24. The bottom of a trough is constructed from a 4 foot by 6 foot rectangular piece of metal by bending it so that the 4 foot width forms an arc of a circle. (See the figure) If the volume of the trough is 5 cubic feet, find the angle 0 subtended by the arc. Use several iterations of Newton's method to solve the resulting equation for the volume (see the hint below) and compare your answer to the one obtained by using fsolve. HINT: Suppose that r is the radius of the circular arc subtended by the angle 6. Explain why the length of the arc is L = r0 = 4 and why the area of an end of the trough is p2 sin(8) A = (Think of the area of the end as the area of the circular sector minus the area of a triangle.) Use these equations to show the volume of the trough is: 48(0 - sin(0)) V = A2

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Cylinders

A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.

Cones

A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.