Derivatives, rates of change (cone)

You forgot the chain rule.In summary, the problem involves finding the rate at which the height of the pile of gravel is increasing when the pile is 10 ft high. Using the formula for the volume of a cone, the problem can be solved by taking the derivative of the volume equation with respect to time. However, in the attempt at a solution, there was a typo in the equation for the radius, which was corrected. The correct answer found by differentiating the volume equation with the chain rule is 6/5π ft/min.
  • #1
physics604
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1. Gravel is being dumped from a conveyor belt at a rate of 30 ft3/min, and its coarseness is such that it forms a pile in the shape of a 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?

Homework Equations


$$V=\frac{\pi}{3}r^2h$$

The Attempt at a Solution



Diameter = height, so $$\frac{h}{2}=r$$
$$V=\frac{\pi}{3}\frac{h^2}{4}h = \frac{\pi}{12}h^3$$
$$\frac{dV}{dt}=\frac{\pi}{12}(3×h)\frac{dh}{dt}$$
$$30=\frac{\pi}{12}(3×10)\frac{dh}{dt}$$
$$\frac{dh}{dt}=\frac{12}{\pi}$$

The textbook's answer is $$\frac{6}{5\pi}$$ What did I do wrong?
 
Last edited:
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  • #2
You incorrectly assumed that the diameter of the pile = half of the radius.
 
  • #3
Sorry that was just a typo. The work should still follow r=h/2.
 
  • #4
Check your differentiation.
 

FAQ: Derivatives, rates of change (cone)

What are derivatives?

Derivatives are mathematical tools used to calculate the instantaneous rate of change of a function with respect to its independent variable. In simpler terms, they help us understand how a function is changing at a specific point.

How are derivatives useful?

Derivatives are used in many fields of science, including physics, engineering, economics, and biology. They are especially useful in calculating motion, optimization, and growth rates of various systems.

What is the process of finding derivatives?

To find the derivative of a function, we use the concept of limits and the derivative rules such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of complicated functions.

What is the relationship between derivatives and rates of change?

Derivatives and rates of change are closely related. In fact, the derivative of a function at a specific point is the instantaneous rate of change of that function at that point. This means that the derivative can tell us how a function is changing at a specific moment.

How are derivatives used to calculate the slope of a tangent line?

The slope of a tangent line is the derivative of a function at a specific point. By finding the derivative of a function at that point, we can determine the slope of the tangent line, which is a line that touches the curve of the function at that point.

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