4914: A gasoline

Question

A gasoline tank has the shape of an inverted right circular cone with base radius 4 meters and height 5 meters. Gasoline is being pumped into the tank at the rate of 8 meters3/sec. Find the rate, in meters/sec, at which the gasoline level is rising when the gas is 4 meters deep. Give 2 decimal places for your answer. Type your answer in the space below. If your answer is a number less than 1, place a leading "0" before the decimal point (ex: 0.35).

Answer 1

Hi Joyce,

 

Picture the inverted cone with the radius, the height, and the slant height forming a right triangle. The surface of the gasoline has a radius, a height, and a slant height each smaller than the cone which also form a right triangle similar to the one in the cone. Since the triangles are similar, a proportion can be set up with x(radius of the cone) and y(height of the cone) to the corresponding sides in the smaller right triangle resulting in x/y = 4/5. It may not be apparent why this is needed at this time, but it is necessary. To begin solving the problem we know that dV/dt = 8 m³/sec and the level of the gasoline is at 4 meters.We want to find dy/dt or how fast the y-value or height of the gasoline is rising when y = 4 meters. 

since we are given that the change in the volume is dV/dt = 8, we must use the formula for the volume of a cone

 

               V = (1/3)πr²h or in terms of x and y        V = (1/3)πx²y    Since we do not know a value for x, this is where the similar triangles are needed.We can use the proportion  x/y = 4/5 and solve for x to get  x = 4y/5. Now

 

                V = (1/3)π(4y/5)² y = (16πy³/75    Differentiating with respect to  t  we have

 

                 dV/dt = (16π/75) 3y² (dy/dt)  =  (48π/75) y² (dy/dt)    Since dV/dt = 8  and y = 4, substitute these values into the equation above and get

                                                 8 = (48π/75)(4²) (dy/dt)  Solve for dy/dt

 

                                          dy/dt = 8(75/48•16π)  = 0.25 m/sec

Answer 2

0.25 m/sec.

 

Watch this video to see why: https://youtu.be/LrGlTpQWGEQ