Math Problem Statement

You just got a part-time job at a home improvement centre—your job is to help customers determine how many tiles they need. The first customer wants to tile a room with an irregular shape and needs help calculating the shaded floor plan area, determining the number of tiles required (including 10% waste), and the total cost if tiles are sold in boxes.

Solution

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:

  1. A rectangular base: 12ft×11ft12 \, \text{ft} \times 11 \, \text{ft}
  2. A semicircle (on the bottom): Radius r=5ftr = 5 \, \text{ft}
  3. Two excluded areas:
    • The bench: 5ft×2ft5 \, \text{ft} \times 2 \, \text{ft}
    • The storage closet: 11ft×2ft11 \, \text{ft} \times 2 \, \text{ft}

Step 1: Area of the main rectangle

Area of rectangle=12×11=132sq ft\text{Area of rectangle} = 12 \times 11 = 132 \, \text{sq ft}

Step 2: Area of the semicircle

The formula for the area of a semicircle is: Area of semicircle=12πr2\text{Area of semicircle} = \frac{1}{2} \pi r^2 Area of semicircle=12π(5)2=12π(25)39.27sq ft\text{Area of semicircle} = \frac{1}{2} \pi (5)^2 = \frac{1}{2} \pi (25) \approx 39.27 \, \text{sq ft}

Step 3: Total area before subtracting exclusions

Total area=Area of rectangle+Area of semicircle\text{Total area} = \text{Area of rectangle} + \text{Area of semicircle} Total area=132+39.27=171.27sq ft\text{Total area} = 132 + 39.27 = 171.27 \, \text{sq ft}

Step 4: Excluded areas

  1. Bench area: Bench area=5×2=10sq ft\text{Bench area} = 5 \times 2 = 10 \, \text{sq ft}
  2. Storage closet area: Storage closet area=11×2=22sq ft\text{Storage closet area} = 11 \times 2 = 22 \, \text{sq ft} Total excluded area=10+22=32sq ft\text{Total excluded area} = 10 + 22 = 32 \, \text{sq ft}

Step 5: Final shaded area

Shaded area=Total areaTotal excluded area\text{Shaded area} = \text{Total area} - \text{Total excluded area} Shaded area=171.2732=139.27sq ft\text{Shaded area} = 171.27 - 32 = 139.27 \, \text{sq ft}


Part (b): Number of tiles required

Each tile is 2ft×2ft2 \, \text{ft} \times 2 \, \text{ft}, so the area of one tile is: Area of one tile=2×2=4sq ft\text{Area of one tile} = 2 \times 2 = 4 \, \text{sq ft}

The number of tiles needed (before accounting for waste) is: Tiles required=Shaded areaArea of one tile\text{Tiles required} = \frac{\text{Shaded area}}{\text{Area of one tile}} Tiles required=139.27434.82\text{Tiles required} = \frac{139.27}{4} \approx 34.82 Rounding up: Tiles required (rounded up)=35\text{Tiles required (rounded up)} = 35

Adding 10% extra for waste: Total tiles (including waste)=35×1.1=38.5\text{Total tiles (including waste)} = 35 \times 1.1 = 38.5 Rounding up again: Total tiles required=39\text{Total tiles required} = 39


Part (c): Cost of tiles

Each box contains 8 tiles, so the number of boxes needed is: Number of boxes=Total tiles requiredTiles per box\text{Number of boxes} = \lceil \frac{\text{Total tiles required}}{\text{Tiles per box}} \rceil Number of boxes=398=4.875=5\text{Number of boxes} = \lceil \frac{39}{8} \rceil = \lceil 4.875 \rceil = 5

The cost of each box is $13.99. The total cost is: Total cost=5×13.99=69.95USD\text{Total cost} = 5 \times 13.99 = 69.95 \, \text{USD}


Final Answers

  1. Shaded area: 139.27sq ft139.27 \, \text{sq ft}
  2. Total tiles required: 3939
  3. Cost of tiles: 69.95USD69.95 \, \text{USD}

Would you like me to explain any of these steps further or explore related concepts? Here are some related questions:

  1. How is the formula for the area of a semicircle derived?
  2. What is the impact of increasing tile size on the total cost?
  3. How does the cost change if the waste percentage is reduced?
  4. How can irregular shapes like this be divided into simpler geometrical shapes?
  5. How do rounding rules affect final material estimates?

Tip: Always round up when calculating materials to ensure sufficient supply

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Math Problem Analysis

Mathematical Concepts

Area calculation
Geometry
Rounding
Cost estimation

Formulas

Area of a rectangle = length × width
Area of a semicircle = (1/2)πr^2
Tiles required = Total area / Area of one tile
Waste adjustment = Tiles × 1.1
Number of boxes = Ceiling(Tiles / Tiles per box)
Cost = Number of boxes × Cost per box

Theorems

Properties of areas
Rounding rules for material estimates

Suitable Grade Level

Grades 8-10