To determine which tile Orchid should choose to minimize costs, we need to follow these steps:
1. Calculate the area of each type of tile:
- For the square tile:
[tex]\[
\text{Area} = \left(\frac{1}{2}\ \text{foot}\right)^2 = \frac{1}{4}\ \text{square feet}
\][/tex]
- For the rectangular tile:
[tex]\[
\text{Area} = 2\ \text{feet} \times \frac{1}{4}\ \text{foot} = \frac{1}{2}\ \text{square feet}
\][/tex]
2. Determine the number of tiles needed to cover the bathroom floor:
- Total area of the bathroom = 40 square feet.
- For the square tiles:
[tex]\[
\text{Number of square tiles} = \frac{\text{Total Area of Bathroom}}{\text{Area of one square tile}} = \frac{40\ \text{square feet}}{\frac{1}{4}\ \text{square feet per tile}} = 160\ \text{tiles}
\][/tex]
- For the rectangular tiles:
[tex]\[
\text{Number of rectangular tiles} = \frac{\text{Total Area of Bathroom}}{\text{Area of one rectangular tile}} = \frac{40\ \text{square feet}}{\frac{1}{2}\ \text{square feet per tile}} = 80\ \text{tiles}
\][/tex]
3. Calculate the total cost for each type of tile:
- Square tiles cost [tex]$\$[/tex] 0.45$ each:
[tex]\[
\text{Total cost for square tiles} = \text{Number of square tiles} \times \text{Cost per tile} = 160 \times 0.45 = 72\ \text{dollars}
\][/tex]
- Rectangular tiles cost [tex]$\$[/tex] 0.80$ each:
[tex]\[
\text{Total cost for rectangular tiles} = \text{Number of rectangular tiles} \times \text{Cost per tile} = 80 \times 0.80 = 64\ \text{dollars}
\][/tex]
4. Compare the total costs to determine the more economical option:
- Total cost for square tiles = \$72.
- Total cost for rectangular tiles = \$64.
- Difference in cost between the two options:
[tex]\[
\text{Difference} = 72\ \text{dollars} - 64\ \text{dollars} = 8\ \text{dollars}
\][/tex]
Since the total cost for the rectangular tiles is less than the total cost for the square tiles by \$8, Orchid should choose the rectangular tiles.
Conclusion: She should choose the rectangular tiles because the total cost will be \$8.00 less.
