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Two similar rectangles, P and Q, are shown below. Rectangle P has a length of 6 cm and a width of 4 cm. Rectangle Q has a length of 15 cm and a width of x cm. Calculate the scale factor of enlargement

Mathematics

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3 parts
b

10 cm\text{10 cm}

c

24 cm2\text{24 cm}^2

d

150 cm2\text{150 cm}^2

Step-by-step solution

5 steps

Part a· 1 step

To find the scale factor of enlargement from Rectangle P to Rectangle Q, we divide a dimension of Rectangle Q by the corresponding dimension of Rectangle P.

  1. Calculate the scale factor using the lengths.

    Scale factor=Length of Rectangle QLength of Rectangle P=15 cm6 cm\text{Scale factor} = \frac{\text{Length of Rectangle Q}}{\text{Length of Rectangle P}} = \frac{15 \text{ cm}}{6 \text{ cm}} Scale factor=156=52=2.5\text{Scale factor} = \frac{15}{6} = \frac{5}{2} = 2.5

    The scale factor is 52\frac{5}{2} or 2.5.

Part b

10 cm\text{10 cm}

Since the rectangles are similar, the ratio of their widths must be equal to the scale factor.

  1. Set up the proportion for the widths.

    Width of Rectangle QWidth of Rectangle P=Scale factor\frac{\text{Width of Rectangle Q}}{\text{Width of Rectangle P}} = \text{Scale factor} x4 cm=52\frac{x}{4 \text{ cm}} = \frac{5}{2}

  2. Solve for xx.

    x=52×4 cmx = \frac{5}{2} \times 4 \text{ cm} x=5×2 cmx = 5 \times 2 \text{ cm} x=10 cmx = 10 \text{ cm}

    The value of xx is 10 cm\boxed{\text{10 cm}}.

Part c

24 cm2\text{24 cm}^2

The area of a rectangle is calculated by multiplying its length by its width.

  1. Calculate the area of Rectangle P.

    Area of Rectangle P=Length×Width\text{Area of Rectangle P} = \text{Length} \times \text{Width} Area of Rectangle P=6 cm×4 cm\text{Area of Rectangle P} = 6 \text{ cm} \times 4 \text{ cm} Area of Rectangle P=24 cm2\text{Area of Rectangle P} = 24 \text{ cm}^2

    The area of Rectangle P is 24 cm2\boxed{\text{24 cm}^2}.

Part d

150 cm2\text{150 cm}^2

Calculate the area of Rectangle Q using its length and the value of xx found in part (b).

  1. Calculate the area of Rectangle Q.

    Area of Rectangle Q=Length×Width\text{Area of Rectangle Q} = \text{Length} \times \text{Width} Area of Rectangle Q=15 cm×10 cm\text{Area of Rectangle Q} = 15 \text{ cm} \times 10 \text{ cm} Area of Rectangle Q=150 cm2\text{Area of Rectangle Q} = 150 \text{ cm}^2

    The area of Rectangle Q is 150 cm2\boxed{\text{150 cm}^2}.

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