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P3 2023 Oct Q6

A Level / Edexcel / P3

IAL 2023 Oct Paper · Question 6

题目

Problem

An area of sea floor is being monitored.

The area of the sea floor, SS km2^2, covered by coral reefs is modelled by the equation

S=pqtS=pq^t

where pp and qq are constants and tt is the number of years after monitoring began.

Given that

log⁡10S=4.5−0.006t\log_{10}S=4.5-0.006t

(a) find, according to the model, the area of sea floor covered by coral reefs when t=2t=2

(2)

(b) find a complete equation for the model in the form

S=pqtS=pq^t

giving the value of pp and the value of qq each to 3 significant figures.

(3)

(c) With reference to the model, interpret the value of the constant qq

(1)
题目中文翻译

某片海床区域正在被监测。

被珊瑚礁覆盖的海床面积为 SS km2^2，其模型方程为

S=pqtS=pq^t

其中 pp 和 qq 是常数，tt 是开始监测后的年数。

已知

log⁡10S=4.5−0.006t\log_{10}S=4.5-0.006t

(a) 根据该模型，求当 t=2t=2 时被珊瑚礁覆盖的海床面积。

(b) 求该模型的完整方程

S=pqtS=pq^t

并将 pp 和 qq 的值都保留 3 位有效数字。

(c) 结合该模型，解释常数 qq 的含义。

解答

(a)

解法一

思路

展开

把 t=2t=2 代入对数模型，先求出 log⁡10S\log_{10}S，再以 1010 为底取幂求出 SS。答案需要带面积单位。

答题过程

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When t=2t=2,

log⁡10S= 4.5−0.006(2)= 4.488.\begin{align*} \log_{10}S =&\,4.5-0.006(2)\\[4mm] =&\,4.488. \end{align*}

Therefore,

S= 104.488= 30760.1…\begin{align*} S=&\,10^{4.488}\\[4mm] =&\,30760.1\ldots \end{align*}

Hence, according to the model, the area covered by coral reefs is approximately

30800 km2.\boxed{30800\text{ km}^2}.

(b)

解法一

思路

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由 log⁡10S\log_{10}S 的表达式直接写出 SS，再利用指数定律把含 tt 的部分整理成某个常数的 tt 次方，便可与 S=pqtS=pq^t 对照。

答题过程

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From

log⁡10S=4.5−0.006t,\log_{10}S=4.5-0.006t,

it follows that

S= 104.5−0.006t= 104.5×10−0.006t= 104.5(10−0.006)t.\begin{align*} S=&\,10^{4.5-0.006t}\\[4mm] =&\,10^{4.5}\times 10^{-0.006t}\\[4mm] =&\,10^{4.5}\bigl(10^{-0.006}\bigr)^t. \end{align*}

Now,

104.5=31622.7…10^{4.5}=31622.7\ldots

and

10−0.006=0.986279…10^{-0.006}=0.986279\ldots

Therefore, to 33 significant figures,

p=31600andq=0.986.p=31600 \qquad\text{and}\qquad q=0.986.

Hence the complete model is

S=31600(0.986)t.\boxed{S=31600(0.986)^t}.

(c)

解法一

思路

展开

在指数模型 S=pqtS=pq^t 中，qq 是时间每增加一年时面积所乘的比例。由于 q=0.986q=0.986，每年保留上一年的 98.6%98.6\%，等价于每年减少约 1.4%1.4\%。

答题过程

展开

The value q=0.986q=0.986 means that, according to the model, the area of sea floor covered by coral reefs is multiplied by 0.9860.986 each year.

Therefore, approximately 98.6%98.6\% of the previous year’s covered area remains each year, which is equivalent to an annual decrease of approximately

100%−98.6%=1.4%.100\%-98.6\%=1.4\%.