We are given the following information about the right circular cone:

• Slant height ($l$) = 13 ft
• Lateral area ($A_{L}$) = $91\pi \, \text{ft}^2$

We need to find the volume of the cone. Let's go step by step:

Step 1: Find the radius ($r$) of the base

The formula for the lateral area of a cone is: $A_{L} = \pi r l$ where:

• $A_{L}$ is the lateral area
• $r$ is the radius of the base
• $l$ is the slant height

We are given $A_{L} = 91\pi$ and $l = 13$. Substituting these into the formula: $91\pi = \pi r \cdot 13$ Now, divide both sides by $\pi$: $91 = 13r$ Solve for $r$: $r = \frac{91}{13} = 7 \, \text{ft}$

Step 2: Find the height ($h$) of the cone

To find the volume, we need the height of the cone. We can use the Pythagorean theorem, since we have a right triangle formed by the radius, the slant height, and the height of the cone. The relation is: $l^2 = r^2 + h^2$ Substitute the known values $l = 13$ and $r = 7$: $13^2 = 7^2 + h^2$ $169 = 49 + h^2$ Solve for $h^2$: $h^2 = 169 - 49 = 120$ Now, take the square root of both sides: $h = \sqrt{120} \approx 10.95 \, \text{ft}$

Step 3: Find the volume ($V$) of the cone

The formula for the volume of a cone is: $V = \frac{1}{3} \pi r^2 h$ Substitute the values $r = 7$ and $h \approx 10.95$: $V = \frac{1}{3} \pi (7)^2 (10.95)$ $V = \frac{1}{3} \pi (49) (10.95)$ $V = \frac{1}{3} \pi (536.55)$ $V \approx 178.85\pi \, \text{ft}^3$ Approximating $\pi$ as 3.1416: $V \approx 561.56 \, \text{ft}^3$

Thus, the volume of the cone is approximately 561.56 cubic feet.

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Would you like more details on any of the steps, or do you have any further questions? Here are some related questions to explore:

1. How would the volume change if the slant height were increased to 15 ft?
2. What would the lateral area be if the radius were 10 ft?
3. How does the height relate to the slant height in different types of cones?
4. How would you derive the formula for the lateral area of a cone from its surface area formula?
5. How can you use the volume of a cone to calculate the amount of material needed to construct it?

Tip: For cones, knowing any two of the following — radius, slant height, lateral area, or height — can help you find the rest!