Math Problem Statement

Solve each problem by constructing and then solving a quadratic equation. a A rectangular area can be completely tiled with 200 square tiles. If the side length of each tile was increased by 1 cm, it would take only 128 tiles to tile the area. Find the side length of each tile.

Solution

To solve this problem, let's denote the side length of each tile as xx cm.

Step 1: Construct the Quadratic Equation

  1. Initial Condition:

    • The total area of the rectangular region is 200×x2200 \times x^2 square cm, where xx is the side length of each tile.
    • Therefore, the area of the rectangular region is 200x2200x^2 square cm.
  2. After Increasing the Side Length by 1 cm:

    • The side length of each tile becomes x+1x + 1 cm.
    • The number of tiles required becomes 128, so the total area of the rectangular region is now 128×(x+1)2128 \times (x + 1)^2 square cm.

Since the area of the rectangular region remains the same in both cases, we can equate the two expressions for the area:

200x2=128(x+1)2200x^2 = 128(x + 1)^2

Step 2: Solve the Quadratic Equation

  1. Expand the right side of the equation:

200x2=128(x2+2x+1)200x^2 = 128(x^2 + 2x + 1)

  1. Distribute 128:

200x2=128x2+256x+128200x^2 = 128x^2 + 256x + 128

  1. Move all terms to one side to set the equation to zero:

200x2128x2256x128=0200x^2 - 128x^2 - 256x - 128 = 0

Simplify the equation:

72x2256x128=072x^2 - 256x - 128 = 0

  1. Divide the entire equation by 8 to simplify further:

9x232x16=09x^2 - 32x - 16 = 0

Step 3: Use the Quadratic Formula

The quadratic equation is in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=9a = 9, b=32b = -32, and c=16c = -16.

The quadratic formula is given by:

x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substitute the values of aa, bb, and cc:

x=(32)±(32)24(9)(16)2(9)x = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(9)(-16)}}{2(9)}

Simplify:

x=32±1024+57618x = \frac{32 \pm \sqrt{1024 + 576}}{18}

x=32±160018x = \frac{32 \pm \sqrt{1600}}{18}

x=32±4018x = \frac{32 \pm 40}{18}

This gives two possible solutions:

x=7218=4orx=818=49x = \frac{72}{18} = 4 \quad \text{or} \quad x = \frac{-8}{18} = -\frac{4}{9}

Step 4: Interpret the Result

Since the side length xx must be positive, we discard the negative value. Therefore, the side length of each tile is x=4x = 4 cm.

Final Answer:

The side length of each tile is 4 cm.

Would you like any further details or have any questions? Here are some related problems you might find interesting:

  1. If the number of tiles increased by 50% instead of 64%, what would the new side length be?
  2. How would the area of the entire rectangular region change if the side length increased by 2 cm?
  3. Find the perimeter of the rectangular area in the original tiling.
  4. If the tiles were triangular instead of square, how would the equation change?
  5. What happens to the equation if the increase in the side length is 2 cm instead of 1 cm?

Tip: When working with quadratic equations, always check if a simpler form can be obtained by factoring before applying the quadratic formula.

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Area Calculations

Formulas

Quadratic formula

Theorems

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Suitable Grade Level

Grades 7-9