题目

Problem

The number of dormice and the number of voles on an island are being monitored.

Initially there are $2000$ dormice on the island.

A model predicts that the number of dormice will increase by $3\%$ each year, so that the numbers of dormice on the island at the end of each year form a geometric sequence.

(a) Find, according to the model, the number of dormice on the island $6$ years after monitoring began. Give your answer to $3$ significant figures.

(2)

The number of voles on the island is being monitored over the same period of time.

Given that

$4$ years after monitoring began there were $3690$ voles on the island
$7$ years after monitoring began there were $3470$ voles on the island
the number of voles on the island at the end of each year is modelled as a geometric sequence

(b) find the equation of this model in the form

N=ab^t

where $N$ is the number of voles, $t$ years after monitoring began and $a$ and $b$ are constants.

Give the value of $a$ and the value of $b$ to $2$ significant figures.

(3)

When $t=T$ , the number of dormice on the island is equal to the number of voles on the island.

(3)

解答

(a)

解法一

思路

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每年增加 $3\%$ ，所以公比是 $1.03$ 。6 年后是 $2000(1.03)^6$ 。

答题过程

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The model for the dormice is

\begin{align*} D=2000(1.03)^t. \end{align*}

After $6$ years,

\begin{align*} D=&\,2000(1.03)^6\\ =&\,2388.10\ldots. \end{align*}

To $3$ significant figures, the number of dormice is

\begin{align*} 2390. \end{align*}

(b)

解法一

思路

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对田鼠模型 $N=ab^t$ ，把 $t=4$ 和 $t=7$ 代入。两式相除可以先求出 $b^3$ ，再求 $b$ 和 $a$ 。

答题过程

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For the voles,

\begin{align*} N=ab^t. \end{align*}

Using the data,

\begin{align*} 3690=&\,ab^4,\\ 3470=&\,ab^7. \end{align*}

Divide the second equation by the first:

\begin{align*} \frac{3470}{3690}=&\,\frac{ab^7}{ab^4}\\ b^3=&\,\frac{3470}{3690}\\ b=&\,\left(\frac{3470}{3690}\right)^{\frac13}\\ =&\,0.9797\ldots. \end{align*}

Now

\begin{align*} a=&\,\frac{3690}{b^4}\\ =&\,\frac{3690}{(0.9797\ldots)^4}\\ =&\,4000.0\ldots. \end{align*}

To $2$ significant figures,

\begin{align*} a=4000,\qquad b=0.98. \end{align*}

Therefore, the model is

\begin{align*} N=4000(0.98)^t. \end{align*}

(c)

解法一

思路

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令睡鼠模型和田鼠模型相等。因为未知数在指数上，需要取对数。

答题过程

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When the two populations are equal,

\begin{align*} 2000(1.03)^T=4000(0.98)^T. \end{align*}

Divide by $2000(0.98)^T$ :

\begin{align*} \left(\frac{1.03}{0.98}\right)^T=&\,2. \end{align*}

Take logarithms:

\begin{align*} T\log\left(\frac{1.03}{0.98}\right)=&\,\log 2\\ T=&\,\frac{\log 2}{\log\left(\frac{1.03}{0.98}\right)}\\ =&\,13.9\ldots. \end{align*}

Therefore,

\begin{align*} T=13.9. \end{align*}

IAL 2024 May Q10

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程

(c)

解法一

思路

答题过程