A commercial coffee roaster heats a large batch of coffee beans. The temperature T, in deg… — SAT

A commercial coffee roaster heats a large batch of coffee beans. The temperature $T$ , in degrees Celsius ( $^\circ\text{C}$ ), of the beans $m$ minutes after heating begins is modeled by the equation $T = 18m + 31$ , where $m \le 10$ . Which of the following is the best interpretation of $18$ in this context?

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.Answer
B
The initial temperature of the beans, in degrees Celsius, when heating begins.
C
The number of minutes it takes for the temperature of the beans to increase by $1^\circ\text{C}$ .
D
The temperature of the beans, in degrees Celsius, after $18$ minutes of heating.

Answer

The increase in the temperature of the beans, in degrees Celsius, for each minute of heating.

The equation is in the slope-intercept form

T = 18m + 31

, where

18

is the slope and

31

is the y-intercept. In this context,

T

represents the temperature of the beans and

m

represents the time in minutes. The slope,

18

, represents the rate of change of the temperature with respect to time, which means the temperature increases by

18^\circ\text{C}

for each minute of heating.

Step-by-Step Solution

Identify the component of the linear equation being questioned.

The number

18

is the coefficient of the variable

m

in the linear equation

T = 18m + 31

We need to determine what role the coefficient of the independent variable plays in a linear model.

Determine the mathematical meaning of the coefficient in a linear equation of the form

y = mx + b

The coefficient of

m

, which is

18

, represents the slope or the rate of change of

T

with respect to

m

In a linear function, the slope represents the change in the dependent variable per unit change in the independent variable.

Interpret the rate of change in the context of the problem.

Since

T

is the temperature in degrees Celsius (

^\circ\text{C}

) and

m

is the time in minutes, a rate of change of

18

means that the temperature increases by

18^\circ\text{C}

for every

1

minute of heating.

This matches the definition of the slope in the given context.

Key Concept

Interpreting the slope of a linear relationship in context

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic