Question

The bottom part of this block is a rectangular prism. The top part is a square pyramid. You want to cover the block entirely with paper. How much paper do you​ need? Use pencil and paper to explain your reasoning.
5 cm
4 cm
4 cm
3 cm
​(The figure is not to​ scale.)
Question content area bottom
Part 1
You need
  
enter your response here cmsquared of paper.

Answer 1

To find the total surface area of the block (a rectangular prism topped with a square pyramid), we will calculate the surface area of each part separately and then sum them up.

### Step 1: Calculate the Surface Area of the Rectangular Prism

For the rectangular prism:
- **Dimensions:**
- Height = 5 cm
- Length = 4 cm
- Width = 4 cm

**Surface Area (SA) Formula for a Rectangular Prism:**
\[
SA = 2(lw + lh + wh)
\]
Where:
- \(l\) = length
- \(w\) = width
- \(h\) = height

In this case:
- \(l = 4\) cm
- \(w = 4\) cm
- \(h = 5\) cm

Now, substituting the values:
\[
SA_{prism} = 2(4 \times 4 + 4 \times 5 + 4 \times 5)
\]
Calculate the area of each part:
- \(lw = 4 \times 4 = 16\)
- \(lh = 4 \times 5 = 20\)
- \(wh = 4 \times 5 = 20\)

Now, plug these into the surface area formula:
\[
SA_{prism} = 2(16 + 20 + 20) = 2(56) = 112\text{ cm}^2
\]
However, we will later subtract the top face of the prism since it is covered by the pyramid.

### Step 2: Calculate the Surface Area of the Square Pyramid

For the square pyramid:
- **Base side length = Width of the prism = 4 cm**
- **Height = 3 cm**

**Surface Area (SA) Formula for a Square Pyramid:**
\[
SA = B + \frac{1}{2}(P \times l)
\]
Where:
- \(B\) = area of the base
- \(P\) = perimeter of the base
- \(l\) = slant height

**Calculate the base area (\(B\))**:
\[
B = \text{side}^2 = 4^2 = 16 \text{ cm}^2
\]

**Calculate the perimeter (\(P\))**:
\[
P = 4 \times \text{side} = 4 \times 4 = 16 \text{ cm}
\]

**Now we need the slant height (\(l\))**:
To find the slant height, we can use the Pythagorean theorem. The height of the pyramid is 3 cm, and the half of the base side length is 2 cm.
\[
l = \sqrt{(3)^2 + (2)^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.61 \text{ cm}
\]

**Now calculate the lateral surface area**:
\[
\text{Lateral Surface Area} = \frac{1}{2} \times P \times l = \frac{1}{2} \times 16 \times 3.61 \approx 28.88 \text{ cm}^2
\]

**Total Surface Area for the Pyramid**:
\[
SA_{pyramid} = B + \text{Lateral Surface Area} \approx 16 + 28.88 \approx 44.88 \text{ cm}^2
\]

### Step 3: Combine the Areas

Since the top face of the rectangular prism is covered by the base of the pyramid, we need to adjust the surface area of the prism:
- Total Surface Area = Surface Area of Prism + Surface Area of Pyramid - Area of top face of prism
\[
SA_{total} = SA_{prism} + SA_{pyramid} - \text{Area of top face of prism}
\]
- Area of top face of prism = \(4 \times 4 = 16 \text{ cm}^2\)

Now substitute back into the total surface area calculation:
\[
SA_{total} = 112 + 44.88 - 16 = 140.88 \text{ cm}^2
\]

### Conclusion

The total amount of paper needed to cover the block entirely is approximately:
\[
\text{You need } 140.88 \text{ cm}^2 \text{ of paper.}
\]

Rounding to two decimal places, we can state:
**140.88 cm²**.