Question 137·Medium·Linear Equations in One Variable
A meteorologist observed that the air temperature fell at a constant rate of degrees Celsius per hour for consecutive hours. At the end of this period, the temperature was degrees Celsius.
What was the air temperature, in degrees Celsius, at the beginning of the 10-hour period?
(Express the answer as an integer)
For linear word problems with a constant rate, first compute the total change using rate × time, being careful with the sign (a fall or decrease is negative). Then define a variable for the starting amount, write an equation of the form final = start + change, and solve for the start by undoing the operation. Always do a quick reasonableness check: if the quantity is falling over time, the starting value must be greater than the final value.
Hints
Think about the total change
If the temperature falls degrees each hour for hours, what single number represents the total change in temperature over the whole period?
Relate starting and ending temperatures
Let be the starting temperature. How can you write an equation that connects , the total change in temperature, and the final temperature of degrees?
Solve your equation carefully
Once you have an equation involving and constants, isolate by doing the opposite operation (undo the subtraction or addition). Be careful with negative signs when you simplify.
Desmos Guide
Use Desmos to compute the starting temperature
In Desmos, type the expression -2 + 1.8*10 and press Enter. The value that Desmos returns is the temperature at the beginning of the 10-hour period.
Step-by-step Explanation
Find the total temperature change
The temperature fell degrees Celsius each hour for hours.
- A fall means the change is negative: degrees per hour.
- Over hours, the total change is:
So the temperature decreased by degrees Celsius in total.
Write an equation for the starting temperature
Let be the temperature at the beginning of the 10-hour period.
After the temperature falls degrees, the final temperature is degrees, so:
This equation says: starting temperature minus equals the ending temperature of degrees.
Solve the equation for the starting temperature
Solve the equation
Add to both sides:
Now compute :
So the air temperature at the beginning of the 10-hour period was degrees Celsius.