A rectangular garden has a length that is 5 meters longer than its width. If the width is represented by w meters: i. Write an expression for the length of the garden. ii. Write an expression for the perimeter of the garden. iii. Write an expression for the area of the garden. c. If the width of the garden is 8 meters, calculate the perimeter and area.

Mathematics

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Answers

4 parts
i

w+5 metersw + 5 \text{ meters}

ii

4w+10 meters4w + 10 \text{ meters}

iii

w2+5w square metersw^2 + 5w \text{ square meters}

c

42 meters42 \text{ meters}

104 square meters104 \text{ square meters}

Step-by-step solution

10 steps

Part i· 2 steps

Write an expression for the length of the garden.

  1. Define the given width.

    The width of the garden is represented by ww meters.

  2. Formulate the length based on the problem description.

    The length is 5 meters longer than its width.

    Length=width+5\text{Length} = \text{width} + 5 L=w+5L = w + 5

    The expression for the length is w+5 meters\boxed{w + 5 \text{ meters}}.

Part ii

4w+10 meters4w + 10 \text{ meters}

Write an expression for the perimeter of the garden.

  1. Recall the formula for the perimeter of a rectangle.

    The perimeter PP of a rectangle is given by P=2(length+width)P = 2(\text{length} + \text{width}).

  2. Substitute the expressions for length and width into the formula.

    We have L=w+5L = w + 5 and width =w= w.

    P=2((w+5)+w)P = 2((w + 5) + w)
  3. Simplify the expression.

    P=2(2w+5)P = 2(2w + 5) P=4w+10P = 4w + 10

    The expression for the perimeter is 4w+10 meters\boxed{4w + 10 \text{ meters}}.

Part iii

w2+5w square metersw^2 + 5w \text{ square meters}

Write an expression for the area of the garden.

  1. Recall the formula for the area of a rectangle.

    The area AA of a rectangle is given by A=length×widthA = \text{length} \times \text{width}.

  2. Substitute the expressions for length and width into the formula.

    We have L=w+5L = w + 5 and width =w= w.

    A=(w+5)wA = (w + 5)w
  3. Simplify the expression.

    A=w2+5wA = w^2 + 5w

    The expression for the area is w2+5w square meters\boxed{w^2 + 5w \text{ square meters}}.

Part c

42 meters42 \text{ meters}

If the width of the garden is 8 meters, calculate the perimeter and area.

  1. Substitute w=8w = 8 meters into the perimeter expression from part (ii).

    P=4w+10P = 4w + 10 P=4(8)+10P = 4(8) + 10 P=32+10P = 32 + 10 P=42 metersP = 42 \text{ meters}

    The perimeter is 42 meters\boxed{42 \text{ meters}}.

  2. Substitute w=8w = 8 meters into the area expression from part (iii).

    A=w2+5wA = w^2 + 5w A=(8)2+5(8)A = (8)^2 + 5(8) A=64+40A = 64 + 40 A=104 square metersA = 104 \text{ square meters}

    The area is 104 square meters\boxed{104 \text{ square meters}}.

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