A shallow vessel for a chemical process is required in the form of an inverted cone, where the height h (in metres); is related to the radius 7 (in metres) by the formula, h = 1-r. You have been assigned the task of finding the dimensions of r and h which will maximise the capacity (V) of the vessel. (a) Show that V = πr² – ½ær³. (b) Determine the two values of r associated with the stationary points of V. (c) Why can we reject the lesser value for r without further consideration? (d) Show that the other value of r gives the maximum capacity of the vessel. (e) What should the value of h be to give the maximum capacity?

Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places unless otherwise specified. A solid brass casting in the shape of a right circular cone has a base diameter of 4.36 inches and a height of 3.94 inches. Find the weight of the casting. Brass weighs 0.302 pound per cubic inch.

Find the intensity of light at a depth of 12 meter if I0=14 and k=0.7. Round to two decimals.

A soda can has a volume of 25 cubic inches. Let x denote its radius and h its height, both in inches. a. Using the fact that the volume of the can is 25 cubic inches, express h in terms of x. b. Express the total surface area S of the can in terms of x.

Tsunami Waves and BreakwatersThis is a continuation of Exercise 16. Breakwaters affect wave height by reducing energy. See Figure 5.30. If a tsunami wave of height H in a channel of width W encounters a breakwater that narrows the channel to a width w, then the height h of the wave beyond the breakwater is given by h=HR0.5, where R is the width ratio R=w/W. a. Suppose a wave of height 8 feet in a channel of width 5000feet encounters a breakwater that narrows the channel to 3000feet. What is the height of the wave beyond the breakwater? b. If a channel width is cut in half by a breakwater, what is the effect on wave height? 16. Height of Tsunami WavesWhen waves generated by tsunamis approach shore, the height of the waves generally increases. Understanding the factors that contribute to this increase can aid in controlling potential damage to areas at risk. Greens law tells how water depth affects the height of a tsunami wave. If a tsunami wave has height H at an ocean depth D, and the wave travels to a location with water depth d, then the new height h of the wave is given by h=HR0.25, where R is the water depth ratio given by R=D/d. a. Calculate the height of a tsunami wave in water 25feet deep if its height is 3feet at its point of origin in water 15,000feet deep. b. If water depth decreases by half, the depth ratio R is doubled. How is the height of the tsunami wave affected?

Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places unless otherwise specified. A vessel in the shape of a right circular cone has a capacity of 0.690 liter. The base diameter is 12.30 centimeters. What is the height of the vessel? One liter contains 1000 cubic centimeters.

Find length x. All dimensions are in millimeters