Question

Please help, I've attempted this question a million times and had someone explain the solution process to me but I still can't get it. Thank you!!

Answer 1

The solid, a pyramid with a square base (side length 14) and height 7, is sliced into prisms. Integrating the area of each slice gives V = 196h. Substituting h = 7, the volume is 1372 cubic units.

Let's break down the calculation step by step:

1. Identify the solid and draw a diagram:

The solid is a pyramid with a square base and a right triangle as its lateral face. The base is a circle with radius a = 7.

2. Determine the dimensions of the base:

The length of the side of the square base is 2a = 14.

3. Define the height of the solid:

Let the height of the solid be h. The height of the right triangle (lateral face) is equal to the radius of the circle, a = 7.

4. Divide the solid into slices:

Divide the solid into thin slices perpendicular to the base. Each slice is a prism with a square base and a right triangle as its lateral face.

5. Express the dimensions of each slice:

Let the thickness of each slice be dx. The area of the square base is

`\((2a)^2 = 196\)`, and the area of the right triangle is
`\((1)/(2)(2a)(a) = 49\)`.

6. Calculate the volume of each slice:

The volume of each slice is the product of the area of the base and the thickness: 196 , dx.

7. Set up the integral for total volume:

To find the total volume of the solid, integrate the volume of each slice with respect to x from 0 to h:

`\[ V = \int_0^h 196 \, dx \]`

8. Evaluate the integral:

Integrate 196 with respect to x over the interval [0, h]:

V = 196h

9. Substitute the value for h:

The height of the solid is equal to the height of the right triangle, which is a = 7. Substitute this value into the expression for V:

V =
`196 * 7` =1372

Therefore, the volume of the solid is 1372 cubic units.