Answer :
The problem involves finding the correct expression for the volume of a cone given its radius and volume.
* The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.
* Substitute the given values $V = 147 \pi$ and $r = 7$ into the formula: $147 \pi = \frac{1}{3} \pi (7^2) h$.
* Compare the resulting equation with the given options.
* The correct expression is $\boxed{147 \pi = \frac{1}{3} \pi (7^2)(h)}$.
### Explanation
1. State the formula for the volume of a cone and given values.
The volume of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$, where $V$ is the volume, $r$ is the radius, and $h$ is the height. We are given that the volume $V = 147 \pi$ cubic centimeters and the radius $r = 7$ cm. We need to find the expression that can be used to find $h$.
2. Substitute the given values into the formula.
Substitute the given values into the formula: $147 \pi = \frac{1}{3} \pi (7^2) h$.
3. Compare the derived equation with the given options.
Now, let's compare this equation with the given options:
Option 1: $147 \pi = \frac{1}{3}(7)(t)^2$ - This is incorrect because it doesn't have the correct form of the volume formula and uses $t$ instead of $h$.
Option 2: $147 \pi = \frac{1}{3} \pi (7^2)(h)$ - This matches the equation we derived, so it is the correct expression.
Option 3: $147 \pi = \frac{1}{3} kh$ - This is incorrect because it doesn't have the correct form of the volume formula and uses $k$ instead of $\pi r^2$.
Option 4: $147 \pi = \frac{1}{3} \pi (7)(h)$ - This is incorrect because it has $7$ instead of $7^2$ in the formula.
4. Identify the correct expression.
Therefore, the correct expression to find the height $h$ of the cone is $147 \pi = \frac{1}{3} \pi (7^2)(h)$.
### Examples
Understanding the volume of cones is useful in various real-world applications, such as designing ice cream cones, determining the amount of material needed to construct conical roofs, or calculating the capacity of conical containers used in manufacturing and engineering. By knowing the radius and height, you can easily find the volume, which helps in planning and resource allocation.
* The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2 h$.
* Substitute the given values $V = 147 \pi$ and $r = 7$ into the formula: $147 \pi = \frac{1}{3} \pi (7^2) h$.
* Compare the resulting equation with the given options.
* The correct expression is $\boxed{147 \pi = \frac{1}{3} \pi (7^2)(h)}$.
### Explanation
1. State the formula for the volume of a cone and given values.
The volume of a cone is given by the formula $V = \frac{1}{3} \pi r^2 h$, where $V$ is the volume, $r$ is the radius, and $h$ is the height. We are given that the volume $V = 147 \pi$ cubic centimeters and the radius $r = 7$ cm. We need to find the expression that can be used to find $h$.
2. Substitute the given values into the formula.
Substitute the given values into the formula: $147 \pi = \frac{1}{3} \pi (7^2) h$.
3. Compare the derived equation with the given options.
Now, let's compare this equation with the given options:
Option 1: $147 \pi = \frac{1}{3}(7)(t)^2$ - This is incorrect because it doesn't have the correct form of the volume formula and uses $t$ instead of $h$.
Option 2: $147 \pi = \frac{1}{3} \pi (7^2)(h)$ - This matches the equation we derived, so it is the correct expression.
Option 3: $147 \pi = \frac{1}{3} kh$ - This is incorrect because it doesn't have the correct form of the volume formula and uses $k$ instead of $\pi r^2$.
Option 4: $147 \pi = \frac{1}{3} \pi (7)(h)$ - This is incorrect because it has $7$ instead of $7^2$ in the formula.
4. Identify the correct expression.
Therefore, the correct expression to find the height $h$ of the cone is $147 \pi = \frac{1}{3} \pi (7^2)(h)$.
### Examples
Understanding the volume of cones is useful in various real-world applications, such as designing ice cream cones, determining the amount of material needed to construct conical roofs, or calculating the capacity of conical containers used in manufacturing and engineering. By knowing the radius and height, you can easily find the volume, which helps in planning and resource allocation.
