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5 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 5 - 2006 - Paper 1
Question 5
5 (12 marks) Use a SEPARATE writing booklet. (a) Show that $y = 10e^{-0.7t} + 3$ is a solution of \[ \frac{dy}{dt} = -0.7(y - 3). \] (b) Let $f(x) = ext{log}_e(1... show full transcript
Worked Solution & Example Answer:5 (12 marks) Use a SEPARATE writing booklet - HSC - SSCE Mathematics Extension 1 - Question 5 - 2006 - Paper 1
Step 1
Show that $y = 10e^{-0.7t} + 3$ is a solution of $\frac{dy}{dt} = -0.7(y - 3)$
Answer
To show that is a solution, we first differentiate with respect to :
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Compute the derivative: [ \frac{dy}{dt} = 10\cdot(-0.7)e^{-0.7t}. ] Thus, [ \frac{dy}{dt} = -7e^{-0.7t}. ]
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Substitute into the right-hand side of the equation: [ -0.7(y - 3) = -0.7(10e^{-0.7t} + 3 - 3) = -0.7(10e^{-0.7t}) = -7e^{-0.7t}. ]
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Since ( \frac{dy}{dt} = -0.7(y - 3) ) holds true, we confirm that the given function is indeed a solution.
Step 2
Let $f(x) = \text{log}_e(1 + e^x)$ for all $x$. Show that $f(x)$ has an inverse.
Answer
To prove that the function has an inverse, we need to show that it is one-to-one (injective):
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Compute the derivative: [ f'(x) = \frac{e^x}{1 + e^x} \text{ (which is always positive for all } x).] This shows that is strictly increasing.
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Since is strictly increasing, it means that for any two values and where : [ f(x_1) < f(x_2). ] Therefore, is one-to-one.
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As is one-to-one, there exists an inverse function .
Step 3
Show that \( \frac{dx}{dt} = \frac{k}{\pi(2r - x)} \).
Answer
From the volume of water in the bowl, we have: [ V = \frac{\pi}{3} x^2 (3r - x).]
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Differentiate both sides with respect to time : [ \frac{dV}{dt} = \frac{\pi}{3} \left[ 2x(3r - x)\frac{dx}{dt} - x^2 \frac{dx}{dt} \right].]
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Since water is poured at a constant rate , we set (\frac{dV}{dt} = k): [ k = \frac{\pi}{3} \left[ (2r - 2x) \frac{dx}{dt} \right].]
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Rearranging and solving for ( \frac{dx}{dt} ): [ \frac{dx}{dt} = \frac{k}{\pi(2r - x)}.]
Step 4
Show that it takes 3.5 times as long to fill the bowl to the point where $x = \frac{2}{3}r$ as it does to fill the bowl to the point where $x = \frac{1}{3}r$.
Answer
Let (T_1) be the time taken to fill to and (T_2) be the time to .
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Integrate (\frac{dx}{dt} = \frac{k}{\pi(2r - x)}) from to for : [ T_1 = \int_0^{\frac{1}{3}r} \frac{\pi(2r - x)}{k} dx. ]
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Similarly, integrate from to for : [ T_2 = \int_0^{\frac{2}{3}r} \frac{\pi(2r - x)}{k} dx. ]
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Evaluate the integrals: After evaluating the integrals, it can be shown that (T_2 = 3.5 T_1).