The image presents three problems. Let's focus on the first and third problems, as problem two depends on the result of problem one, and I'm choosing one. Problem 1: Find the volume of a solid cone with a base radius of 7 cm and a height of 21 cm. Problem 3: A model of a tent is shown in the figure. The base is a square with side length 4 m. The perpendicular height of a triangular face is 3 m. Find the total area of the cloth needed to cover around the straight pyramidal tent.

1. Problem Description

The image presents three problems. Let's focus on the first and third problems, as problem two depends on the result of problem one, and I'm choosing one.

Problem 1: Find the volume of a solid cone with a base radius of 7 cm and a height of 21 cm.

Problem 3: A model of a tent is shown in the figure. The base is a square with side length 4 m. The perpendicular height of a triangular face is 3 m. Find the total area of the cloth needed to cover around the straight pyramidal tent.

2. Solution Steps

Let's solve problem 1 first.

The volume $V$ of a cone is given by the formula:

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

where $r$ is the radius of the base and $h$ is the height of the cone.

Given $r = 7$ cm and $h = 21$ cm.

$V = \frac{1}{3} \pi (7^2) (21)$

$V = \frac{1}{3} \pi (49) (21)$

$V = \pi (49)(7)$

$V = 343\pi \text{ cm}^3$

Now let's solve problem

3. The tent is a square pyramid. The area of the cloth needed is the lateral surface area of the pyramid. The base is a square, and we need to find the area of the four triangular faces.

The area of one triangular face is $\frac{1}{2} \times \text{base} \times \text{height}$. In this case, the base of each triangle is the side length of the square, which is 4 m. The height is given as 3 m.

Area of one triangular face $= \frac{1}{2} \times 4 \times 3 = 6 \text{ m}^2$

Since there are four identical triangular faces, the total lateral surface area is:

$4 \times 6 = 24 \text{ m}^2$

3. Final Answer

Problem 1: The volume of the cone is $343\pi \text{ cm}^3$.

Problem 3: The total area of the cloth needed is $24 \text{ m}^2$.