Math Problem Statement

The length of a rectangle is four times its width. The rectangle has an area of 1024 cm². Work out the width of the rectangle.

Solution

We are tasked with finding the width of a rectangle where the length is four times the width, and the area of the rectangle is 1024 cm².

Let's define the variables:

  • Let the width of the rectangle be ww cm.
  • Since the length is four times the width, the length of the rectangle will be 4w4w cm.
  • The area AA of a rectangle is given by the formula: A=length×widthA = \text{length} \times \text{width}

Substituting the given values:

1024=w×4w1024 = w \times 4w 1024=4w21024 = 4w^2

Solving for ww:

  1. Divide both sides by 4: w2=10244=256w^2 = \frac{1024}{4} = 256
  2. Take the square root of both sides: w=256=16w = \sqrt{256} = 16

Thus, the width of the rectangle is 16 cm.

Would you like further details or clarifications?


Here are five related questions to deepen your understanding:

  1. How would you calculate the length of the rectangle now that you know the width?
  2. If the area was doubled, what would be the new width?
  3. What would happen to the width if the length was only three times the width?
  4. How do you find the perimeter of the rectangle given the width and length?
  5. Can the area of a rectangle ever be negative? Why or why not?

Tip: Always double-check your equation setup based on the problem conditions to avoid mistakes when solving geometric problems.

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Quadratic Equations

Formulas

Area of a rectangle = length × width
Quadratic equation: ax² = c

Theorems

Properties of Quadratic Equations
Basic Geometry Theorems

Suitable Grade Level

Grades 6-8