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A container is made in the shape of a hollow inverted right circular cone - Edexcel - A-Level Maths: Pure - Question 6 - 2009 - Paper 3
Question 6
A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm, as shown in Figure 2. Water ... show full transcript
Worked Solution & Example Answer:A container is made in the shape of a hollow inverted right circular cone - Edexcel - A-Level Maths: Pure - Question 6 - 2009 - Paper 3
Step 1
Show that V = \frac{4}{27} \pi h^3
Answer
To find the volume V of the water in the conical container, we first establish the relationship between the radius r and height h using similar triangles. The full height of the cone is 24 cm, and the radius of the base is 16 cm. Thus, we have:
From this, we can express r in terms of h:
Substituting this into the volume formula of a cone, we get:
Now expanding this:
This confirms that .
Step 2
Find, in terms of π, the rate of change of h when h = 12.
Answer
Given that water flows into the container at a rate of 8 cm³ s⁻¹, we have:
We need to find the rate of change of height h with respect to time. First, we differentiate the volume formula with respect to time:
Now, we compute the derivative of V with respect to h:
Substituting this back into the rate of change equation:
To find ( \frac{dh}{dt} ) when h = 12:
Calculating ( 12^2 = 144 ):
Solving for ( \frac{dh}{dt} ):
Thus, the rate of change of h when h = 12 is (\frac{1}{8\pi}) cm s⁻¹.