Thermometer X Scale Conversion Calculating Equivalent Readings

by Scholario Team 63 views

Hey guys! Ever wondered how different thermometers stack up against each other? Let's dive into a classic physics problem that involves converting temperatures between Celsius and a quirky thermometer X. We're going to break down how to tackle this type of question step by step, so you'll be a pro at temperature conversions in no time! The key here is understanding the linear relationship between temperature scales and how to use reference points like the freezing and boiling points of water to create a conversion formula. So, buckle up and let’s get started!

Understanding the Problem

Our main goal here is to figure out what a temperature of 30°C would read on thermometer X. We know that thermometer X has its own scale: 40°X for the freezing point of water and 220°X for the boiling point. This is different from the Celsius scale, where water freezes at 0°C and boils at 100°C. This type of problem emphasizes the practical application of thermometry and the importance of understanding different temperature scales.

The core challenge in this physics problem lies in converting a temperature reading from the Celsius scale to a different, non-standard scale (thermometer X). To successfully convert between these scales, we need to establish a relationship between them. This relationship is typically linear, meaning the change in temperature on one scale is directly proportional to the change in temperature on the other scale. The given information—the freezing and boiling points of water on both scales—serves as our anchor points for establishing this relationship. We'll use these points to create a conversion formula that allows us to accurately translate any Celsius temperature to its equivalent on thermometer X.

Setting Up the Conversion

To nail this, we need to set up a conversion formula. Think of it like this: we're mapping one scale onto another. The key is to use the two reference points we have – the freezing and boiling points of water – to create a kind of 'bridge' between the Celsius and thermometer X scales.

  1. Identify the Reference Points:

    • Celsius Scale: Freezing Point = 0°C, Boiling Point = 100°C
    • Thermometer X Scale: Freezing Point = 40°X, Boiling Point = 220°X

  2. Establish the Linear Relationship:

    We assume a linear relationship between the two scales, meaning the temperature change ratio is constant. This can be expressed as:

    (X - X_freezing) / (X_boiling - X_freezing) = (C - C_freezing) / (C_boiling - C_freezing)

    Where:

    • X is the temperature on the X scale
    • X_freezing is the freezing point on the X scale (40°X)
    • X_boiling is the boiling point on the X scale (220°X)
    • C is the temperature in Celsius
    • C_freezing is the freezing point in Celsius (0°C)
    • C_boiling is the boiling point in Celsius (100°C)

Plugging in the Values

Now comes the fun part – let's plug in the values we know into our formula! This will help us simplify things and get closer to finding our answer. This step involves substituting the known values of freezing and boiling points on both the Celsius and thermometer X scales into the linear relationship equation we established. By doing this, we transform the general equation into a specific one tailored to our problem. This specific equation will allow us to directly convert between Celsius temperatures and thermometer X readings. Accuracy in this step is crucial because any errors in substitution will propagate through the rest of the solution. So, let's be meticulous and ensure we've got all the values in the right place!

Our formula looks like this:

(X - 40) / (220 - 40) = (C - 0) / (100 - 0)

Let's simplify it:

(X - 40) / 180 = C / 100

Solving for X

Okay, we're on the home stretch! We need to isolate X, which is the temperature on thermometer X. To do this, we'll do some algebraic magic to get X by itself on one side of the equation. This step is crucial for finding the unknown temperature on the X scale. Isolating X involves a series of algebraic manipulations designed to undo the operations affecting X. We'll start by multiplying both sides of the equation by 180 to eliminate the denominator on the left side. Then, we'll add 40 to both sides to get X completely by itself. Each of these steps must be performed carefully to maintain the equality of the equation and avoid errors. Once we've successfully isolated X, we'll have a formula that directly expresses the thermometer X reading in terms of the Celsius temperature.

  1. Multiply both sides by 180:

    X - 40 = (180/100) * C

  2. Simplify:

    X - 40 = 1.8 * C

  3. Add 40 to both sides:

    X = 1.8 * C + 40

Now we have a neat formula that directly converts Celsius (C) to thermometer X (X).

Calculating the Temperature on Thermometer X

Now for the grand finale! We're given that the substance has a temperature of 30°C. All we need to do is plug this value into our conversion formula and calculate the corresponding temperature on thermometer X. This is the moment where all our previous work comes together to give us the final answer. We'll substitute 30°C for C in our equation and perform the arithmetic operations to solve for X. This calculation demonstrates the practical application of the conversion formula we derived and provides a concrete answer to the problem. It's important to double-check our calculations at this stage to ensure accuracy and avoid any careless mistakes.

Let's do it:

X = 1.8 * 30 + 40

X = 54 + 40

X = 94°

So, a substance with a temperature of 30°C would show 94° on thermometer X.

The Answer

Drumroll, please! The correct answer is B. 94°. We successfully converted the temperature from Celsius to the thermometer X scale using our carefully crafted formula. This final step underscores the importance of accurate calculation and reinforces the link between the mathematical solution and the physical scenario described in the problem. It's always a good idea to review the entire solution one last time to ensure that the answer makes sense in the context of the problem. In this case, 94°X seems reasonable given the scales of the two thermometers, providing a final check on the validity of our result.

Key Takeaways

  • Temperature Scale Conversions: You can convert between temperature scales using a linear relationship and reference points.
  • Setting up Equations: The freezing and boiling points are your friends! Use them to build your conversion formula.
  • Algebra is Your Superpower: Isolating the variable you want is key to solving these problems.

Why This Matters

Understanding temperature conversions isn't just about acing physics exams, guys. It's super practical in many real-world scenarios! Think about scientific research, where precise temperature measurements are crucial. Or even in everyday life, like when you're baking a cake and need to convert oven temperatures. This skill helps in various scientific and practical contexts, making it a valuable tool in your problem-solving arsenal. Being able to accurately convert temperatures ensures that experiments are conducted under the right conditions, recipes turn out perfectly, and various other temperature-sensitive processes are managed effectively. So, mastering this skill not only helps you in academics but also equips you for real-world challenges.

Practice Makes Perfect

Want to become a temperature conversion master? Try tackling similar problems with different scales and reference points. The more you practice, the more confident you'll become! Experimenting with different temperature scales and conversion scenarios will deepen your understanding of the underlying principles. You can find practice problems in textbooks, online resources, or even create your own scenarios. Try changing the freezing and boiling points of the imaginary thermometer and see how it affects the conversion formula. The key is to actively engage with the material and challenge yourself to apply the concepts in various contexts. With consistent practice, you'll develop a strong intuition for temperature conversions and be able to solve these problems with ease.

So there you have it! Temperature conversions demystified. Keep practicing, and you'll be a pro in no time. Stay curious, guys!