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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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d730b9d-41b9-44d3-ac16-130c51dd79cf%2F877f6a7f-40f7-4f31-bf92-33e67a786b25%2F57cdfig_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### 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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