Let's break down and solve the problem step by step.
Part (a): Calculate the shaded area of the floor plan
The floor plan consists of:
- A rectangular base: 12ft×11ft
- A semicircle (on the bottom): Radius r=5ft
- Two excluded areas:
- The bench: 5ft×2ft
- The storage closet: 11ft×2ft
Step 1: Area of the main rectangle
Area of rectangle=12×11=132sq ft
Step 2: Area of the semicircle
The formula for the area of a semicircle is:
Area of semicircle=21πr2
Area of semicircle=21π(5)2=21π(25)≈39.27sq ft
Step 3: Total area before subtracting exclusions
Total area=Area of rectangle+Area of semicircle
Total area=132+39.27=171.27sq ft
Step 4: Excluded areas
- Bench area:
Bench area=5×2=10sq ft
- Storage closet area:
Storage closet area=11×2=22sq ft
Total excluded area=10+22=32sq ft
Step 5: Final shaded area
Shaded area=Total area−Total excluded area
Shaded area=171.27−32=139.27sq ft
Part (b): Number of tiles required
Each tile is 2ft×2ft, so the area of one tile is:
Area of one tile=2×2=4sq ft
The number of tiles needed (before accounting for waste) is:
Tiles required=Area of one tileShaded area
Tiles required=4139.27≈34.82
Rounding up:
Tiles required (rounded up)=35
Adding 10% extra for waste:
Total tiles (including waste)=35×1.1=38.5
Rounding up again:
Total tiles required=39
Part (c): Cost of tiles
Each box contains 8 tiles, so the number of boxes needed is:
Number of boxes=⌈Tiles per boxTotal tiles required⌉
Number of boxes=⌈839⌉=⌈4.875⌉=5
The cost of each box is $13.99. The total cost is:
Total cost=5×13.99=69.95USD
Final Answers
- Shaded area: 139.27sq ft
- Total tiles required: 39
- Cost of tiles: 69.95USD
Would you like me to explain any of these steps further or explore related concepts? Here are some related questions:
- How is the formula for the area of a semicircle derived?
- What is the impact of increasing tile size on the total cost?
- How does the cost change if the waste percentage is reduced?
- How can irregular shapes like this be divided into simpler geometrical shapes?
- How do rounding rules affect final material estimates?
Tip: Always round up when calculating materials to ensure sufficient supply