Hello my name is Sarah. I am a secondary student, actually a 12th grader in high school. I am having a hard time figuring out the formulas for the following problem from my claculus I class.
The problem: Water flows into a conical funnel at a continuous rate of one gallon per minute (One gallon = 231 Cu.In.). The height of the funnel is 5" and the diameter is 8".
The 1st formula: I need to develop a formula that will give the volume, in cubic inches, of the water in the funnel at any time t (in seconds). V = f(t).
The 2nd formula: I need to develop a formula that will give the height of the water in the funnel at any time t (in seconds). h = f(t).
Thank you so much for helping me.
The fact you need is an expression for the volume of a cone if you know its height and the radius of the top. The expression is
A diagram of the funnel with some water in it is below.
I have called the height of the water h(t) and the radius of the top of the water r(t). These are both functions of t and change as water flows into the funnel.
If you imagine a line from the center of the circle at the top of the cone to the vertex of the cone you form two triangles.
These triangles are similar so