Question
Sara is making a model tent. The dimensions are given in the diagram below. How much material will she need to create the tent? Explain your steps to solve this problem and show ALL work.
5 cm
5 cm
3 cm
4 cm
15 cm
5 cm
5 cm
3 cm
4 cm
15 cm
Answers
Answered by
GPT-4o mini
To determine how much material Sara will need to create her model tent, we need to calculate the surface area of the tent. Assuming the tent is a triangular prism, we will calculate the lateral surface area and the area of the base.
Let’s break down the calculations step by step based on the provided dimensions.
1. **Identify the shape and dimensions**:
- The tent appears to have a triangular base with a height of 4 cm, meaning the two sides of the triangle are 5 cm each.
- The length of the tent (the distance from the front to the back) is 15 cm.
2. **Calculating the area of the triangular base**:
The base is a triangle, and we will need to find its area using the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base is 5 cm (a side of the triangle), and the height is 4 cm.
\[
\text{Area} = \frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = \frac{20}{2} = 10 \, \text{cm}^2
\]
3. **Calculating the lateral surface area of the tent**:
The lateral surface area consists of two triangular sides and one rectangular front and one rectangular back.
- The slant height of each triangular side can be calculated using the Pythagorean theorem:
\[
\text{slant height} = \sqrt{\text{height}^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{4^2 + \left(\frac{5}{2}\right)^2}
\]
\[
= \sqrt{16 + 6.25} = \sqrt{22.25} \approx 4.72 \, \text{cm}
\]
- The area of one triangular side is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{slant height}
\]
\[
= \frac{1}{2} \times 5 \times 4.72 \approx 11.8 \, \text{cm}^2
\]
- Since there are two triangular sides:
\[
\text{Total area of triangular sides} = 2 \times 11.8 \approx 23.6 \, \text{cm}^2
\]
- The lateral area from the rectangular front and back, each having an area of:
\[
\text{Area of front or back rectangle} = \text{base} \times \text{length} = 5 \times 15 = 75 \, \text{cm}^2
\]
- Therefore, for both the front and back:
\[
\text{Total area of rectangles} = 75 + 75 = 150 \, \text{cm}^2
\]
4. **Total Surface Area**:
Combine all areas calculated:
\[
\text{Total Surface Area} = \text{base area} + \text{lateral area}
\]
\[
= 10 + 23.6 + 150 = 183.6 \, \text{cm}^2
\]
Therefore, the total amount of material Sara needs to create her model tent is approximately **183.6 cm²**.
**Final Answer: 183.6 cm²**
Let’s break down the calculations step by step based on the provided dimensions.
1. **Identify the shape and dimensions**:
- The tent appears to have a triangular base with a height of 4 cm, meaning the two sides of the triangle are 5 cm each.
- The length of the tent (the distance from the front to the back) is 15 cm.
2. **Calculating the area of the triangular base**:
The base is a triangle, and we will need to find its area using the formula:
\[
\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, the base is 5 cm (a side of the triangle), and the height is 4 cm.
\[
\text{Area} = \frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = \frac{20}{2} = 10 \, \text{cm}^2
\]
3. **Calculating the lateral surface area of the tent**:
The lateral surface area consists of two triangular sides and one rectangular front and one rectangular back.
- The slant height of each triangular side can be calculated using the Pythagorean theorem:
\[
\text{slant height} = \sqrt{\text{height}^2 + \left(\frac{\text{base}}{2}\right)^2} = \sqrt{4^2 + \left(\frac{5}{2}\right)^2}
\]
\[
= \sqrt{16 + 6.25} = \sqrt{22.25} \approx 4.72 \, \text{cm}
\]
- The area of one triangular side is:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{slant height}
\]
\[
= \frac{1}{2} \times 5 \times 4.72 \approx 11.8 \, \text{cm}^2
\]
- Since there are two triangular sides:
\[
\text{Total area of triangular sides} = 2 \times 11.8 \approx 23.6 \, \text{cm}^2
\]
- The lateral area from the rectangular front and back, each having an area of:
\[
\text{Area of front or back rectangle} = \text{base} \times \text{length} = 5 \times 15 = 75 \, \text{cm}^2
\]
- Therefore, for both the front and back:
\[
\text{Total area of rectangles} = 75 + 75 = 150 \, \text{cm}^2
\]
4. **Total Surface Area**:
Combine all areas calculated:
\[
\text{Total Surface Area} = \text{base area} + \text{lateral area}
\]
\[
= 10 + 23.6 + 150 = 183.6 \, \text{cm}^2
\]
Therefore, the total amount of material Sara needs to create her model tent is approximately **183.6 cm²**.
**Final Answer: 183.6 cm²**