A scientist records the temperature of a gas compound inside a chamber as it is heated at … — ACT

A scientist records the temperature of a gas compound inside a chamber as it is heated at a constant rate. At a time of $t = 3$ minutes, the temperature of the compound is $-8^\circ\text{C}$ . At $t = 15$ minutes, the temperature is $28^\circ\text{C}$ . What is the rate of change of the temperature with respect to time, in degrees Celsius per minute, of the compound?

A
$\frac{1}{3}$
B
$-3$
$3$ Answer
D
$\frac{5}{3}$
E
$\frac{3}{5}$

Answer

The rate of change of the temperature of the compound is

3

degrees Celsius per minute.

The rate of change is equivalent to the slope of the line passing through the points

(3, -8)

and

(15, 28)

. Using the slope formula, we compute the ratio of the change in temperature to the change in time:

\frac{28 - (-8)}{15 - 3} = \frac{36}{12} = 3

degrees Celsius per minute.

Step-by-Step Solution

Identify the coordinate points from the given information.

The two points are

(3, -8)

and

(15, 28)

, where the first coordinate represents time

t

in minutes and the second coordinate represents temperature

T

in degrees Celsius.

To find the constant rate of change, we need to determine the change in temperature per unit of time, which corresponds to the slope between these two points.

Apply the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

to calculate the rate of change.

Substituting the coordinates into the formula gives

\frac{28 - (-8)}{15 - 3}

The slope formula measures the vertical change (rise) over the horizontal change (run).

Simplify the expression to find the final value.

\frac{28 + 8}{12} = \frac{36}{12} = 3

Simplifying the numerator and denominator gives the constant rate of change.

Key Concept

The slope of a line represents the constant rate of change of the dependent variable with respect to the independent variable.

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