The image shows a cone with a radius of 3 inches and a slant height of 10 inches. The task is to find both the surface area and the volume of the cone using $\pi = 3.14$, and round to the nearest hundredth.

Step 1: Volume of the cone

The formula for the volume $V$ of a cone is:

$V = \frac{1}{3} \pi r^2 h$

• $r$ is the radius (3 inches),
• $h$ is the height, but it's not given directly. We can calculate it using the Pythagorean theorem because the height $h$, radius $r$, and slant height $l$ form a right triangle.

We can calculate the height $h$ as follows:

$h = \sqrt{l^2 - r^2} = \sqrt{10^2 - 3^2} = \sqrt{100 - 9} = \sqrt{91} \approx 9.53 \, \text{inches}$

Now, we can calculate the volume:

$V = \frac{1}{3} \times 3.14 \times 3^2 \times 9.53 \approx \frac{1}{3} \times 3.14 \times 9 \times 9.53 \approx 89.69 \, \text{cubic inches}$

Step 2: Surface Area of the cone

The surface area $A$ of a cone is given by the formula:

$A = \pi r (r + l)$

Where $l$ is the slant height (10 inches), and $r$ is the radius (3 inches).

$A = 3.14 \times 3 \times (3 + 10) = 3.14 \times 3 \times 13 = 122.46 \, \text{square inches}$

Final Answers:

• The volume of the cone is approximately 89.69 cubic inches.
• The surface area of the cone is approximately 122.46 square inches.

Let me know if you need more details or explanations!

Here are five related questions to further your understanding:

1. How does changing the radius affect the volume and surface area of a cone?
2. What is the relationship between the slant height and the height of a cone?
3. How would the volume change if the height of the cone were doubled?
4. Can you find the lateral surface area of the cone without the base area?
5. How does the formula for the surface area of a cone relate to that of a cylinder?

Tip: When solving for the height using the Pythagorean theorem, make sure to always double-check your slant height and radius values for accuracy.