If I were Tyrese, I would choose to build a rectangular prism for my three-dimensional project. The rectangular prism is versatile and allows for a variety of dimensions, which can make it visually interesting and provide a solid base for construction.
Sketch and Dimensions
Let's say I choose a rectangular prism with the following dimensions:
- Length (L) = 4 units
- Width (W) = 3 units
- Height (H) = 5 units
I would sketch the rectangular prism and label these dimensions clearly.
Surface Area Calculation
The formula for calculating the surface area (SA) of a rectangular prism is: \[ SA = 2(LW + LH + WH) \]
Plugging in the values: \[ SA = 2(4 \cdot 3 + 4 \cdot 5 + 3 \cdot 5) \] \[ SA = 2(12 + 20 + 15) \] \[ SA = 2(47) \] \[ SA = 94 \text{ square units} \]
Miguel's Construction
Miguel will need to double all of the dimensions of the rectangular prism:
- New Length (L) = 4 * 2 = 8 units
- New Width (W) = 3 * 2 = 6 units
- New Height (H) = 5 * 2 = 10 units
New Surface Area Calculation
Now, we can calculate the surface area of Miguel's rectangular prism using the same formula: \[ SA = 2(LW + LH + WH) \] For Miguel’s dimensions: \[ SA = 2(8 \cdot 6 + 8 \cdot 10 + 6 \cdot 10) \] \[ SA = 2(48 + 80 + 60) \] \[ SA = 2(188) \] \[ SA = 376 \text{ square units} \]
Understanding the Change in Surface Area
When we compare the surface areas:
- Tyrese's surface area = 94 square units
- Miguel's surface area = 376 square units
It can be observed that Miguel's surface area is 4 times that of Tyrese’s: \[ \frac{376}{94} = 4 \]
Explanation
When the dimensions of a three-dimensional shape (like our rectangular prism) are doubled, the surface area doesn’t just double—it increases by a factor of four. This occurs because surface area is dependent on the square of the linear dimensions. Specifically, if the dimensions are scaled by a factor of k (in this case, k=2), the surface area scales by a factor of \( k^2 \): \[ SA_{\text{new}} = k^2 \cdot SA_{\text{original}} \] Thus, \( 2^2 = 4 \), which explains why Miguel’s rectangular prism has a surface area 4 times larger than Tyrese's.
This demonstrates the powerful effect that scaling dimensions has on surface area in three-dimensional geometry.