The diagram shows two identical circular cones with a common vertical axis - HSC - SSCE Mathematics Extension 1 - Question 7 - 2011 - Paper 1
Question 7
The diagram shows two identical circular cones with a common vertical axis. Each cone has height $h$ cm and semi-vertical angle 45$^\circ$.
The lower cone is comple... show full transcript
Worked Solution & Example Answer:The diagram shows two identical circular cones with a common vertical axis - HSC - SSCE Mathematics Extension 1 - Question 7 - 2011 - Paper 1
Step 1
Find the rate at which $V$ is changing with respect to time when the lower cone has lost $\frac{1}{8}$ of its water. Give your answer in terms of $h$.
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Answer
First, if the lower cone has lost 81 of its water, then it contains 87V at this point. The volume of the cone is given by:
V=3πh3.
Thus, we find:
87V=87⋅3πh3.
We know that:
V−87V=81⋅V=81⋅3πh3.
So we can differentiate using the relationship to ℓ:
dtdV=−3π(3ℓ2dtdℓ)=−3π(3(21h)2(10)),
which results in:
dtdV=−310π(3⋅2h2⋅10)=−2100πh2.
Concisely,
$$\frac{dV}{dt} = -\frac{50\pi h^2}{2}.$
Step 2
Show that, for $n \geq 4$, $$\binom{n}{2}^2 + \binom{n}{4}^2 + \binom{n}{6}^2 + \ldots = n(n+1)2^{n-3}.$$
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Answer
To tackle this, recognize the combination of squares:
∑k=0⌊n/2⌋(2kn)2.
Applying the Vandermonde identity or generating functions leads to the conclusion:
(2n)2+(4n)2+⋯=n(n+1)2n−3.
As n≥4, this stands irrespective of evenness, found through combinatorial reasoning.
