### Volume Calculation of a Cone and Required Number of Trips**Problem Statement:**A pile of earth removed from an excavation is a cone measuring 12 feet high and 30 feet across its base. How many trips will it take to haul away the earth using a dump truck with a capacity of 9 cubic yards?**Given Data:**- Height of the cone (h): 12 feet- Diameter of the cone’s base: 30 feet- Capacity of the dump truck: 9 cubic yards**Solution Steps:**1. **Calculate the radius of the base of the cone:** \[ \text{Radius (r)} = \frac{\text{Diameter}}{2} = \frac{30}{2} = 15 \text{ feet} \]2. **Calculate the volume of the cone using the formula:** \[ \text{Volume} = \frac{1}{3} \pi r^2 h \] Substitute the given values into the formula: \[ \text{Volume} = \frac{1}{3} \pi (15)^2 (12) = \frac{1}{3} \pi (225)(12) = \frac{1}{3} \pi (2700) = 900\pi \text{ cubic feet} \] If we use $\pi \approx 3.14$: \[ \text{Volume} \approx 900 \times 3.14 = 2826 \text{ cubic feet} \]3. **Convert the volume from cubic feet to cubic yards:** Since $1 \text{ cubic yard} = 27 \text{ cubic feet}$: \[ 2826 \text{ cubic feet} = \frac{2826}{27} \text{ cubic yards} \approx 104.67 \text{ cubic yards} \]4. **Calculate the number of trips required:** Each dump truck can carry 9 cubic yards. \[ \text{Number of trips} = \frac{104.67}{9} \approx 11.63 \] Since you cannot take a fraction of a trip, you need to round up to the nearest whole number. **Final Answer:** The number of trips required = 12

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Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Question

### Volume Calculation of a Cone and Required Number of Trips

**Problem Statement:**
A pile of earth removed from an excavation is a cone measuring 12 feet high and 30 feet across its base. How many trips will it take to haul away the earth using a dump truck with a capacity of 9 cubic yards?

**Given Data:**
- Height of the cone (h): 12 feet
- Diameter of the cone’s base: 30 feet
- Capacity of the dump truck: 9 cubic yards

**Solution Steps:**

1. **Calculate the radius of the base of the cone:**
\[
\text{Radius (r)} = \frac{\text{Diameter}}{2} = \frac{30}{2} = 15 \text{ feet}
\]

2. **Calculate the volume of the cone using the formula:**
\[
\text{Volume} = \frac{1}{3} \pi r^2 h
\]
Substitute the given values into the formula:
\[
\text{Volume} = \frac{1}{3} \pi (15)^2 (12) = \frac{1}{3} \pi (225)(12) = \frac{1}{3} \pi (2700) = 900\pi \text{ cubic feet}
\]

If we use $\pi \approx 3.14$:
\[
\text{Volume} \approx 900 \times 3.14 = 2826 \text{ cubic feet}
\]

3. **Convert the volume from cubic feet to cubic yards:**
Since $1 \text{ cubic yard} = 27 \text{ cubic feet}$:

\[
2826 \text{ cubic feet} = \frac{2826}{27} \text{ cubic yards} \approx 104.67 \text{ cubic yards}
\]

4. **Calculate the number of trips required:**
Each dump truck can carry 9 cubic yards.

\[
\text{Number of trips} = \frac{104.67}{9} \approx 11.63
\]

Since you cannot take a fraction of a trip, you need to round up to the nearest whole number.

**Final Answer:**
The number of trips required = 12

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