91Ó°ÊÓ

The face of a gate of a dam is vertical and in the shape of an isosceles trapezoid \(3 \mathrm{ft}\) wide at the top, \(4 \mathrm{ft}\) wide at the bottom, and \(3 \mathrm{ft}\) high. If the upper base is \(20 \mathrm{ft}\) below the surface of the water, find the total force due to liquid pressure on the gate.

The length of a rod is \(L \mathrm{ft}\) and the center of mass of the rod is at the point \(\frac{3}{4} L \mathrm{ft}\) from the left end. If the measure of the linear density at a point is proportional to a power of the measure of the distance of the point from the left end and the linear density at the right end is 20 slugs/ft, find the linear density at a point \(x \mathrm{ft}\) from the left end. Assume the mass is measured in slugs.

Prove that the distance from the centroid of a triangle to any side of the triangle is equal to one-third the length of the altitude to that side.

The total mass of a rod of length \(L \mathrm{ft}\) is \(M\) slugs and the measure of the linear density at a point \(x \mathrm{ft}\) from the left end is proportional to the measure of the distance of the point from the right end. Show that the linear density at a point on the rod \(x \mathrm{ft}\) from the left end is \(2 M(L-x) / L^{2}\) slugs \(/ \mathrm{ft}\).

A tank in the form of a rectangular parallelepiped 6 ft deep, 4 ft wide, and \(12 \mathrm{ft}\) long is full of oil weighing \(50 \mathrm{lb} / \mathrm{ft}^{3} .\) When one-third of the work necessary to pump the oil to the top of the tank has been done, find by how much the surface of the oil is lowered.

Find the center of mass of the lamina bounded by the parabola \(2 y^{2}=18-3 x\) and the \(y\) axis if the area density at any point \((x, y)\) is \(\sqrt{6-x}\) slugs \(/ \mathrm{ft}^{2}\)

Use the theorem of Pappus to find the volume of the torus (doughnut-shaped) generated by revolving a circle with a radius of \(r\) units about a line in its plane at a distance of \(b\) units from its center, where \(b>r\).

Use the theorem of Pappus to find the centroid of the region bounded by a semicircle and its diameter.

Use the theorem of Pappus to find the volume of a sphere with a radius of \(r\) units.

Determine \(m\) so that the region above the line \(y=m x\) and below the parabola \(y=2 x-x^{2}\) has an area of 36 square units.