Question

The temperature at 5:00 a.m. was \(-11\) degrees. By 8:00 a.m., the temperature is \(-2\) degrees. If the temperature continues to rise at th same rate, what will the temperature be at 2:00 p.m.? (A) 7 degrees (B) 16 degrees (C) 21 degrees (D) 27 degrees

Short answer

The temperature at 2:00 p.m. will be 16 degrees.

Step-by-step solution

Calculate the rate of change in temperature between 5:00 a.m. and 8:00 a.m.

To do this, we will subtract the temperature at 5:00 a.m. from the temperature at 8:00 a.m. and divide that by the change in time (3 hours). The formula for the rate of change in temperature is: \[rate = \frac{Temperature_{8:00am} - Temperature_{5:00am}}{Time_{8:00am} - Time_{5:00am}}\]

Use the data to find the rate

Now let's plug in the given data: \[rate = \frac{-2 - (-11)}{8 - 5}\] Simplify the equation: \[rate = \frac{9}{3}\]

Calculate the rate

Solve for the rate: \[rate = 3\] So the temperature is rising at a rate of 3 degrees per hour.

Calculate the change in time between 8:00 a.m. and 2:00 p.m.

To find the change in time between 8:00 a.m. and 2:00 p.m., we need to subtract the current time (8:00 a.m.) from the desired time (2:00 p.m.): \[Time_{change} = Time_{2:00pm} - Time_{8:00am}\] Transform the given times into the 24-hour format: \[Time_{change} = 14 - 8\]

Calculate the change in time

Solve for the change in time: \[Time_{change} = 6\] The change in time between 8:00 a.m. and 2:00 p.m. is 6 hours.

Calculate the temperature change during the 6-hour period

Now we know that the temperature is rising at a rate of 3 degrees per hour. We can use this rate to find the temperature change during the 6-hour period: \[Temperature_{change} = Rate \times Time_{change} = 3 \times 6\]

Calculate the temperature change

Solve for the temperature change: \[Temperature_{change} = 18\] During the 6-hour period, the temperature will rise by 18 degrees.

Calculate the temperature at 2:00 p.m.

Now we can find the temperature at 2:00 p.m. by adding the temperature change to the temperature at 8:00 a.m.: \[Temperature_{2:00pm} = Temperature_{8:00am} + Temperature_{change} = -2 + 18\]

Find the temperature at 2:00 p.m.

Solve for the temperature at 2:00 p.m. : \[Temperature_{2:00pm} = 16\] The temperature at 2:00 p.m. will be 16 degrees. The correct answer is (B) 16 degrees.

Key concepts

GED Math Practice

GED math practice exercises often include real-world applications such as interpreting graphs or calculating rates of change like in temperature problems. It’s essential for students preparing for their GED exam to understand how to apply mathematical concepts to everyday scenarios.

To master these problems, you should start by familiarizing yourself with the basics of arithmetic operations, algebra, and reading graphs. Then, as you progress, moving onto more complex problems, like those involving temperature changes over time, can help strengthen your understanding and build your confidence.

As part of your practice, always ensure that you understand each step and the reasoning behind it. When solving temperature rate of change problems, for instance, consider using units that make sense to you (like degrees per hour) and translate the problem into a form that resembles something you’ve already practiced.

Rate of Change Calculation

Understanding rate of change is fundamental in mathematics and it applies to various fields such as physics, economics, and environmental science. It is essentially a way of measuring how much one quantity changes in relation to another. In this context, the rate of change is the speed at which temperature increases or decreases.

To calculate the rate of change, you can use the formula \[rate = \frac{\Delta Temperature}{\Delta Time}\] where \(\Delta\) symbolizes ‘change in’. Always double-check to ensure you're using consistent units for both temperature and time so that the rate is correctly determined. Through regular practice, the calculation of rates of change can become second nature, providing a powerful tool for solving a myriad of real-world problems.

Solving for Temperature Increase

Understanding Temperature Change Over Time

When solving for temperature increases, it’s important to understand that the rate of temperature change, combined with the amount of time passed, gives us the amount of change in temperature. If you have the initial temperature, you can add this change to find the final temperature after a certain period.

To perform the calculations, use the formula: \[Final\ Temperature = Initial\ Temperature + (Rate\ of\ Temperature Change \times Time)\] Using this approach enables us to predict future temperatures or calculate the temperature change for past events. In any case, when you’re practicing these problems, always watch for the positive or negative signs in temperatures, especially when dealing with degrees below zero.

Algebraic Problem-Solving

Breaking Down the Steps

Algebraic problem-solving is a process that involves identifying the unknowns, establishing relationships between them, and then systematically solving for those unknowns. Take for example our temperature problem: we began by defining the rate of change as an algebraic expression, then we found the time difference and finally applied these values to get our answer.

Effective problem-solving requires careful reasoning and logical progression through the steps. Practice identifying what you're solving for, isolating that variable, and manipulating the equation to get your answer. Remember, algebra is a tool for abstracting real-world problems into mathematical terms and arriving at a solution through logical reasoning.