Problem 28 - Prealgebra

Question

During a 10 -hour period, the temperature in Browning, Montana, changed at a rate of \(-10^{\circ} \mathrm{F}\) per hour, starting at \(44^{\circ} \mathrm{F}\). What was the ending temperature? (Lesson \(2-4\) )

Step-by-Step Solution

Answer

The ending temperature was \(-56^{\circ} \mathrm{F}\).

Step 1: Identify Initial Conditions

The initial temperature at the beginning of the 10-hour period is given as \(44^{\circ} \mathrm{F}\). The rate of temperature change is \(-10^{\circ} \mathrm{F}\) per hour.

Step 2: Calculate Total Temperature Change

The total change in temperature can be calculated by multiplying the rate of change by the number of hours. The rate is \(-10^{\circ} \mathrm{F}\) per hour and the time period is 10 hours. Thus, the total change is \(-10^{\circ} \mathrm{F/hour} \times 10 \mathrm{hours} = -100^{\circ} \mathrm{F}\).

Step 3: Determine Ending Temperature

To find the ending temperature, subtract the total temperature change from the initial temperature. Starting at \(44^{\circ} \mathrm{F}\) and subtracting \(100^{\circ} \mathrm{F}\) results in \(44^{\circ} \mathrm{F} - 100^{\circ} \mathrm{F} = -56^{\circ} \mathrm{F}\).

Key Concepts

Rate of Change

Understanding the rate of change is key to solving problems involving temperature variations over time. The "rate of change" is a measure of how much a quantity, such as temperature, changes over a specific period. It is expressed as a change in degrees per unit of time, usually per hour. In this exercise, the rate of change is given as

• \(-10^{\circ} \mathrm{F}\) per hour.

This means that for every hour passed, the temperature decreases by 10 degrees Fahrenheit. Grasping this concept helps predict how a temperature will fluctuate throughout a certain period. Knowing just the rate, we can't determine the starting or ending temperature, but we can understand how significant the change is going to be within a time frame.
To calculate the total change from the rate, you multiply the rate by the time period considered, hence leading you to the next step of calculating either the current state or endpoint of the value in question.

Initial Conditions

Initial conditions serve as the starting point for understanding how a situation will evolve over time. In problems dealing with temperature change, the initial condition specifically refers to the starting temperature, which we must know before applying any calculation based on the rate of change.

• For this exercise, our initial temperature is \(44^{\circ} \mathrm{F}\).

Knowledge of the initial conditions allows you to apply the rate of change to determine future temperatures.
So, when working with any problem regarding change, be it temperature, speed, or another variable, always start by identifying where you are starting from. This provides a reference point for applying the rate of change, ensuring that calculations remain accurate.
Remember, without the information of initial conditions, any calculations for ending values would only be hypothetical.

Ending Temperature

Calculating the ending temperature involves applying the initial conditions with the rate of change over a known period. Starting with the initial temperature of \(44^{\circ} \mathrm{F}\), you must account for how the rate of change impacts this over the specified time. In our example,

• The rate of change is \(-10^{\circ} \mathrm{F}\) per hour over 10 hours.
• Multiply: \(-10^{\circ} \mathrm{F/hour} \times 10 \mathrm{hours} = -100^{\circ} \mathrm{F}\).
• Apply this change to the initial temperature: \(44^{\circ} \mathrm{F} - 100^{\circ} \mathrm{F}\).
• Resulting in an ending temperature of \(-56^{\circ} \mathrm{F}\).

Recognizing the ending temperature signifies a complete shift from where the process began and provides a clear outcome that can be immediately understood.
This kind of calculation is important for planning purposes, as it helps predict future conditions and adjust as needed based on the anticipated change.