Author Topic: An easy/challenging paper question (Read 1092 times) Tweet Share

Mohammad130no · « **on:** April 13, 2017, 10:25:41 pm »

Some help please?

Its quite challenging

RuiAce · « **Reply #1 on:** April 14, 2017, 07:18:05 am »

$\text{Because the exponential has a horizontal asymptote }y=0\\ \text{It will take the form }y=a^{bx+c}\\ \text{Since the 'doubling' time is constant, we have }a=2\\ \text{Hence }y=2^{bx+c}$
$\text{The points }\left(3, -\frac12\right)\text{ and }(2,4)\text{ lie on this exponential}\\ \text{Upon substitution, we arrive at a pair of simultaneous equations}\\ \begin{align*}\frac12=2^{-1}&=2^{-3b+c}\\ 4=2^2&=2^{2b+c}\end{align*}\\\text{Equating exponents, we have }-1=-3b+c\text{ and }2=2b+c$
$\text{The solution to these simultaneous equations is }b=\frac35, \, c=\frac45\\ \text{Hence the exponential has formula}\\ y=2^{\frac35x+\frac45}$
____________________
$\text{The doubling time can then be found by comparing any other possible }y\text{ value}\\ \text{Here, I will find }x\text{ when }y=1,\text{ and compare to }\left(-3, \frac12\right)$
$\text{When }y=1\\ \begin{align*}1&=2^{\frac35x+\frac45}\\ 0&=\frac35x+\frac45\\ x&= -\frac{4}{3}\end{align*}$
$\text{Hence the doubling time is the time taken to go from }-3\text{ to }-\frac43\\ \text{which is }\frac53$
Note that 5/3 could've been attained by inspection.

Mohammad130no · « **Reply #2 on:** April 14, 2017, 12:02:59 pm »

But what could have been written for a) for the estimate?

jamonwindeyer · « **Reply #3 on:** April 14, 2017, 12:37:28 pm »

Quote from: Mohammad130no on April 14, 2017, 12:02:59 pm

But what could have been written for a) for the estimate?

Probably 2 - If you look at the graph, the $y$-value roughly doubles from $x=0$ to $x=2$, so 2 is a solid estimate of the doubling time (close to $\frac{5}{3}$ too)

RuiAce · « **Reply #4 on:** April 14, 2017, 02:52:39 pm »

Quote from: Mohammad130no on April 14, 2017, 12:02:59 pm

But what could have been written for a) for the estimate?

It's an estimate. You can do it in a variety of ways.

One way would be to grab your ruler and measure it. Your ruler isn't precise but it gives an idea.

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Author Topic: An easy/challenging paper question (Read 1092 times) Tweet Share

Mohammad130no

An easy/challenging paper question

RuiAce

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Mohammad130no

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