题目

Problem

The weed on the surface of a pond is being monitored. The surface area of the pond covered by the weed, $A$ m $^2$ , is modelled by the equation

\log_{10} A=1+0.03t

where $t$ is the number of weeks after monitoring began.

Use the equation of the model to answer parts (a) and (b).

(a) Find the surface area of the pond initially covered by the weed.

(1)

After $T$ weeks, $25$ m $^2$ of the pond is covered by the weed.

(b) Find the value of $T$ , giving your answer to 2 decimal places.

(2)

题目中文翻译

正在监测池塘表面的水草。池塘中被水草覆盖的表面积为 $A$ m $^2$ ，模型方程为

\log_{10} A=1+0.03t

其中 $t$ 是从监测开始后经过的周数。

使用该模型方程回答 (a) 和 (b)。

(a) 求最初被水草覆盖的池塘表面积。

经过 $T$ 周后，池塘中有 $25$ m $^2$ 被水草覆盖。

(b) 求 $T$ 的值，答案精确到小数点后 2 位。

解答

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监测开始时 $t=0$ 。代入模型后，把常用对数方程 $\log_{10}A=1$ 改写成指数形式。

展开

Initially, $t=0$ . Therefore,

\log_{10}A=1.

Hence

A=10^1=10.

The initial area covered by the weed is

\boxed{10\ \mathrm{m}^2}.

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把 $A=25$ 和 $t=T$ 代入模型，直接把 $T$ 移到等号一边，再将所得数值四舍五入到小数点后两位。

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When $A=25$ and $t=T$ ,

\log_{10}25=1+0.03T.

Therefore,

\begin{align*} T=&\,\frac{\log_{10}25-1}{0.03}\\ =&\,13.264\ldots. \end{align*}

Hence, to two decimal places,

\boxed{T=13.26\text{ weeks}}.