题目

Problem

A tree was planted.

Exactly 3 years after it was planted, the height of the tree was $2$ m.

Exactly 5 years after it was planted, the height of the tree was $2.4$ m.

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

\begin{align*} H^3=pt^2+q \end{align*}

where $p$ and $q$ are constants,

(a) find, to 3 significant figures where necessary, the value of $p$ and the value of $q$ .

(4)

Exactly $T$ years after the tree was planted, its height was $5$ m.

(b) Find the value of $T$ according to the model, giving your answer to one decimal place.

(2)

解答

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把两个给定时间和高度分别代入模型，得到两个关于 $p,q$ 的一次方程，再联立求解。

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When $t=3$ , $H=2$ :

\begin{align*} 2^3=&\,p(3)^2+q\\ 8=&\,9p+q. \end{align*}

When $t=5$ , $H=2.4$ :

\begin{align*} 2.4^3=&\,p(5)^2+q\\ 13.824=&\,25p+q. \end{align*}

Subtract the first equation from the second:

\begin{align*} 13.824-8=&\,16p\\ 5.824=&\,16p\\ p=&\,0.364. \end{align*}

Then

\begin{align*} 8=&\,9(0.364)+q\\ q=&\,4.724. \end{align*}

So, to 3 significant figures,

\begin{align*} p=0.364,\qquad q=4.72. \end{align*}

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高度为 $5$ 时，令 $H=5$ ，用 (a) 的模型求 $T$ 。

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When $H=5$ ,

\begin{align*} 5^3=&\,pT^2+q\\ 125=&\,0.364T^2+4.724\\ T^2=&\,\frac{125-4.724}{0.364}\\ T=&\,18.18\ldots. \end{align*}

Therefore

\begin{align*} T=18.2 \end{align*}

to one decimal place.