# Finding the Rate of Change of a Cone's Height Using Related Rates

• Burjam
In summary, the height of a mound of grit is increasing at a rate of 2 cm per minute, and the radius of the mound is increasing at the same rate.
Burjam

## Homework Statement

Grit, which is spread on roads in winter, is stored in mounds which are the shape of a cone. As grit is added to the top of a mound at 2 cubic meters per minute, the angle between the slant side of the cone and the vertical remains 45º. How fast is the height of the mound increasing when it is half a meter high?

V=πr2/3

## The Attempt at a Solution

So I need to solve for dh/dt. I know dV/dt=2 and I know the height, but not the radius. So I draw a right triangle. Since the angle is 45º, r=h. So r=0.5. Now time to take d/dt of each side.

dV/dt=d/dt[πr2/3]
2=1/3πr2*dh/dt

I treated r as a constant and h as a function of time here. I applied the product rule and the chain rule.

dh/dt=6/πr2

Substitue 0.5 for r and I get

dh/dt=24/π

The correct answer is 8/π. Where did I go wrong?

You can't treat r as a constant. As grit is added to the cone, both h and r are changing w.r.t. time.

If r stays constant and h increases, then the angle of repose of the grit (45 degrees) will increase, which it cannot do. The angle stays constant.

Here's the derivative I get now:

2=1/3πr2*dh/dt + 2/3πrh*dr/dt

This is great and all, but what am I supposed to sub in for dr/dt to solve the equation? I have two unknowns now.

Hi Burjam!
Burjam said:
I have two unknowns now.

No, you have only one unknown (h).

r is not an indpendent unknown, it is a function of h.

You know the angle of repose of the grit (45 degrees). For each cm the height increases, how many cm does the radius increase? (Hint: draw a diagram.)

The radius will increase at the same rate as the height because the angle is 45º and constant. I actually drew a diagram initially to get r so this is partially from that. So dr/dt=dh/dt. From here I can solve it:

2=1/3πr2*dh/dt + 2/3πrh*dh/dt
2=dh/dt(1/3πr2 + 2/3πrh)
dh/dt=9/(πr2 + πrh)

When I plug in 0.5 for r and h I get:

dh/dt=18/π

Still not the right answer. What's wrong?

Last edited:
Burjam said:
The radius will increase at the same rate as the height because the angle is 45º and constant. I actually drew a diagram initially to get r so this is partially from that. So dr/dt=dh/dt. From here I can solve it:

2=1/3πr2*dh/dt + 2/3πrh*dh/dt
2=dh/dt(1/3πr2 + 2/3πrh)
dh/dt=9/(πr2 + πrh)

When I plug in 0.5 for r and h I get:

dh/dt=18/π

Still not the right answer. What's wrong?

It seems to me that there's an algebra issue going between the two bolded lines. I found that using ##r = h## made it fairly simple, personally.

## 1. What is a cone related rates problem?

A cone related rates problem is a type of mathematical problem that involves finding the rate of change of one variable with respect to another variable, using the relationship between the two variables in a cone shape.

## 2. How do you set up a cone related rates problem?

To set up a cone related rates problem, you need to identify the variables involved and their relationships in the cone. Then, you can use the chain rule to differentiate the equation and solve for the unknown rate of change.

## 3. What are the key concepts to understand in solving a cone related rates problem?

The key concepts to understand in solving a cone related rates problem are the chain rule, the relationship between the variables in a cone (such as the radius, height, and volume), and how to use implicit differentiation to solve for the unknown rate of change.

## 4. What are some real-life applications of cone related rates problems?

Cone related rates problems have many real-life applications, such as calculating the rate at which water is being poured into a cone-shaped tank or the rate at which the height of a tree is changing as it grows in a conical shape.

## 5. What are some tips for solving cone related rates problems?

Some tips for solving cone related rates problems include clearly defining the variables, drawing a diagram to visualize the problem, and carefully applying the chain rule and implicit differentiation. It can also be helpful to check your answer by plugging it back into the original equation.

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