Question

a researcher is cooling a metal. she cools the metal so that the temperature of the metal drops at a constant rate. after 20 minutes of cooling, the metal is 390 °c. after 40 minutes, the metal is 110 °c. (a) choose the statement that best describes how the time and the temperature of the metal are related. then fill in the blank. as time increases, the temperature of the metal decreases. the temperature of the metal decreases at a rate of 14 °c per minute. as time increases, the temperature of the metal increases. the temperature of the metal increases at a rate of °c per minute. (b) what was the temperature of the metal when the researcher started cooling it? °c

Explanation:

Step1: Calculate the rate of temperature change

The change in temperature is $390 - 110= 280^{\circ}C$, and the change in time is $40 - 20 = 20$ minutes. The rate of change of temperature $r=\frac{390 - 110}{40 - 20}=\frac{280}{20}=14^{\circ}C$ per minute. Since the metal is being cooled, as time increases, the temperature of the metal decreases.

Step2: Find the initial temperature

Let the initial temperature be $T_0$. Using the point - slope form of a linear equation $T=T_0-rt$, where $T$ is the temperature at time $t$, $r$ is the rate of temperature decrease. We know that when $t = 20$ minutes, $T = 390^{\circ}C$ and $r = 14^{\circ}C$ per minute. Substitute into the equation: $390=T_0-14\times20$.

Step3: Solve for the initial temperature

First, calculate $14\times20 = 280$. Then, rewrite the equation as $T_0=390 + 280$. So $T_0 = 670^{\circ}C$.

Answer:

(a) As time increases, the temperature of the metal decreases. The temperature of the metal decreases at a rate of $14^{\circ}C$ per minute.
(b) $670$