Question 108·Hard·Linear Functions
The function gives the temperature, in degrees Celsius, of a reactor when the control knob is at position millimeters from the fully closed position. During a calibration, the technician slowly moves the knob from millimeters to millimeters. According to the model, by how many degrees Celsius does the reactor temperature change?
For linear function questions asking how much the output changes between two inputs, you can usually skip finding both function values separately. Instead, recognize that the coefficient of the variable (the slope) is the rate of change: multiply this rate by the difference between the two input values to get the total change, then interpret the sign and magnitude to match the wording of the question (for example, "by how many" often means use the absolute value).
Hints
Focus on the change in knob position
You are not being asked for the temperature at one position, but how much the temperature changes as the knob moves. First, find how many millimeters the knob moves by subtracting the starting position from the ending position.
Use the rate from the linear function
In , the number tells you how many degrees Celsius the temperature changes for each 1 millimeter change in knob position. How can you use this with the total distance the knob moves?
Connect total change to rate × distance
Once you know the knob moves a certain number of millimeters, multiply that distance by to get the total temperature change. Then think about what "by how many degrees" is asking you to report.
Desmos Guide
Compute the temperature change directly
In Desmos, type the expression -4.7*(27.5 - 12) and look at the output; this value is the change in temperature (it will be negative because the temperature decreases), and the number of degrees the temperature changes is the absolute value of that result.
Step-by-step Explanation
Interpret the function and the rate
The temperature function is .
- The coefficient is the rate of change: for each millimeter increase in , the temperature changes by degrees Celsius (it decreases by degrees).
- So if the knob position changes by some amount , then the temperature change is given by
Find how much the knob position changes
The technician moves the knob from millimeters to millimeters.
Compute the change in position:
Compute the change in temperature
Use the rate of change with :
Multiply:
The negative sign means the temperature decreases by degrees Celsius, so the reactor temperature changes by degrees Celsius.
Answer: 72.85.