Calculate Hourly Temperature Change

Hey guys, let's dive into a cool math problem that's super relevant, especially when you're dealing with those wild temperature swings! We're going to break down how to figure out the change in temperature per hour. This isn't just about numbers; it's about understanding how quickly things cool down or warm up. So, grab your notebooks, and let's get this sorted!

Understanding the Initial Scenario

Alright, picture this: the temperature took a nosedive. It started at a chilly $0^{\circ} F$. But it didn't stop there; it plummeted further to 15 rac{1}{2}{ }^{\circ} F below zero. That's a significant drop, folks! This whole temperature shift happened over a period of 5 rac{3}{4} hours. Our mission, should we choose to accept it (and we totally should!), is to find out how much the temperature changed each hour. This involves a bit of arithmetic, specifically with fractions and negative numbers, which can be a bit tricky, but that's what we're here to master. The key is to accurately represent the starting and ending temperatures and the duration over which this change occurred. When we talk about temperatures below zero, we're dealing with negative values. So, $0^{\circ} F$ is our starting point, and the ending temperature is -15 rac{1}{2}{ }^{\circ} F. The time elapsed is 5 rac{3}{4} hours. The initial thought might be to just add or subtract, but we need to be careful with the signs. The total change in temperature is the final temperature minus the initial temperature. This will give us the net change. After finding the total change, we'll divide that by the time taken to get the rate of change per hour. This is a fundamental concept in understanding rates of change, applicable in many scientific and real-world scenarios beyond just weather.

Wen's Approach: A Closer Look

Now, let's examine Wen's work. She started with the expression: -15 rac{1}{2} + 5 rac{3}{4}. This looks like she might be trying to combine the final temperature and the time duration, but that's not quite the right way to find the change in temperature per hour. To find the change in temperature, we first need to calculate the total difference between the final temperature and the initial temperature. The initial temperature was $0^{\circ} F$, and the final temperature was -15 rac{1}{2}{ }^{\circ} F. So, the total change in temperature is \text{Final Temperature} - \text{Initial Temperature} = -15 rac{1}{2} - 0 = -15 rac{1}{2}{ }^{\circ} F. This means the temperature dropped by a total of 15 rac{1}{2}{ }^{\circ} F. Wen's calculation of -15 rac{1}{2} + 5 rac{3}{4} doesn't directly represent this total change. It seems like she might be mixing up the values or the operation. Remember, the rate of change is $\frac{\text{Total Change}}{\text{Time Taken}}$. Wen's expression doesn't fit this formula. Let's convert the mixed numbers to improper fractions to make calculations easier. -15 rac{1}{2} becomes $-\frac{(15 \times 2) + 1}{2} = -\frac{31}{2}$. And 5 rac{3}{4} becomes $\frac{(5 \times 4) + 3}{4} = \frac{23}{4}$. If Wen was trying to do something with these numbers, she needs to apply them in the correct context. Her current setup doesn't lead to the desired hourly rate. It's crucial to correctly identify what each number represents: initial state, final state, and the duration. By misapplying the operations, the result will be incorrect, no matter how accurately the arithmetic itself is performed. We need to ensure that our mathematical model accurately reflects the physical situation.

Correcting the Calculation: Step-by-Step

Okay guys, let's fix this and get the right answer. The first step is always to figure out the total change in temperature. We know the temperature started at $0^{\circ} F$ and ended at -15 rac{1}{2}{ }^{\circ} F. So, the total change is:

$\text{Total Change} = \text{Final Temperature} - \text{Initial Temperature}$ \text{Total Change} = -15 rac{1}{2}{ }^{\circ} F - 0^{\circ} F \text{Total Change} = -15 rac{1}{2}{ }^{\circ} F

This tells us the temperature decreased by 15 rac{1}{2}{ }^{\circ} F. Now, we need to find out how much this change happened per hour. To do this, we divide the total change by the total time taken. The time taken was 5 rac{3}{4} hours. So, the change in temperature per hour is:

$\text{Change per Hour} = \frac{\text{Total Change}}{\text{Time Taken}}$ \text{Change per Hour} = \frac{-15 rac{1}{2}}{5 rac{3}{4}}

Now, let's convert these mixed numbers into improper fractions to make the division easier.

-15 rac{1}{2} = -\frac{(15 \times 2) + 1}{2} = -\frac{31}{2}

5 rac{3}{4} = \frac{(5 \times 4) + 3}{4} = \frac{23}{4}

So, the division becomes:

$\text{Change per Hour} = \frac{-\frac{31}{2}}{\frac{23}{4}}$

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

$\text{Change per Hour} = -\frac{31}{2} \times \frac{4}{23}$

Now, we can multiply the numerators together and the denominators together:

$\text{Change per Hour} = -\frac{31 \times 4}{2 \times 23}$

$\text{Change per Hour} = -\frac{124}{46}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$\text{Change per Hour} = -\frac{124 \div 2}{46 \div 2}$ $\text{Change per Hour} = -\frac{62}{23}$

To make this more understandable, we can convert this improper fraction back into a mixed number. Divide 62 by 23:

$62 \div 23 = 2$ with a remainder of $62 - (23 \times 2) = 62 - 46 = 16$.

So, the change per hour is:

\text{Change per Hour} = -2 rac{16}{23}{ }^{\circ} F per hour.

This means that, on average, the temperature dropped by approximately 2 rac{16}{23} degrees Fahrenheit every hour during that 5 rac{3}{4} hour period. It's important to note that this is an average rate; the actual temperature might have fluctuated more rapidly or slowly at different points within that time frame. Understanding this calculation process is key to solving similar rate problems.

Final Answer and Takeaways

So, the correct change in temperature per hour is -2 rac{16}{23}{ }^{\circ} F per hour. Wen's initial calculation didn't reflect the correct steps needed to solve this problem. Remember, guys, the key steps are: first, find the total change in temperature (final minus initial), and second, divide that total change by the total time taken. Using the wrong operations or mixing up the numbers will lead you down the wrong path, just like Wen's initial attempt. This problem highlights the importance of setting up the equation correctly before diving into the calculations. Negative numbers and fractions can be a bit daunting, but by breaking them down into improper fractions and using the rules of fraction division (multiplying by the reciprocal), we can tackle them effectively. The result, -2 rac{16}{23}{ }^{\circ} F per hour, tells us a clear story about how rapidly the temperature was falling. This kind of analysis is crucial in meteorology, climate science, and even in fields like engineering where understanding rates of change is vital for design and prediction. Always double-check your setup to ensure you're modeling the real-world situation accurately. Practice makes perfect, so try working through similar problems to build your confidence! Keep those math skills sharp!