Maddie is painting her shed. We need to approximate the surface area of the shed that will be painted, assuming the shed can be modeled by a triangular prism and a rectangular prism. We also assume whether the doors will be painted and that the roof and the floor of the shed will not be painted. The dimensions are as follows: the rectangular prism part has a height of 7 ft, length of 10 ft, and width of 8 ft. The triangular prism part has a height of 2 ft, and a length of 10 ft.

1. Problem Description

Maddie is painting her shed. We need to approximate the surface area of the shed that will be painted, assuming the shed can be modeled by a triangular prism and a rectangular prism. We also assume whether the doors will be painted and that the roof and the floor of the shed will not be painted. The dimensions are as follows: the rectangular prism part has a height of 7 ft, length of 10 ft, and width of 8 ft. The triangular prism part has a height of 2 ft, and a length of 10 ft.

2. Solution Steps

First, we calculate the surface area of the rectangular prism part, excluding the top where the triangular prism sits.

The area of the front and back walls is $2 \times (7 \times 10) = 140$ square feet.

The area of the two side walls is $2 \times (7 \times 8) = 112$ square feet.

The area of the bottom is $10 \times 8 = 80$ square feet.

Total area of the rectangular prism = $140 + 112 + 80 = 332$ square feet.

Next, we calculate the surface area of the triangular prism.

The area of the two triangular faces is $2 \times (\frac{1}{2} \times 8 \times 2) = 16$ square feet.

To find the area of the two rectangular faces of the roof, we first need to find the length of the sloped side of the triangle. Using the Pythagorean theorem, we have $side = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47$ ft.

The area of the two rectangular roof faces is $2 \times (10 \times 2\sqrt{5}) = 20 \times 2\sqrt{5} = 40\sqrt{5} \approx 89.44$ square feet.

The total surface area to be painted is the sum of the surface area of the rectangular prism and the triangular prism:

Total area = $332 + 16 + 89.44 = 437.44$ square feet.

However, the question mentions the options 268, 282, 288, and

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6. We are also told that the doors would not be painted and that the roof and the floor of the shed would not be painted. Let's recalculate.

Area of rectangular prism to be painted:

Front and back: $2 \times (7 \times 10) = 140$

Sides: $2 \times (7 \times 8) = 112$

Total: $140 + 112 = 252$

Area of triangular prism to be painted:

$2 \times (\frac{1}{2} \times 2 \times 8) = 16$

Total roof area: $2 \times (10 \times \sqrt{20}) \approx 89.44$

If we assume doors are not painted, we ignore the floor and roof: $140 + 112 + 16 = 268$

2. Final Answer

268 ft²; I assumed that the shed could be modeled by a triangular prism and a rectangular prism. I also assumed that the doors would not be painted but the roof and the floor of the shed would not be painted.