The problem asks us to find the surface area of a prism, given that the lateral area of the prism is $140 ft^2$ and a side of the base is $4 ft$.

1. Problem Description

The problem asks us to find the surface area of a prism, given that the lateral area of the prism is $140 ft^2$ and a side of the base is $4 ft$.

2. Solution Steps

The surface area of a prism is given by the formula:

$Surface Area = Lateral Area + 2 \times Base Area$

We are given the lateral area as $140 ft^2$. We need to find the base area. From the image, the base appears to be a pentagon formed by a rectangle and an isosceles triangle. However, since we are given one side as 4ft, and it appears as though the side lengths around the base are all 4 ft, then the pentagon base consists of a rectangle (4x4) and an equilateral triangle with side

4. Area of a rectangle = $l \times w = 4 \times 4 = 16 ft^2$.

Area of an equilateral triangle with side $a$ is given by $\frac{\sqrt{3}}{4}a^2$.

Area of equilateral triangle = $\frac{\sqrt{3}}{4}(4^2) = \frac{\sqrt{3}}{4}(16) = 4\sqrt{3} ft^2$.

Base Area = Area of Rectangle + Area of Equilateral Triangle $= 16 + 4\sqrt{3}$.

The surface area is:

$Surface Area = 140 + 2(16 + 4\sqrt{3}) = 140 + 32 + 8\sqrt{3} = 172 + 8\sqrt{3}$

$Surface Area \approx 172 + 8(1.732) = 172 + 13.856 = 185.856$

Rounding to the nearest tenth:

$Surface Area \approx 185.9 ft^2$

3. Final Answer

185.9 ft²