SSCE HSC Mathematics Extension 1 Solving differential equations using separation of variables: 5 (12 marks) Use a SEPARATE writ

Question

5 (12 marks) Use a SEPARATE writing booklet.

(a) Show that $y = 10 e^{-0.07t} + 3$ is a solution of 
$$rac{dy}{dt} = -0.7(y - 3).$$

(b) Let $f(x) = 	ext{log}_e(1 + e^x)$ for all $x$. Show that $f(x)$ has an inverse.

(c) A hemispherical bowl of radius $r 	ext{ cm}$ is initially empty. Water is poured into it at a constant rate of $k 	ext{ cm}^3$ per minute. When the depth of water in the bowl is $x 	ext{ cm}$, the volume, $V 	ext{ cm}^3$, of water in the bowl is given by

$$ V = rac{rac{m{π}}{3}x^2(3r - x)}{(r^2 - x^2)}. 	ext{ (Do NOT prove this.)} $$

(i) Show that $$rac{dx}{dt} = rac{k}{rac{m{π}}{1 - x}}$$ for $(2r - x)$.

(ii) Hence, or otherwise, show that it takes 3.5 times as long to fill the bowl to the point where $x = rac{2}{3} r$ as it does to fill the bowl to the point where $x = rac{1}{3} r$.

(d) (i) Use the fact that $	an(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$ to show that

$$1 + \tan \theta \tan(n + 1)\theta = \cot\left( (n + 1)\theta \right) - \tan(n + 1)\theta.$$

(ii) Use mathematical induction to prove that, for all integers $n \geq 1$,

$$ \tan 10\theta + 20\tan 30\theta + \ldots + \tan(n + 1)\theta = -(n + 1)\theta + \cot(n + 1)\theta.$

5 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 5 - 2006 - Paper 1

Worked Solution & Example Answer:5 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 5 - 2006 - Paper 1

Show it takes 3.5 times as long to fill the bowl to $x = \frac{2}{3}r$ compared to $x = \frac{1}{3}r$

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