When sawdust is poured into a pile, it takes the shape of a cone where the proportion of height to base diameter is always the same. The volume of a cone can be found using the formula , where  is the radius of the base circle, and  is the height of the cone.

A particular sawdust pile has a base diameter of 20 feet when the height is 15 feet. Find the volume of this sawdust pile when it is 35 feet high.

**Hint**: Begin by finding the new diameter by setting up a proportion between height and diameter. Then find the new radius.

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Give your answer as an expression (using pi for ), or as a value accurate to the nearest whole number

To find the volume of the sawdust pile when its height is 35 feet, let's follow the steps outlined in the hint:

### Step 1: Set up a proportion for the height and diameter.
We know the relationship between the height and base diameter remains constant. For the initial pile:
- Height \( h_1 = 15 \) feet
- Base diameter \( d_1 = 20 \) feet

This gives us the ratio:
\[
\frac{h_1}{d_1} = \frac{15}{20} = \frac{3}{4}
\]

Since the ratio stays the same, we can set up a proportion to find the new diameter \( d_2 \) when the height is \( h_2 = 35 \) feet:
\[
\frac{h_2}{d_2} = \frac{3}{4}
\]

Now, solve for \( d_2 \):
\[
d_2 = \frac{4}{3} \times h_2 = \frac{4}{3} \times 35 = 46.67 \text{ feet}
\]

### Step 2: Find the new radius.
The radius \( r_2 \) is half of the diameter:
\[
r_2 = \frac{d_2}{2} = \frac{46.67}{2} = 23.33 \text{ feet}
\]

### Step 3: Find the volume of the cone.
The formula for the volume of a cone is:
\[
V = \frac{1}{3} \pi r^2 h
\]
Substituting \( r_2 = 23.33 \) feet and \( h_2 = 35 \) feet into the formula:
\[
V = \frac{1}{3} \pi (23.33)^2 (35)
\]

First, calculate \( (23.33)^2 \):
\[
(23.33)^2 = 544.29
\]

Now, substitute this back into the volume formula:
\[
V = \frac{1}{3} \pi (544.29) (35) = \frac{1}{3} \pi (19050.15)
\]
\[
V = \pi (6350.05)
\]

Thus, the exact volume expression is:
\[
V = 6350.05\pi \text{ cubic feet}
\]

### Step 4: Approximate the volume.
Using \( \pi \approx 3.1416 \):
\[
V \approx 6350.05 \times 3.1416 = 19955.6 \text{ cubic feet}
\]
Rounding to the nearest whole number, the volume is approximately:
\[
V \approx 19956 \text{ cubic feet}
\]

### Final Answer:
The volume of the sawdust pile when it is 35 feet high is approximately \( 19956 \) cubic feet.

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Would you like more details or have any questions?

Here are five related questions for further practice:
1. What would the volume be if the height was 25 feet instead of 35 feet?
2. How would the volume change if the base diameter stayed the same, but the height doubled?
3. What is the ratio of the volume when the height is 15 feet to when it is 35 feet?
4. How would the formula change if the cone was only partially filled with sawdust?
5. If the pile was in a different material, how would density impact the overall weight?

**Tip**: Always ensure that units are consistent (feet for length, cubic feet for volume) when calculating.

When sawdust is poured into a pile, it takes the shape of a cone where the proportion of height to base diameter is always the same. The volume of a cone can be found using the formula, where r is the radius of the base circle, and h is the height of the cone.

A particular sawdust pile has a base diameter of 20 feet when the height is 15 feet. Find the volume of this sawdust pile when it is 35 feet high.

Hint: Begin by finding the new diameter by setting up a proportion between height and diameter. Then find the new radius.

Give your answer as an expression (using pi for π), or as a value accurate to the nearest whole number.

Find out how to calculate the volume of a sawdust pile shaped like a cone with a changing height. This problem demonstrates key geometric concepts like proportions and the volume formula for cones. It is suitable for students in Grades 9-11 and provides a step-by-step solution.

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