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 \\(\square\\) °c per minute.

○ as time increases, the temperature of the metal increases.
the temperature of the metal increases at a rate of \\(\square\\) °c per minute.

(b) what was the temperature of the metal when the researcher started cooling it?
\\(\square\\) °c

Explanation:

Step1: Identify temperature trend

The problem states the metal is cooling, so temperature decreases as time increases. We select the first statement.

Step2: Calculate cooling rate

Find temperature change over time change.
$\text{Rate} = \frac{110 - 390}{40 - 20} = \frac{-280}{20} = -14$
The negative sign means a decrease of $14$ °C per minute.

Step3: Find initial temperature

Use the rate to find temperature at 0 minutes. Start from 20 minutes (390 °C) and add the temperature lost in 20 minutes.
$\text{Initial Temp} = 390 + (14 \times 20) = 390 + 280 = 670$

Answer:

(a) As time increases, the temperature of the metal decreases.
The temperature of the metal decreases at a rate of $14$ °C per minute.
(b) $670$ °C