题目

Problem

A tree was planted in the ground.

Exactly 2 years after it was planted, the height of the tree was $1.85$ m.

Exactly 7 years after it was planted, the height of the tree was $3.45$ m.

Given that the height, $H$ metres, of the tree, $t$ years after it was planted in the ground, can be modelled by the equation

\begin{align*} H=at+b \end{align*}

where $a$ and $b$ are constants,

(a) find the value of $a$ and the value of $b$ .

(4)

(b) State, according to the model, the height of the tree when it was planted.

(1)

解答

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把两个已知时间和高度代入线性模型，得到两个关于 $a,b$ 的方程，再联立求解。

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Using $H=at+b$ ,

\begin{align*} 1.85=&\,2a+b,\\ 3.45=&\,7a+b. \end{align*}

Subtract the first equation from the second:

\begin{align*} 1.60=&\,5a\\ a=&\,0.32. \end{align*}

Then

\begin{align*} 1.85=&\,2(0.32)+b\\ b=&\,1.21. \end{align*}

Therefore

\begin{align*} a=0.32,\qquad b=1.21. \end{align*}

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刚种下时 $t=0$ ，所以高度就是模型中的截距 $b$ 。

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When the tree was planted, $t=0$ , so

\begin{align*} H=0.32(0)+1.21=1.21. \end{align*}

The height was $1.21$ m.