The base of a pyramid is a rectangle 40 m long and 20 m wide. The slant height of the pyramid from the mid-point of the shorter side of the base to the apex is 29 m. What is the volume of pyramid ?

Options

A. a. `5600 m^(3) `
B. b. `400 m^(3) `
C. c. `6500 m^(3) `
D. d. `1753sqrt(110) m^(3)`

Answer

To find the volume of the pyramid, we will follow these steps:

### Step 1: Identify the dimensions of the base
The base of the pyramid is a rectangle with the following dimensions:
- Length (l) = 40 m
- Width (w) = 20 m

### Step 2: Calculate the area of the base
The area (A) of the rectangular base can be calculated using the formula:
\[ A = l \times w \]
Substituting the values:
\[ A = 40 \, \text{m} \times 20 \, \text{m} = 800 \, \text{m}^2 \]

### Step 3: Determine the slant height and find the height of the pyramid
We are given the slant height (l_s) from the midpoint of the shorter side of the base to the apex of the pyramid, which is 29 m. We need to find the vertical height (h) of the pyramid.

To do this, we can use the Pythagorean theorem. The base of the triangle formed by the height, the slant height, and half of the length of the base is:
\[ \text{Base} = \frac{l}{2} = \frac{40}{2} = 20 \, \text{m} \]

Using the Pythagorean theorem:
\[ l_s^2 = h^2 + \left(\frac{l}{2}\right)^2 \]
Substituting the known values:
\[ 29^2 = h^2 + 20^2 \]
\[ 841 = h^2 + 400 \]
Now, solve for h:
\[ h^2 = 841 - 400 \]
\[ h^2 = 441 \]
\[ h = \sqrt{441} = 21 \, \text{m} \]

### Step 4: Calculate the volume of the pyramid
The volume (V) of the pyramid can be calculated using the formula:
\[ V = \frac{1}{3} \times \text{Area of base} \times \text{Height} \]
Substituting the values we have:
\[ V = \frac{1}{3} \times 800 \, \text{m}^2 \times 21 \, \text{m} \]
\[ V = \frac{1}{3} \times 16800 \, \text{m}^3 \]
\[ V = 5600 \, \text{m}^3 \]

### Final Answer
The volume of the pyramid is **5600 m³**.

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