题目

Problem

The number of subscribers to an online video streaming service, $N$ , is modelled by the equation

N=ab^t

where $a$ and $b$ are constants and $t$ is the number of years since monitoring began.

The line in Figure 1 shows the linear relationship between $t$ and $\log_{10}N$

The line passes through the points $(0,3.08)$ and $(5,3.85)$

Using this information,

(a) find an equation for this line.

(2)

(b) Find the value of $a$ and the value of $b$ , giving your answers to 3 significant figures.

(3)

When $t=T$ the number of subscribers is $500\,000$

According to the model,

(2)

题目中文翻译

某在线视频流服务的订阅人数 $N$ 由方程

N=ab^t

建模，其中 $a$ 和 $b$ 是常数， $t$ 是开始监测后的年数。

图 1 中的直线表示 $t$ 与 $\log_{10}N$ 之间的线性关系。

该直线经过点 $(0,3.08)$ 和 $(5,3.85)$ 。

利用这些信息，

(a) 求这条直线的方程。

(b) 求 $a$ 和 $b$ 的值，答案都保留 3 位有效数字。

当 $t=T$ 时，订阅人数为 $500\,000$ 。

根据该模型，

解答

(a)

The line passes through

(0,3.08)

and

(5,3.85)

Its gradient is

\frac{3.85-3.08}{5-0} =\frac{0.77}{5} =0.154

Since the intercept is $3.08$ , the equation of the line is

\boxed{\log_{10}N=3.08+0.154t}

(b)

We are given

N=ab^t

Taking $\log_{10}$ of both sides,

\log_{10}N=\log_{10}a+t\log_{10}b

Compare this with

\log_{10}N=3.08+0.154t

Therefore

\log_{10}a=3.08

a=10^{3.08}=1202.26\ldots

Hence

\boxed{a=1200}

to 3 significant figures.

Also,

\log_{10}b=0.154

b=10^{0.154}=1.425\ldots

Hence

\boxed{b=1.43}

to 3 significant figures.

(c)

When

N=500000

use

\log_{10}N=3.08+0.154t

\log_{10}(500000)=3.08+0.154T

Therefore

0.154T=\log_{10}(500000)-3.08

and hence

T=\frac{\log_{10}(500000)-3.08}{0.154}

Thus

T=17.0\ldots

\boxed{T=17}

IAL 2022 June Q4

题目

解答

(a)

(b)

(c)