Related Rates Problem

[playfulpanda]

No idea where to start.. please help

[FONT=&quot]John creates dual purpose birthday hats. The hats are cone-shaped and can be taken off and flipped over to hold fruit punch. His friends decided to try out his invention over the summer. It was hot out, and the fruit punch began to evaporate at a constant rate. The hats are 10 in tall and have a 10 in diameter. The fruit punch is evaporating at a rate of -3/10 in/hr.[/FONT]
[FONT=&quot]a.) Find the volume of the fruit punch when the height of the fruit punch is 5 inches.[/FONT]
[FONT=&quot]b.) Find the rate of change of the volume of fruit punch in the hat, with respect to time when the height of the fruit punch is 5 in.[/FONT]
[FONT=&quot]c.) Show that the rate of change of the volume of fruit punch in the hat due to evaporation is directly proportional to exposed surface area of the fruit punch.[/FONT]

 

[Dr.Peterson]

No idea where to start.. please help

John creates dual purpose birthday hats. The hats are cone-shaped and can be taken off and flipped over to hold fruit punch. His friends decided to try out his invention over the summer. It was hot out, and the fruit punch began to evaporate at a constant rate. The hats are 10 in tall and have a 10 in diameter. The fruit punch is evaporating at a rate of -3/10 in/hr.
a.) Find the volume of the fruit punch when the height of the fruit punch is 5 inches.
b.) Find the rate of change of the volume of fruit punch in the hat, with respect to time when the height of the fruit punch is 5 in.
c.) Show that the rate of change of the volume of fruit punch in the hat due to evaporation is directly proportional to exposed surface area of the fruit punch.

Let's start with part (a). What is the formula for the volume of a cone? What is the volume of the full cone? Now, if you look at a side view, you'll see similar triangles formed by the whole cone, and the punch within it. Can you write a formula for the radius of the surface of the punch as a function of depth, and use that to write a formula for the volume of punch as a function of depth?

Please show any work you can do, because that will serve as a starting point for helping you. If all we know is that you can do nothing, we'll have to start at 1+1!

 

[Steven G]

No idea where to start.. please help

John creates dual purpose birthday hats. The hats are cone-shaped and can be taken off and flipped over to hold fruit punch. His friends decided to try out his invention over the summer. It was hot out, and the fruit punch began to evaporate at a constant rate. The hats are 10 in tall and have a 10 in diameter. The fruit punch is evaporating at a rate of -3/10 in/hr.
a.) Find the volume of the fruit punch when the height of the fruit punch is 5 inches.
b.) Find the rate of change of the volume of fruit punch in the hat, with respect to time when the height of the fruit punch is 5 in.
c.) Show that the rate of change of the volume of fruit punch in the hat due to evaporation is directly proportional to exposed surface area of the fruit punch.

1) The hats are cone-shaped--maybe you need a formula for the volume or maybe the surface area of a cone (read on and you'll see which one to use!)
2)The hats are 10 in tall and have a 10 in diameter. So the height is 10 and the radius is 5 for the cone. Using similar triangles for a height h we get h/r = 10/10 or h=r. So we can get rid of h or r if needed.
3)The fruit punch is evaporating at a rate of -3/10 in/hr. Inch per hr, in this case is the change in height. So dh/dt = (-3/10)in/hr
a.) Find the volume of the fruit punch when the height of the fruit punch is 5 inches. No calculus needed. Just use the formula for the volume of a cone.
b.) Find the rate of change of the volume of fruit punch in the hat, with respect to time (which is dV/dt) when the height of the fruit punch is 5 in. So compute dV/dt from V

 

[Harmless Drudge]

No idea where to start.. please help

[FONT=&quot]John creates dual purpose birthday hats. The hats are cone-shaped and can be taken off and flipped over to hold fruit punch. His friends decided to try out his invention over the summer. It was hot out, and the fruit punch began to evaporate at a constant rate. The hats are 10 in tall and have a 10 in diameter. The fruit punch is evaporating at a rate of -3/10 in/hr.[/FONT]
[FONT=&quot]a.) Find the volume of the fruit punch when the height of the fruit punch is 5 inches.[/FONT]
[FONT=&quot]b.) Find the rate of change of the volume of fruit punch in the hat, with respect to time when the height of the fruit punch is 5 in.[/FONT]
[FONT=&quot]c.) Show that the rate of change of the volume of fruit punch in the hat due to evaporation is directly proportional to exposed surface area of the fruit punch.[/FONT]

The key to all basic related rates problems is to remember that derivatives are simply rates of change. Identify what things are changing, then find the basic formula of those things in general. Then find the concept by which they change.

In this case, the question prompts you along the way. Start with the formula for the cone, and look at the components. What does it change according to? What are you measuring the change of one part by? Does it change according to another feature? Do both change according to time? Whatever the change is measured by is what you differentiate in relation to.