To calculate the surface area of this triangular pyramid, we need to account for the areas of all its triangular faces. Here's a breakdown of the steps:

1. Identify the faces:

• There is one base triangle and three lateral triangular faces.

2. Area of the base triangle:

• The base is an equilateral triangle with side lengths of 4 ft.
• To calculate the area of an equilateral triangle, the formula is: $A = \frac{\sqrt{3}}{4} s^2$ where $s$ is the length of the side. Substituting $s = 4$ ft: $A_{\text{base}} = \frac{\sqrt{3}}{4} (4)^2 = \frac{\sqrt{3}}{4} \times 16 \approx 6.93 \, \text{ft}^2$

3. Area of the lateral faces:

• Each lateral triangular face has a base of 4 ft and a slant height of 7 ft.
• The area of one triangular face is calculated as: $A = \frac{1}{2} \times \text{base} \times \text{height}$ Substituting the values: $A_{\text{lateral}} = \frac{1}{2} \times 4 \times 7 = 14 \, \text{ft}^2$ Since there are three identical lateral faces, the total lateral area is: $A_{\text{total lateral}} = 3 \times 14 = 42 \, \text{ft}^2$

4. Total surface area:

Now, add the base area and the lateral areas: $A_{\text{total}} = A_{\text{base}} + A_{\text{total lateral}} = 6.93 + 42 = 48.93 \, \text{ft}^2$

So, the surface area of the triangular pyramid is approximately 48.93 square feet.

Let me know if you'd like further clarification or details.

Here are five related questions to explore further:

1. How do you calculate the volume of a triangular pyramid?
2. What is the difference between lateral and total surface area?
3. Can this method apply to pyramids with different base shapes?
4. How would the surface area change if the slant height was different?
5. What is the significance of using the slant height versus the perpendicular height?

Tip: Remember to distinguish between slant height and perpendicular height when solving problems with pyramids or cones!