Answer :
* The volume of a cone is given by $V = \frac{1}{3} \pi r^2 h$.
* Rearrange the formula to solve for the radius: $r = \sqrt{\frac{3V}{\pi h}}$.
* Substitute the given values $V = 225$ and $h = 8.5$ into the formula and approximate $\pi$ as 3.14: $r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \approx 5.02895$.
* Round the result to the nearest hundredth: $\boxed{5.03}$ feet.
### Explanation
1. State the formula for the volume of a cone
We are given the volume and height of a conical water reservoir and need to find its radius. 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.
2. Rearrange the formula to solve for the radius
We need to rearrange the formula to solve for the radius $r$. Multiplying both sides by 3, we get:
$$3V = \pi r^2 h$$
Dividing both sides by $\pi h$, we get:
$$\frac{3V}{\pi h} = r^2$$
Taking the square root of both sides, we get:
$$r = \sqrt{\frac{3V}{\pi h}}$$
This is the formula we will use to calculate the radius.
3. Substitute the given values into the formula
Now, we substitute the given values into the formula. We have $V = 225$ cubic feet and $h = 8.5$ feet. We will approximate $\pi$ as 3.14.
$$r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}}$$
$$r = \sqrt{\frac{675}{26.69}}$$
$$r = \sqrt{25.29037}$$
$$r \approx 5.02895$$
4. Round the calculated value to the nearest hundredth
Rounding the calculated value of $r$ to the nearest hundredth of a foot, we get:
$$r \approx 5.03$$
Therefore, the radius of the water reservoir is approximately 5.03 feet.
### Examples
Understanding the volume and dimensions of conical shapes is crucial in various real-world applications. For instance, engineers use these calculations to design storage tanks, construction cones, and even the tips of certain projectiles. Knowing how to determine the radius from the volume and height helps in optimizing designs for capacity and stability.
* Rearrange the formula to solve for the radius: $r = \sqrt{\frac{3V}{\pi h}}$.
* Substitute the given values $V = 225$ and $h = 8.5$ into the formula and approximate $\pi$ as 3.14: $r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}} \approx 5.02895$.
* Round the result to the nearest hundredth: $\boxed{5.03}$ feet.
### Explanation
1. State the formula for the volume of a cone
We are given the volume and height of a conical water reservoir and need to find its radius. 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.
2. Rearrange the formula to solve for the radius
We need to rearrange the formula to solve for the radius $r$. Multiplying both sides by 3, we get:
$$3V = \pi r^2 h$$
Dividing both sides by $\pi h$, we get:
$$\frac{3V}{\pi h} = r^2$$
Taking the square root of both sides, we get:
$$r = \sqrt{\frac{3V}{\pi h}}$$
This is the formula we will use to calculate the radius.
3. Substitute the given values into the formula
Now, we substitute the given values into the formula. We have $V = 225$ cubic feet and $h = 8.5$ feet. We will approximate $\pi$ as 3.14.
$$r = \sqrt{\frac{3 \times 225}{3.14 \times 8.5}}$$
$$r = \sqrt{\frac{675}{26.69}}$$
$$r = \sqrt{25.29037}$$
$$r \approx 5.02895$$
4. Round the calculated value to the nearest hundredth
Rounding the calculated value of $r$ to the nearest hundredth of a foot, we get:
$$r \approx 5.03$$
Therefore, the radius of the water reservoir is approximately 5.03 feet.
### Examples
Understanding the volume and dimensions of conical shapes is crucial in various real-world applications. For instance, engineers use these calculations to design storage tanks, construction cones, and even the tips of certain projectiles. Knowing how to determine the radius from the volume and height helps in optimizing designs for capacity and stability.
