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Two bottles are mathematically similar - OCR - GCSE Maths - Question 15 - 2023 - Paper 6

Question 15

Two bottles are mathematically similar. The small bottle holds 0.5 litres and has a height of 35 cm. The large bottle holds 2 litres. Calculate the height of the l... show full transcript

Worked Solution & Example Answer:Two bottles are mathematically similar - OCR - GCSE Maths - Question 15 - 2023 - Paper 6

Step 1

Calculate the Volume Ratio

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Answer

The volume of the small bottle is 0.5 litres, and the volume of the large bottle is 2 litres. The ratio of their volumes can be calculated as follows:

$\text{Volume Ratio} = \frac{\text{Volume of large bottle}}{\text{Volume of small bottle}} = \frac{2}{0.5} = 4$

Step 2

Relate the Volume Ratio to the Height Ratio

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Answer

Since the bottles are mathematically similar, the ratio of their volumes is equal to the cube of the ratio of their corresponding dimensions (heights).

Let ( h_s ) be the height of the small bottle (35 cm), and ( h_l ) be the height of the large bottle. Therefore:

$\frac{\text{Volume Ratio}}{1} = \left(\frac{h_l}{h_s}\right)^3$

This can be expressed as:

$4 = \left(\frac{h_l}{35}\right)^3$

Step 3

Solve for the Height of the Large Bottle

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Taking the cube root of both sides gives:

$\frac{h_l}{35} = \sqrt[3]{4}$

This means:

$h_l = 35 \cdot \sqrt[3]{4}$

Using a calculator, ( \sqrt[3]{4} \approx 1.5874 ), we can find:

$h_l \approx 35 \cdot 1.5874 \approx 55.56 \text{ cm}$

Thus, the height of the large bottle is approximately 55.56 cm.

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