题目

Problem

A leaking vessel is modelled by

\begin{align*} V=a\sqrt{t}+b \end{align*}

where $V$ is the volume of water in the vessel, in $\text{m}^3$ , after $t$ minutes, and $a$ and $b$ are constants.

Given that the volume of water in the vessel is $6\text{ m}^3$ after $25$ minutes and $3.3\text{ m}^3$ after $64$ minutes,

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

(4)

(b) Using the model, find

(i) the initial volume of water in the vessel,

(ii) the time taken for the vessel to become empty, giving your answer to the nearest minute.

(3)

解答

展开

把两个时间和体积分别代入模型。注意 $\sqrt{25}=5$ ， $\sqrt{64}=8$ ，所以会得到两个关于 $a,b$ 的一次方程。

展开

Using $t=25$ and $V=6$ ,

\begin{align*} 6=&\,a\sqrt{25}+b\\ 6=&\,5a+b. \end{align*}

Using $t=64$ and $V=3.3$ ,

\begin{align*} 3.3=&\,a\sqrt{64}+b\\ 3.3=&\,8a+b. \end{align*}

Subtract the first equation from the second:

\begin{align*} 3.3-6=&\,(8a+b)-(5a+b)\\ -2.7=&\,3a\\ a=&\,-0.9. \end{align*}

Substitute into $6=5a+b$ :

\begin{align*} 6=&\,5(-0.9)+b\\ 6=&\,-4.5+b\\ b=&\,10.5. \end{align*}

\begin{align*} a=-0.9,\qquad b=10.5. \end{align*}

展开

初始体积对应 $t=0$ 。

容器空了对应 $V=0$ ，把 (a) 的模型代入后解 $\sqrt t$ ，最后平方得到 $t$ 。

展开

The model is

\begin{align*} V=-0.9\sqrt t+10.5. \end{align*}

When $t=0$ ,

\begin{align*} V=&\,-0.9\sqrt0+10.5\\ =&\,10.5. \end{align*}

So the initial volume is

\begin{align*} 10.5\text{ m}^3. \end{align*}

For the vessel to be empty, set $V=0$ :

\begin{align*} -0.9\sqrt t+10.5=&\,0\\ 0.9\sqrt t=&\,10.5\\ \sqrt t=&\,\frac{10.5}{0.9}\\ =&\,\frac{35}{3}. \end{align*}

Therefore

\begin{align*} t=&\,\left(\frac{35}{3}\right)^2\\ =&\,\frac{1225}{9}\\ =&\,136.111\ldots \end{align*}

The vessel becomes empty after

\begin{align*} 136\text{ minutes}. \end{align*}