Question
A toy building is made of a rectangular prism with a height of 10cm, width of 5 cm, and depth of 3 cm. On top of the building is an isosceles triangular prism roof with a base of 5 cm, height of 6 cm, and a slant height of 4 cm. what is the total surface area of the toy building, including the roof and the sides?
Answers
GPT-4o mini
To find the total surface area of the toy building, we need to calculate the surface areas of both the rectangular prism and the isosceles triangular prism roof, and then sum them up.
### Step 1: Calculate the surface area of the rectangular prism
The surface area \(A\) of a rectangular prism can be calculated using the formula:
\[
A = 2(lw + lh + wh)
\]
where \(l\) is the length (height), \(w\) is the width, and \(h\) is the height.
In our case:
- Length (height) \(= 10 \, \text{cm}\)
- Width \(= 5 \, \text{cm}\)
- Depth \(= 3 \, \text{cm}\)
Calculating:
\[
A = 2(10 \times 5 + 10 \times 3 + 5 \times 3)
\]
\[
= 2(50 + 30 + 15)
\]
\[
= 2(95)
\]
\[
= 190 \, \text{cm}^2
\]
### Step 2: Calculate the surface area of the isosceles triangular prism roof
The surface area of the isosceles triangular prism consists of:
1. Two triangular bases.
2. Three rectangular sides.
#### 2.1. Area of the triangular bases
The area \(A\) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular prism:
- Base \(= 5 \, \text{cm}\)
- Height \(= 6 \, \text{cm}\)
Calculating:
\[
A = \frac{1}{2} \times 5 \times 6 = \frac{30}{2} = 15 \, \text{cm}^2
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 15 = 30 \, \text{cm}^2
\]
#### 2.2. Area of the rectangular sides
The rectangular sides of the prism are:
1. Two sides with dimensions \(h\) and slant height.
2. One side with dimensions \(b\) and the height of the prism.
The dimensions are:
- Height of the roof (prism's height) \(= 10 \, \text{cm}\) (same as the prism),
- Slant height \(= 4 \, \text{cm}\),
- Base height \(= 5 \, \text{cm}\).
Calculating the three rectangular sides:
1. Two sides with base height and slant height:
- Area = \(2 \times \text{height} \times \text{slant height} = 2 \times 6 \times 4 = 48 \, \text{cm}^2\)
2. One side with the base height and prism height:
- Area = \( \text{base} \times \text{height} = 5 \times 10 = 50 \, \text{cm}^2\)
So, the total area of the rectangular sides is:
\[
\text{Total rectangular sides} = 48 + 50 = 98 \, \text{cm}^2
\]
### Step 3: Total surface area of the roof
Now, summing the areas of the components of the roof:
\[
\text{Total area of the roof} = 30 + 98 = 128 \, \text{cm}^2
\]
### Final Step: Total surface area of the toy building
Add the surface area of the rectangular prism with the surface area of the roof:
\[
\text{Total surface area} = \text{Area of rectangular prism} + \text{Area of roof}
\]
\[
= 190 + 128 = 318 \, \text{cm}^2
\]
Thus, the total surface area of the toy building, including the roof and the sides, is:
\[
\boxed{318 \, \text{cm}^2}
\]
### Step 1: Calculate the surface area of the rectangular prism
The surface area \(A\) of a rectangular prism can be calculated using the formula:
\[
A = 2(lw + lh + wh)
\]
where \(l\) is the length (height), \(w\) is the width, and \(h\) is the height.
In our case:
- Length (height) \(= 10 \, \text{cm}\)
- Width \(= 5 \, \text{cm}\)
- Depth \(= 3 \, \text{cm}\)
Calculating:
\[
A = 2(10 \times 5 + 10 \times 3 + 5 \times 3)
\]
\[
= 2(50 + 30 + 15)
\]
\[
= 2(95)
\]
\[
= 190 \, \text{cm}^2
\]
### Step 2: Calculate the surface area of the isosceles triangular prism roof
The surface area of the isosceles triangular prism consists of:
1. Two triangular bases.
2. Three rectangular sides.
#### 2.1. Area of the triangular bases
The area \(A\) of a triangle is given by:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
For the triangular prism:
- Base \(= 5 \, \text{cm}\)
- Height \(= 6 \, \text{cm}\)
Calculating:
\[
A = \frac{1}{2} \times 5 \times 6 = \frac{30}{2} = 15 \, \text{cm}^2
\]
Since there are two triangular bases:
\[
\text{Total area of triangular bases} = 2 \times 15 = 30 \, \text{cm}^2
\]
#### 2.2. Area of the rectangular sides
The rectangular sides of the prism are:
1. Two sides with dimensions \(h\) and slant height.
2. One side with dimensions \(b\) and the height of the prism.
The dimensions are:
- Height of the roof (prism's height) \(= 10 \, \text{cm}\) (same as the prism),
- Slant height \(= 4 \, \text{cm}\),
- Base height \(= 5 \, \text{cm}\).
Calculating the three rectangular sides:
1. Two sides with base height and slant height:
- Area = \(2 \times \text{height} \times \text{slant height} = 2 \times 6 \times 4 = 48 \, \text{cm}^2\)
2. One side with the base height and prism height:
- Area = \( \text{base} \times \text{height} = 5 \times 10 = 50 \, \text{cm}^2\)
So, the total area of the rectangular sides is:
\[
\text{Total rectangular sides} = 48 + 50 = 98 \, \text{cm}^2
\]
### Step 3: Total surface area of the roof
Now, summing the areas of the components of the roof:
\[
\text{Total area of the roof} = 30 + 98 = 128 \, \text{cm}^2
\]
### Final Step: Total surface area of the toy building
Add the surface area of the rectangular prism with the surface area of the roof:
\[
\text{Total surface area} = \text{Area of rectangular prism} + \text{Area of roof}
\]
\[
= 190 + 128 = 318 \, \text{cm}^2
\]
Thus, the total surface area of the toy building, including the roof and the sides, is:
\[
\boxed{318 \, \text{cm}^2}
\]