Chapter 24

A water tank is the shape of a rectangular prism having length \(2 \mathrm{~m}\), breadth \(75 \mathrm{~cm}\) and height \(50 \mathrm{~cm}\). Determine the capacity of the tank in (a) \(\mathrm{m}^{3}\) (b) \(\mathrm{cm}^{3}\) (c) litres.

Find the volume and total surface area of a cylinder of length \(15 \mathrm{~cm}\) and diameter \(8 \mathrm{~cm}\).

Determine the volume and total surface area of a cone of radius \(5 \mathrm{~cm}\) and perpendicular height \(12 \mathrm{~cm}\).

Find the volume and surface area of a sphere of diameter \(8 \mathrm{~cm}\).

A pyramid has a rectangular base \(3.60 \mathrm{~cm}\) by \(5.40 \mathrm{~cm}\). Determine the volume and total surface area of the pyramid if each of its sloping edges is \(15.0 \mathrm{~cm}\).

Calculate the volume and total surface area of a hemisphere of diameter \(5.0 \mathrm{~cm}\).

A rectangular piece of metal having dimensions \(4 \mathrm{~cm}\) by \(3 \mathrm{~cm}\) by \(12 \mathrm{~cm}\) is melted down and recast into a pyramid having a rectangular base measuring \(2.5 \mathrm{~cm}\) by \(5 \mathrm{~cm}\). Calculate the perpendicular height of the pyramid.

A rivet consists of a cylindrical head, of diameter \(1 \mathrm{~cm}\) and depth \(2 \mathrm{~mm}\), and a shaft of diameter \(2 \mathrm{~mm}\) and length \(1.5 \mathrm{~cm}\). Determine the volume of metal in 2000 such rivets.

A solid metal cylinder of radius \(6 \mathrm{~cm}\) and height \(15 \mathrm{~cm}\) is melted down and recast into a shape comprising a hemisphere surmounted by a cone. Assuming that \(8 \%\) of the metal is wasted in the process, determine the height of the conical portion, if its diameter is to be \(12 \mathrm{~cm}\).

A block of copper having a mass of \(50 \mathrm{~kg}\) is drawn out to make \(500 \mathrm{~m}\) of wire of uniform cross-section. Given that the density of copper is \(8.91 \mathrm{~g} / \mathrm{cm}^{3}\), calculate (a) the volume of copper, (b) the cross-sectional area of the wire, and (c) the diameter of the cross-section of the wire.

A boiler consists of a cylindrical section of length \(8 \mathrm{~m}\) and diameter \(6 \mathrm{~m}\), on one end of which is surmounted a hemispherical section of diameter \(6 \mathrm{~m}\), and on the other end a conical section of height \(4 \mathrm{~m}\) and base diameter \(6 \mathrm{~m}\). Calculate the volume of the boiler and the total surface area.

Determine the volume of a frustum of a cone if the diameter of the ends are \(6.0 \mathrm{~cm}\) and \(4.0 \mathrm{~cm}\) and its perpendicular height is \(3.6 \mathrm{~cm}\).

A storage hopper is in the shape of a frustum of a pyramid. Determine its volume if the ends of the frustum are squares of sides \(8.0 \mathrm{~m}\) and \(4.6 \mathrm{~m}\), respectively, and the perpendicular height between its ends is \(3.6 \mathrm{~m}\).

A lampshade is in the shape of a frustum, of a cone. The vertical height of the shade is \(25.0 \mathrm{~cm}\) and the diameters of the ends are \(20.0 \mathrm{~cm}\) and \(10.0 \mathrm{~cm}\), respectively. Determine the area of the material needed to form the lampshade, correct to 3 significant figures.

A car has a mass of \(1000 \mathrm{~kg}\). A model of the car is made to a scale of 1 to 50 . Determine the mass of the model if the car and its model are made of the same material.