Module: Foundation Mathematics I (ITE3705)

Unit: GCNU303A – Solve problems related to simple exponential and logarithmic functions

Supplementary Tutorial Exercise / 補充導修練習題

---

Questions / 題目

Section 1: Rational Indices / 有理指數

Question 1

Simplify the following expressions and give your answers with positive indices:

化簡下列各式，並以正指數表示答案：

• (a) ${\displaystyle \frac{a^3\cdot a^{-7}}{a^{-2}}}$a−2a3⋅a−7​
• (b) ${\displaystyle \frac{(x^{-2}{y}^4{)}^3}{x^2{y}^{-5}}}$x2y−5(x−2y4)3​
• (c) ${\left({\displaystyle \frac{16u^8}{v^{-4}}}\right)}^{-\frac{3}{4}}$(v−416u8​)−43​

Question 2

In an anime combat game, a hero’s damage amplification formula is modeled by the expression ${\displaystyle \frac{(p^2{q}^{-3}{)}^{-2}\cdot p^{-1}{q}^4}{p^{-3}{q}^2}}$p−3q2(p2q−3)−2⋅p−1q4​. Simplify this expression and write the answer with positive indices.

在一款動漫風格的戰鬥遊戲中，英雄的傷害增幅公式以算式 ${\displaystyle \frac{(p^2{q}^{-3}{)}^{-2}\cdot p^{-1}{q}^4}{p^{-3}{q}^2}}$p−3q2(p2q−3)−2⋅p−1q4​ 表示。請化簡此算式，並以正指數表示答案。

---

Section 2: Exponential Equations / 指數方程

Question 3

Solve the equation: $4^{x+1}={\left({\displaystyle \frac{1}{32}}\right)}^{x-2}$4x+1=(321​)x−2.

解方程：$4^{x+1}={\left({\displaystyle \frac{1}{32}}\right)}^{x-2}$4x+1=(321​)x−2。

Question 4

Solve the equation: $3^{2x-1}\times {27}^{x+2}=81$32x−1×27x+2=81.

解方程：$3^{2x-1}\times {27}^{x+2}=81$32x−1×27x+2=81。

---

Section 3: Population & Environmental Growth Models / 人口與環境增長模型

Question 5

The active player count $P$P (in thousands) of a Hong Kong mobile esports game $n$n months after its official launch is approximated by $P=120(1.05{)}^n$P=120(1.05)n.

一款香港手遊電競項目在正式推出 $n$n 個月後的活躍玩家人數 $P$P（以千人計）由公式 $P=120(1.05{)}^n$P=120(1.05)n 估算。

• (a) Predict the active player count, in thousands, after 12 months. (Give your answer correct to 1 decimal place.)

預測 12 個月後的活躍玩家人數（以千人計）。（答案須準確至小數點後 1 位。）

• (b) In how many months will the active player count double its initial value? (Give your answer correct to 2 decimal places.)

多少個月後活躍玩家人數會增加至初始值的一倍（即翻倍）？（答案須準確至小數點後 2 位。）

Question 6

The height $h$h (in meters) of an ornamental bamboo planted outside the Hong Kong Cultural Centre $t$t years after planting is given by $h(t)=16-A{e}^{-0.25t}$h(t)=16−Ae−0.25t, where $A$A is a constant. Its initial planted height was 2 m.

種植於香港文化中心外的一株觀賞竹，在種植 $t$t 年後的高度 $h$h（米）由公式 $h(t)=16-A{e}^{-0.25t}$h(t)=16−Ae−0.25t 給出，其中 $A$A 為常數。該竹樹初種植時的高度為 2 米。

• (a) Find the value of $A$A.

求 $A$A 的值。

• (b) Complete the following table (correct to 1 decimal place):

完成下表（答案準確至小數點後 1 位）：

$t$t	0	2	4	6	8	10
$h(t)$h(t)	2

• (c) In how many years will the bamboo reach a height of 12 m? (Give your answer correct to 2 decimal places.)

多少年後竹樹的高度會生長至 12 米？（答案須準確至小數點後 2 位。）

---

Section 4: Financial Applications & Continuous Decay / 財務應用與連續衰減

Question 7

An engineering student bought a high-end graphic workstation in Sham Shui Po for HK$24,000 for 3D animation rendering. Its market value $V$V depreciates continuously according to the formula $V(t)=C{e}^{-0.28t}$V(t)=Ce−0.28t, where $t$t is the time in years and $C$C is a constant.

一名工程系學生在深水埗以 24,000 港元購入一部高階繪圖工作站作 3D 動畫渲染之用。其市場價值 $V$V 依公式 $V(t)=C{e}^{-0.28t}$V(t)=Ce−0.28t 持續折舊，其中 $t$t 為年數，而 $C$C 為常數。

• (a) State the value of $C$C.

寫出 $C$C 的值。

• (b) Find the market value of the workstation after 3 years. (Give your answer correct to the nearest integer.)

求該工作站 3 年後的市場價值。（答案須準確至最接近的整數。）

• (c) After how many years will the market value drop to 25% of its original purchase price? (Give your answer correct to 2 decimal places.)

多少年後該工作站的市場價值會下降至原價的 25%？（答案須準確至小數點後 2 位。）

Question 8

A commuter deposits HK$15,000 into an automated Octopus-linked high-yield savings scheme that pays 3.6% annual interest compounded monthly ($A=P{(1+{\displaystyle \frac{r\mathrm{\%}}{12}})}^{12t}$A=P(1+12r%​)12t)[cite: 1]. How many years will it take for the account balance to reach HK$20,000? (Give your answer correct to 1 decimal place.)

一名通勤者將 15,000 港元存入一項與八達通自動增值關聯的高息儲蓄計劃，年利率為 3.6%，按月計算複利（$A=P{(1+{\displaystyle \frac{r\mathrm{\%}}{12}})}^{12t}$A=P(1+12r%​)12t）。該戶口結餘需要多少年才能累積至 20,000 港元？（答案須準確至小數點後 1 位。）

---