In this page rate of change question9 we are going to see solution of some practice question of the worksheet.

**Question 9:**

Gravel is being dumped from a conveyor belt at a rate of 30
ft³/min and its coarsened such that it forms a pile in the shape of cone
whose base diameter and height are always equal. How fast is the height
of the pile increasing when the pile is 10 ft high?

**Solution:**

Let "V" be the volume of the cone at a time "t"**.** Let r and h are the radius and height of the cone respectively.

Here we have a information that the base diameter and height are always equal. Form this we come to know that 2 r = h

volume of the gravel is being changed at the rate of 30 ft³/min

rate of change of volume (dV/dt) = 30

now we have to find how fast is the height of the pile increasing when the pile is 10 ft high

h = 10

Volume of cone (V) = (1/3) π r² h

V = (1/3) π (h/2)² h

V = (1/3) π (h³/4)

dV/dt = (1/3) π (3 h²/4) (dh/dt)

substitute h = 10 and dV/dt = 30

30 = (1/3) π [3 (10)²/4] (dh/dt)

(30 x 3 x 4)/300 π = (dh/dt)

6/5π = dh/dt

dh/dt = (6/5 π) ft/min

Therefore the height is increasing at the rate of (6/5 π) ft/min.

- Back to worksheet
- First Principles
- Implicit Function
- Parametric Function
- Substitution Method
- logarithmic function
- Product Rule
- Chain Rule
- Quotient Rule
- Rolle's theorem
- Lagrange's theorem
- Finding increasing or decreasing interval
- Increasing function
- Decreasing function
- Monotonic function
- Maximum and minimum
- Examples of maximum and minimum