Answer :
- Substitute the given volume $V = 147 \pi$ and radius $r = 7$ into the cone volume formula $V = \frac{1}{3} \pi r^2 h$.
- This gives $147 \pi = \frac{1}{3} \pi (7^2) h$.
- Compare the resulting equation with the provided options.
- The correct expression is $\boxed{147 \pi = \frac{1}{3} \pi (7^2)(h)}$.
### Explanation
1. Problem Analysis and Setup
We are given the volume of a cone, $V = 147
\pi$ cubic centimeters, and its radius, $r = 7$ cm. We need to find the expression that can be used to find the height, $h$, of the cone. The formula for the volume of a cone is given by:
$$V = \frac{1}{3} \pi r^2 h$$
We will substitute the given values into this formula and then compare the result with the given options to find the correct expression.
2. Substitution of Values
Substitute the given values, $V = 147 \pi$ and $r = 7$, into the volume formula:
$$147 \pi = \frac{1}{3} \pi (7^2) h$$
$$147 \pi = \frac{1}{3} \pi (49) h$$
3. Comparison with Options
Now, let's compare this equation with the given options:
Option 1: $147 \pi = \frac{1}{3}(7)(h)^2$
This is incorrect because it has $h^2$ and lacks a $\pi$ term and $7^2 = 49$.
Option 2: $147 \pi = \frac{1}{3} \pi (7^2)(h)$
This is the same as $147 \pi = \frac{1}{3} \pi (49) h$, which matches our equation.
Option 3: $147 \pi = \frac{1}{3} \pi h$
This is incorrect because it's missing the $7^2 = 49$ term.
Option 4: $147 \pi = \frac{1}{3} \pi (7)(h)$
This is incorrect because it has 7 instead of $7^2 = 49$.
4. Final Answer
The correct expression is:
$$147 \pi = \frac{1}{3} \pi (7^2) h$$
### Examples
Cones are common shapes in everyday life, from ice cream cones to traffic cones. Understanding how to calculate the volume of a cone can be useful in various practical situations. For example, if you're filling ice cream cones at a shop, you can use this formula to ensure each cone has the right amount of ice cream. Similarly, engineers use this formula to calculate the amount of material needed to construct conical structures.
- This gives $147 \pi = \frac{1}{3} \pi (7^2) h$.
- Compare the resulting equation with the provided options.
- The correct expression is $\boxed{147 \pi = \frac{1}{3} \pi (7^2)(h)}$.
### Explanation
1. Problem Analysis and Setup
We are given the volume of a cone, $V = 147
\pi$ cubic centimeters, and its radius, $r = 7$ cm. We need to find the expression that can be used to find the height, $h$, of the cone. The formula for the volume of a cone is given by:
$$V = \frac{1}{3} \pi r^2 h$$
We will substitute the given values into this formula and then compare the result with the given options to find the correct expression.
2. Substitution of Values
Substitute the given values, $V = 147 \pi$ and $r = 7$, into the volume formula:
$$147 \pi = \frac{1}{3} \pi (7^2) h$$
$$147 \pi = \frac{1}{3} \pi (49) h$$
3. Comparison with Options
Now, let's compare this equation with the given options:
Option 1: $147 \pi = \frac{1}{3}(7)(h)^2$
This is incorrect because it has $h^2$ and lacks a $\pi$ term and $7^2 = 49$.
Option 2: $147 \pi = \frac{1}{3} \pi (7^2)(h)$
This is the same as $147 \pi = \frac{1}{3} \pi (49) h$, which matches our equation.
Option 3: $147 \pi = \frac{1}{3} \pi h$
This is incorrect because it's missing the $7^2 = 49$ term.
Option 4: $147 \pi = \frac{1}{3} \pi (7)(h)$
This is incorrect because it has 7 instead of $7^2 = 49$.
4. Final Answer
The correct expression is:
$$147 \pi = \frac{1}{3} \pi (7^2) h$$
### Examples
Cones are common shapes in everyday life, from ice cream cones to traffic cones. Understanding how to calculate the volume of a cone can be useful in various practical situations. For example, if you're filling ice cream cones at a shop, you can use this formula to ensure each cone has the right amount of ice cream. Similarly, engineers use this formula to calculate the amount of material needed to construct conical structures.
