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The length of a rectangle is four meters less than twice its width.

If the area of the rectangle is 96 m^2, what is the length and the width?

An expression for the length of the rectangle in terms of the width would be Response area

The formula for the area of a rectangle is Response area

Using trial and error, if the area is 96 m^2, then the length and width are Response area

options for response areas:

area = 1/2 (l x w)

l = 4 - 2w

l = 12 & w = 8

l = 8 & w = 12

area = 2l + 2w

l = 2w - 4

w = 2l - 4

l = 24 & w = 4

area = l x w

l = 4 & w = 24

1 answer

Let's solve the problem step by step.

  1. Expression for the length in terms of the width:

    • The length (l) is four meters less than twice the width (w).
    • This can be expressed as:
      \( l = 2w - 4 \)

  2. Formula for the area of a rectangle:

    • The area (A) of a rectangle is given by the formula:
      \( \text{area} = l \times w \)

  3. Set up the equation using the area:

    • We know the area is 96 m², so we can set up the equation:
      \( l \times w = 96 \)
    • Substituting the expression for l from step 1 into this equation:
      \( (2w - 4) \times w = 96 \)

  4. Simplifying the equation:

    • Expanding this gives:
      \( 2w^2 - 4w = 96 \)
    • Rearranging the equation:
      \( 2w^2 - 4w - 96 = 0 \)
    • Dividing all terms by 2:
      \( w^2 - 2w - 48 = 0 \)

  5. Factoring or using the quadratic formula:

    • Factoring the quadratic:
      \( (w - 8)(w + 6) = 0 \)
    • This gives us the solutions:
      \( w = 8 \) or \( w = -6 \) (we reject the negative solution as width cannot be negative)

  6. Finding the length:

    • Using \( w = 8 \) in the expression for length:
      \( l = 2(8) - 4 = 16 - 4 = 12 \)

  7. Final dimensions:

    • Therefore, the dimensions are:
      \( l = 12 \) meters and \( w = 8 \) meters.

Responses:

  • An expression for the length of the rectangle in terms of the width would be:
    l = 2w - 4

  • The formula for the area of a rectangle is:
    area = l x w

  • Using trial and error, if the area is 96 m², then the length and width are:
    l = 12 & w = 8