Question
Bradley made a house for his dog, Bowser, out of wood with a cube base and a triangular prism top. The dimensions of the dog house are a = 5 feet, b = 1 foot, and c = 2.7 feet.
Note: Figure is not drawn to scale.
If Bradley plans to paint the outside of the dog house blue, not including the bottom, how many square feet of paint will he use?
A.
182 square feet
B.
132 square feet
C.
129.5 square feet
D.
157 square feet
Note: Figure is not drawn to scale.
If Bradley plans to paint the outside of the dog house blue, not including the bottom, how many square feet of paint will he use?
A.
182 square feet
B.
132 square feet
C.
129.5 square feet
D.
157 square feet
Answers
GPT-4o mini
To find the total area that Bradley will paint, we need to calculate the surface area of the dog house. The dog house consists of two parts:
1. The cube base
2. The triangular prism top
### 1. Area of the Cube Base
Given that the cube has a side length \(a = 5\) feet, we can calculate the surface area of the cube (which consists of 5 sides, as the bottom will not be painted).
- Area of 1 face = \(a^2 = 5^2 = 25\) square feet.
- Total area of 5 faces = \(5 \times 25 = 125\) square feet.
### 2. Area of the Triangular Prism Top
The top of the dog house is a triangular prism. The base of the prism is a triangle with base \(b = 1\) foot and height \(c = 2.7\) feet.
#### Area of the Triangle Base
The area \(A_{triangle}\) is calculated as:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 2.7 = 1.35 \text{ square feet}
\]
#### Lateral Surface Area of the Prism
The lateral surface area of the triangular prism consists of three rectangular faces corresponding to each side of the triangle.
1. Two rectangles with height \(c\) and the base \(b\):
- Area 1 = Base \(b\) x Height \(c\):
\[
A_1 = 1 \times 2.7 = 2.7 \text{ square feet}
\]
2. The side adjacent to height \(c\) will be the slant height of the triangular prism. To find the lengths of the sides of the triangle:
- We have a triangle with base \(b = 1\) and height \(c = 2.7\).
- The hypotenuse (side) length can be calculated using Pythagoras' theorem:
\[
\text{Hypotenuse} = \sqrt{(b/2)^2 + c^2} = \sqrt{(0.5)^2 + (2.7)^2} \approx \sqrt{0.25 + 7.29} = \sqrt{7.54} \approx 2.74
\]
So, each of the two slant sides has an area:
\[
A_2 = \text{Hypotenuse} \times \text{Height} = 2.74 \times 5 \approx 13.7 \text{ square feet} \text{ (for one side)}
\]
Total for both sides:
\[
2 \times A_2 = 2 \times 13.7 \approx 27.4 \text{ square feet}
\]
### Summarizing the Areas
- Area of the cube base (not including bottom) = 125 square feet
- Area of the triangular prism top (including both triangle faces and lateral sides):
- Area of the triangle base = 1.35 square feet
- Lateral area for two slant sides = 27.4 square feet
- Total area of the prism = \(1.35 + 2.7 + 27.4 \approx 31.45\) square feet (only lateral, excluding bottom)
### Total Area Painted
Combining the areas:
\[
Total \text{ Area} = 125 + 31.45 = 156.45 \text{ square feet}
\]
Rounding to one decimal point provides \(156.5\) square feet. Thus, approximately \(157\) square feet.
### Final Answer
The answer is \(D. 157\) square feet.
1. The cube base
2. The triangular prism top
### 1. Area of the Cube Base
Given that the cube has a side length \(a = 5\) feet, we can calculate the surface area of the cube (which consists of 5 sides, as the bottom will not be painted).
- Area of 1 face = \(a^2 = 5^2 = 25\) square feet.
- Total area of 5 faces = \(5 \times 25 = 125\) square feet.
### 2. Area of the Triangular Prism Top
The top of the dog house is a triangular prism. The base of the prism is a triangle with base \(b = 1\) foot and height \(c = 2.7\) feet.
#### Area of the Triangle Base
The area \(A_{triangle}\) is calculated as:
\[
A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 2.7 = 1.35 \text{ square feet}
\]
#### Lateral Surface Area of the Prism
The lateral surface area of the triangular prism consists of three rectangular faces corresponding to each side of the triangle.
1. Two rectangles with height \(c\) and the base \(b\):
- Area 1 = Base \(b\) x Height \(c\):
\[
A_1 = 1 \times 2.7 = 2.7 \text{ square feet}
\]
2. The side adjacent to height \(c\) will be the slant height of the triangular prism. To find the lengths of the sides of the triangle:
- We have a triangle with base \(b = 1\) and height \(c = 2.7\).
- The hypotenuse (side) length can be calculated using Pythagoras' theorem:
\[
\text{Hypotenuse} = \sqrt{(b/2)^2 + c^2} = \sqrt{(0.5)^2 + (2.7)^2} \approx \sqrt{0.25 + 7.29} = \sqrt{7.54} \approx 2.74
\]
So, each of the two slant sides has an area:
\[
A_2 = \text{Hypotenuse} \times \text{Height} = 2.74 \times 5 \approx 13.7 \text{ square feet} \text{ (for one side)}
\]
Total for both sides:
\[
2 \times A_2 = 2 \times 13.7 \approx 27.4 \text{ square feet}
\]
### Summarizing the Areas
- Area of the cube base (not including bottom) = 125 square feet
- Area of the triangular prism top (including both triangle faces and lateral sides):
- Area of the triangle base = 1.35 square feet
- Lateral area for two slant sides = 27.4 square feet
- Total area of the prism = \(1.35 + 2.7 + 27.4 \approx 31.45\) square feet (only lateral, excluding bottom)
### Total Area Painted
Combining the areas:
\[
Total \text{ Area} = 125 + 31.45 = 156.45 \text{ square feet}
\]
Rounding to one decimal point provides \(156.5\) square feet. Thus, approximately \(157\) square feet.
### Final Answer
The answer is \(D. 157\) square feet.