**Problem:** If a spherical tank 6 m in diameter can be filled with a liquid for $200, find the cost to fill a tank 18 m in diameter. **Solution:** To solve this problem, we need to understand that the cost to fill a spherical tank is proportional to its volume. The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. **Steps:** 1. **Calculate the volume of the 6 m diameter tank:** - Radius \( r_1 = \frac{6}{2} = 3 \) m - Volume \( V_1 = \frac{4}{3} \pi (3)^3 = 36\pi \) cubic meters 2. **Calculate the volume of the 18 m diameter tank:** - Radius \( r_2 = \frac{18}{2} = 9 \) m - Volume \( V_2 = \frac{4}{3} \pi (9)^3 = 972\pi \) cubic meters 3. **Find the proportional cost:** - Since cost is proportional to volume, use the ratio \( \frac{V_2}{V_1} = \frac{972\pi}{36\pi} = 27 \). - Therefore, the cost for the 18 m tank is \( 27 \times 200 = 5400 \). **Answer:** The cost to fill the 18 m tank is $5400.
**Problem:** If a spherical tank 6 m in diameter can be filled with a liquid for $200, find the cost to fill a tank 18 m in diameter. **Solution:** To solve this problem, we need to understand that the cost to fill a spherical tank is proportional to its volume. The volume \( V \) of a sphere is given by the formula: \[ V = \frac{4}{3} \pi r^3 \] where \( r \) is the radius of the sphere. **Steps:** 1. **Calculate the volume of the 6 m diameter tank:** - Radius \( r_1 = \frac{6}{2} = 3 \) m - Volume \( V_1 = \frac{4}{3} \pi (3)^3 = 36\pi \) cubic meters 2. **Calculate the volume of the 18 m diameter tank:** - Radius \( r_2 = \frac{18}{2} = 9 \) m - Volume \( V_2 = \frac{4}{3} \pi (9)^3 = 972\pi \) cubic meters 3. **Find the proportional cost:** - Since cost is proportional to volume, use the ratio \( \frac{V_2}{V_1} = \frac{972\pi}{36\pi} = 27 \). - Therefore, the cost for the 18 m tank is \( 27 \times 200 = 5400 \). **Answer:** The cost to fill the 18 m tank is $5400.
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
Related questions
Question
![**Problem:**
If a spherical tank 6 m in diameter can be filled with a liquid for $200, find the cost to fill a tank 18 m in diameter.
**Solution:**
To solve this problem, we need to understand that the cost to fill a spherical tank is proportional to its volume. The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.
**Steps:**
1. **Calculate the volume of the 6 m diameter tank:**
- Radius \( r_1 = \frac{6}{2} = 3 \) m
- Volume \( V_1 = \frac{4}{3} \pi (3)^3 = 36\pi \) cubic meters
2. **Calculate the volume of the 18 m diameter tank:**
- Radius \( r_2 = \frac{18}{2} = 9 \) m
- Volume \( V_2 = \frac{4}{3} \pi (9)^3 = 972\pi \) cubic meters
3. **Find the proportional cost:**
- Since cost is proportional to volume, use the ratio \( \frac{V_2}{V_1} = \frac{972\pi}{36\pi} = 27 \).
- Therefore, the cost for the 18 m tank is \( 27 \times 200 = 5400 \).
**Answer:**
The cost to fill the 18 m tank is $5400.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b94b20b-b5c6-4187-a980-d088872702e5%2F0a601beb-6b8e-45ed-b9ef-40480e1323d6%2Ftie4quj.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
If a spherical tank 6 m in diameter can be filled with a liquid for $200, find the cost to fill a tank 18 m in diameter.
**Solution:**
To solve this problem, we need to understand that the cost to fill a spherical tank is proportional to its volume. The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.
**Steps:**
1. **Calculate the volume of the 6 m diameter tank:**
- Radius \( r_1 = \frac{6}{2} = 3 \) m
- Volume \( V_1 = \frac{4}{3} \pi (3)^3 = 36\pi \) cubic meters
2. **Calculate the volume of the 18 m diameter tank:**
- Radius \( r_2 = \frac{18}{2} = 9 \) m
- Volume \( V_2 = \frac{4}{3} \pi (9)^3 = 972\pi \) cubic meters
3. **Find the proportional cost:**
- Since cost is proportional to volume, use the ratio \( \frac{V_2}{V_1} = \frac{972\pi}{36\pi} = 27 \).
- Therefore, the cost for the 18 m tank is \( 27 \times 200 = 5400 \).
**Answer:**
The cost to fill the 18 m tank is $5400.
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