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A cylindrical pipe with a radius of 12.5 cm is filled with water to a depth, d cm, as shown - HSC - SSCE Mathematics Standard - Question 14 - 2023 - Paper 1

Question 14

A cylindrical pipe with a radius of 12.5 cm is filled with water to a depth, d cm, as shown. The surface of the water has a width of 20 cm. What is the depth of wa... show full transcript

Worked Solution & Example Answer:A cylindrical pipe with a radius of 12.5 cm is filled with water to a depth, d cm, as shown - HSC - SSCE Mathematics Standard - Question 14 - 2023 - Paper 1

Step 1

Determine the Geometry of the Pipe

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Answer

The pipe has a circular cross-section with a radius of 12.5 cm. The width of the water's surface is 20 cm, which indicates that it spans across the pipe's diameter.

From geometry, we know that if a horizontal line cuts through a circle creating a chord, the relationship between the radius (r), the depth of water (d), and half the chord length (c) is:

$c^2 + d^2 = r^2$

Here, the chord length is 20 cm, so half of that is 10 cm, thus we have: $c = 10 ext{ cm}$ .

Step 2

Calculate the Depth of Water (d)

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Answer

Using the formula:

$10^2 + d^2 = 12.5^2$

Substituting the values:

$100 + d^2 = 156.25$

This simplifies to:

$d^2 = 156.25 - 100$ $d^2 = 56.25$

Taking the square root gives:

$d = ext{sqrt}(56.25) = 7.5 ext{ cm}$ .

Thus, the depth of water in the pipe is 7.5 cm, corresponding to option C.

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