AMC8 2022

AMC8 2022 · Q9

AMC8 2022 · Q9. It mainly tests Interest / growth (simple).

A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. Suppose the difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?

一杯沸水（$212^{\circ}\text{F}$）被放置在室温恒定为$68^{\circ}\text{F}$的房间中冷却。假设水温与室温的差每$5$分钟减半。$15$分钟后水温是多少华氏度？

(A) 77 77

(B) 86 86

(D) 98 98

(E) 104 104

Answer

Correct choice: (B)

正确答案：（B）

Solution

Initially, the difference between the water temperature and the room temperature is $212-68=144$ degrees Fahrenheit. After $5$ minutes, the difference between the temperatures is $144\div2=72$ degrees Fahrenheit. After $10$ minutes, the difference between the temperatures is $72\div2=36$ degrees Fahrenheit. After $15$ minutes, the difference between the temperatures is $36\div2=18$ degrees Fahrenheit. At this point, the water temperature is $68+18=\boxed{\textbf{(B) } 86}$ degrees Fahrenheit. Remark Alternatively, we can condense the solution above into the following equation: \[68+(212-68)\cdot\left(\frac12\right)^{\tfrac{15}{5}}=86.\]

最初，水温与室温的差是$212-68=144$华氏度。 $5$分钟后，温差为$144\div2=72$华氏度。 $10$分钟后，温差为$72\div2=36$华氏度。 $15$分钟后，温差为$36\div2=18$华氏度。此时，水温为$68+18=\boxed{\textbf{(B) } 86}$华氏度。备注或者，可以将上述解法浓缩为以下方程：\[68+(212-68)\cdot\left(\frac12\right)^{\tfrac{15}{5}}=86.\]

Topics

AR.010 Interest / growth (simple)