Differentiating related functions: Sand falls from a conveyor belt and forms a conical pile....
- Thread starter Jkacson
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I would begin with the volume of a cone where the height and radius are the same:
[MATH]V=\frac{1}{3}\pi h^3[/MATH]
Differentiate with respect to time \(t\):
[MATH]\d{V}{t}=\pi h^2\d{h}{t}[/MATH]
Plug in the given data:
[MATH]\d{V}{t}=\pi (12\text{ cm})^2\left(2\,\frac{\text{cm}}{\text{s}}\right)=288\pi\frac{\text{cm}^3}{\text{s}}[/MATH]
Do you see where you went wrong?
[MATH]V=\frac{1}{3}\pi h^3[/MATH]
Differentiate with respect to time \(t\):
[MATH]\d{V}{t}=\pi h^2\d{h}{t}[/MATH]
Plug in the given data:
[MATH]\d{V}{t}=\pi (12\text{ cm})^2\left(2\,\frac{\text{cm}}{\text{s}}\right)=288\pi\frac{\text{cm}^3}{\text{s}}[/MATH]
Do you see where you went wrong?
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The rate at which the sand falls off the conveyor belt is equivalent to the rate at which the volume of the pile changes, because the sand falls from the belt onto the pile. Because we are given the height of the pile and the rate at which the height is changing at a specific point in time, then expressing the volume of the pile solely as a function of the height simplifies things, since we know the relationship between the height and radius of the pile.
Steven G
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V stands for Volume yet you thought that dV/dt is the change in height, h.
One way to catch this mistake is that Volume is ALWAYS in cubic units and length (like radius and height) are ALWAYS in linear units,
Since you were told that something changed at the rate of 2 cm/sec, and cm is linear it can NOT be the volume that is changing.
One way to catch this mistake is that Volume is ALWAYS in cubic units and length (like radius and height) are ALWAYS in linear units,
Since you were told that something changed at the rate of 2 cm/sec, and cm is linear it can NOT be the volume that is changing.