stretching the spring from 12 in. to 14 in.

The base of a solid is a circle with a radius of \(r\) units, and all plane sections perpendicular to a fixed diameter of the base are isosceles right triangles having the hypotenuse in the plane of the base. Find the volume of the solid.

A plate in the shape of an isosceles right triangle is submerged vertically in a tank of water, with one leg lying in the surface. The legs are each \(6 \mathrm{ft}\) long. Find the force due to liquid pressure on one side of the plate.

Find the volume of the tetrahedron having 3 mutually perpendicular faces and three mutually perpendicular edges whose lengths have measures \(a, b\), and \(c\).

The face of a dam adjacent to the water is vertical, and its shape is in the form of an isosceles triangle \(250 \mathrm{ft}\) wide across the top and \(100 \mathrm{ft}\) high in the center. If the water is \(10 \mathrm{ft}\) deep in the center, find the total force on the dam due to liquid pressure.

Two right-circular cylinders, each having a radius of \(r\) units, have axes that intersect at right angles. Find the volume of the solid common to the two cylinders.

A right-circular cylindrical tank with a depth of \(12 \mathrm{ft}\) and a radius of \(4 \mathrm{ft}\) is half full of oil weighing \(60 \mathrm{lb} / \mathrm{ft}^{3}\). Find the work done in pumping the oil to a height \(6 \mathrm{ft}\) above the tank.

An oil tank is in the shape of a right-circular cylinder \(4 \mathrm{ft}\) in diameter, and its axis is horizontal. If the tank is half full of oil weighing \(50 \mathrm{lb} / \mathrm{ft}^{3}\), find the total force on one end due to liquid pressure.

The face of the gate of a dam is in the shape of an isosceles triangle \(4 \mathrm{ft}\) wide at the top and \(3 \mathrm{ft}\) high. If the upper edge of the face of the gate is \(15 \mathrm{ft}\) below the surface of the water, find the total force due to liquid pressure on the gate.

A cable \(200 \mathrm{ft}\) long and weighing \(4 \mathrm{lb} / \mathrm{ft}\) is hanging vertically down a well. If a weight of \(100 \mathrm{lb}\) is suspended from the lower end of the cable, find the work done in pulling the cable and weight to the top of the well.