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The areas of two mathematically similar shapes are in the ratio
49:81
The length of the smaller shape is
24.5cm
Work out the length of the larger shape.

The areas of two mathematically similar shapes are in the ratio $49:81$ . The length of the smaller shape is $24.5\,\text{cm}$ . Work out the length of the larger shape.

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Convert between customary and metric systems

Full solution

Q. The areas of two mathematically similar shapes are in the ratio

49:81

. The length of the smaller shape is

24.5\,\text{cm}

. Work out the length of the larger shape.

Understand Relationship: Understand the relationship between the areas of similar shapes and their corresponding lengths. $\newline$ The ratio of the areas of similar shapes is equal to the square of the ratio of their corresponding lengths. This means that if the areas are in the ratio $49:81$ , then the lengths are in the ratio $\sqrt{49}:\sqrt{81}$ , which simplifies to $7:9$ .
Calculate Length of Larger Shape: Calculate the length of the larger shape using the ratio of lengths. $\newline$ We know the length of the smaller shape is $24.5\,\text{cm}$ , which corresponds to the smaller number in the ratio of lengths ( $7$ ). To find the length of the larger shape, we set up a proportion: $\newline$ $\frac{\text{smaller length}}{\text{larger length}} = \frac{\text{smaller ratio number}}{\text{larger ratio number}}$ $\newline$ $\frac{24.5\,\text{cm}}{\text{larger length}} = \frac{7}{9}$
Solve Proportion: Solve the proportion for the length of the larger shape. $\newline$ $(\text{larger length}) = \frac{24.5 \, \text{cm} \times 9}{7}$ $\newline$ $(\text{larger length}) = \frac{220.5 \, \text{cm}}{7}$ $\newline$ $(\text{larger length}) = 31.5 \, \text{cm}$

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