题目

Problem

Figure 1 shows a linear relationship between $\log_{10}y$ and $\log_{10}x$ .

The line passes through the points $(0,4)$ and $(6,0)$ as shown.

(a) Find an equation linking $\log_{10}y$ with $\log_{10}x$ .

(2)

(b) Hence, or otherwise, express $y$ in the form $px^q$ , where $p$ and $q$ are constants to be found.

(3)

题目中文翻译

图 1 显示了 $\log_{10}y$ 与 $\log_{10}x$ 的线性关系。

如图所示，该直线经过点 $(0,4)$ 和 $(6,0)$ 。

(a) 求连接 $\log_{10}y$ 与 $\log_{10}x$ 的方程。

(b) 由此，或用其他方法，将 $y$ 写成 $px^q$ 的形式，其中 $p$ 和 $q$ 为待求常数。

解答

The line passes through

(0,4)\quad \text{and}\quad (6,0)

The vertical axis is $\log_{10}y$ and the horizontal axis is $\log_{10}x$ .

The gradient is

\frac{0-4}{6-0}=-\frac{4}{6}=-\frac23

The intercept on the $\log_{10}y$ axis is $4$ .

Therefore

\boxed{\log_{10}y=4-\frac23\log_{10}x}

From part (a),

\log_{10}y=4-\frac23\log_{10}x

Use the log rule

a\log_{10}x=\log_{10}(x^a)

\log_{10}y=4+\log_{10}\left(x^{-\frac23}\right)

Also,

4=\log_{10}(10^4)

Therefore

\log_{10}y=\log_{10}(10^4)+\log_{10}\left(x^{-\frac23}\right)

\log_{10}y=\log_{10}\left(10^4x^{-\frac23}\right)

Hence

y=10^4x^{-\frac23}

Therefore

\boxed{y=10000x^{-\frac23}}

\boxed{p=10000,\qquad q=-\frac23}