Lindsey made a model of a house in construction class. The block of wood for the base measures 4 in. by 13 in. and is 8 in. tall. She used a triangular prism for the roof that hangs over the base by half an inch on all sides and is 4 in. in height. Lindsey would like to paint the entire model. For the surface area of the model house to determine how much paint would be needed. Assume the roof is the same on both sides. (Hint don't forget about painting underneath the overhang and house bottom) Round your answer up to the nearest whole number.

To determine the surface area of Lindsey's model house, we'll calculate the surface area of the base (rectangular prism) and the roof (triangular prism), and then add them together. 

### Step 1: Calculate the Surface Area of the Rectangular Prism (Base)
The dimensions of the base are:
- Length = 13 inches
- Width = 4 inches
- Height = 8 inches

**Surface area of the rectangular prism:**
\[
\text{Surface Area}_{\text{rectangular prism}} = 2lw + 2lh + 2wh
\]
Substitute the values:
\[
= 2(13 \times 4) + 2(13 \times 8) + 2(4 \times 8)
\]
\[
= 2(52) + 2(104) + 2(32)
\]
\[
= 104 + 208 + 64 = 376 \text{ square inches}
\]

### Step 2: Calculate the Surface Area of the Triangular Prism (Roof)
The roof is a triangular prism that hangs over the base by 0.5 inches on all sides. 

The new dimensions for the roof are:
- Length = \(13 + 2 \times 0.5 = 14\) inches
- Width (base of the triangle) = \(4 + 2 \times 0.5 = 5\) inches
- Height of the triangle = 4 inches
- Height of the prism (length of the roof) = 14 inches

**Find the area of one triangular face:**
Using the formula for the area of a triangle \( A = \frac{1}{2} \times \text{base} \times \text{height} \),
\[
A_{\text{triangle}} = \frac{1}{2} \times 5 \times 4 = 10 \text{ square inches}
\]
Since there are two triangular faces,
\[
\text{Area of both triangles} = 2 \times 10 = 20 \text{ square inches}
\]

**Find the area of the three rectangular faces (sides):**
Each of these faces is a rectangle with area \( \text{length} \times \text{width} \):
1. First rectangular face: \( 14 \times 4 = 56 \) square inches
2. Second rectangular face: \( 14 \times \sqrt{4^2 + 2.5^2} \) (the slant height)
   
   Calculate the slant height using Pythagoras' theorem:
   \[
   \text{Slant height} = \sqrt{4^2 + 2.5^2} = \sqrt{16 + 6.25} = \sqrt{22.25} \approx 4.72 \text{ inches}
   \]
   Then the area is:
   \[
   14 \times 4.72 \approx 66.08 \text{ square inches}
   \]
3. The third face has the same area as the second, so it also contributes approximately 66.08 square inches.

Thus, the total area of the three rectangular faces is:
\[
56 + 66.08 + 66.08 = 188.16 \text{ square inches}
\]

**Total surface area of the triangular prism:**
\[
\text{Surface Area}_{\text{triangular prism}} = 20 + 188.16 = 208.16 \text{ square inches}
\]

### Step 3: Combine the Surface Areas
Finally, the total surface area of the house is:
\[
\text{Total Surface Area} = 376 + 208.16 = 584.16 \text{ square inches}
\]
Rounding up to the nearest whole number, the total surface area is approximately **585 square inches**.

This is the amount of surface area Lindsey will need to paint.

Would you like any further details or clarifications?

---

Here are 5 related questions:
1. How would the surface area change if the height of the house base was increased by 2 inches?
2. What would the volume of the entire house model be?
3. How would the surface area calculation differ if the roof did not overhang the base?
4. What is the area of the roof alone without the overhang?
5. How would changing the overhang distance affect the total surface area?

**Tip:** When working with models or real objects, always remember to account for all visible surfaces, including the undersides of overhangs or bases when calculating the surface area.

Learn how to calculate the surface area of a model house constructed with a rectangular base and a triangular prism roof. This problem involves detailed geometric calculations, including the surface area formulas for rectangular and triangular prisms. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation