Mixing Water Temperatures: A Math Problem
Hey guys! Let's dive into a cool math problem that's all about mixing water at different temperatures. Imagine you're trying to create the perfect bathwater. You've got ice-cold water at 11 degrees Celsius and piping hot water at 66 degrees Celsius. The challenge? To mix them just right to get 110 liters of water at a comfy 36 degrees Celsius. Sounds like fun, right? Don't worry, we'll break it down step-by-step, so it's super easy to understand. We'll even whip up the formula so you can play around with different temperatures and volumes. It's like a fun science experiment with numbers! This problem is a classic example of a mixture problem in mathematics, often encountered in algebra or basic physics. The key concept here is that the heat gained by the cold water must equal the heat lost by the hot water, assuming no heat is lost to the environment. This principle allows us to set up an equation to solve for the unknown quantities of cold and hot water needed. Let's get started, shall we?
Understanding the Problem
Okay, so the core of the problem is this: we need to find out how much of each water temperature (cold and hot) to mix together to achieve a final temperature of 36 degrees Celsius, with a total volume of 110 liters. This involves using the principle of weighted averages. Each volume of water contributes to the final temperature based on its own temperature. The colder water will lower the overall temperature, while the hotter water will raise it. The final temperature is, essentially, a balance between these two extremes. To solve this, we can set up a system of equations. One equation represents the total volume of water, and the other represents the heat balance. This approach allows us to determine the unknown volumes accurately. Think of it like a seesaw; the point where the seesaw balances is the final temperature we're aiming for. The volumes of cold and hot water act as the weights on each side of the seesaw. Let's make sure we're on the right track! We have a mix of cold water, a final temperature, and the total volume of the mixture to find out the required volume. Now that we know what we need, let's start the solution! Keep going, we are almost there, guys!
Setting Up the Equations
Alright, let's get our math hats on and create some equations. First, let's say the volume of cold water we need is 'x' liters, and the volume of hot water is 'y' liters. Since we want a total of 110 liters, our first equation is super simple: x + y = 110. This is the volume equation. Next up, we have the temperature equation, which is a bit more involved. We'll use the principle that the total heat in the mixture is a weighted average of the temperatures. The heat contributed by the cold water is 11x (11 degrees times the volume of cold water), and the heat contributed by the hot water is 66y (66 degrees times the volume of hot water). The total heat in the final mixture is 36 * 110 (36 degrees times the total volume of 110 liters). So, our second equation becomes: 11x + 66y = 36 * 110. Now, we have a system of two equations with two variables: x + y = 110 and 11x + 66y = 3960. Solving this system will give us the values for 'x' and 'y', which are the volumes of cold and hot water we need. This approach is fundamental in various applications, from chemistry, where solutions are mixed to achieve a specific concentration, to financial modeling, where assets are combined to achieve a target return. Setting up these equations is crucial; it's the foundation upon which the entire solution is built. Let's solve them now, guys, and find out the final values!
Solving the Equations
Now for the fun part: solving our equations! We have: x + y = 110 and 11x + 66y = 3960. There are several ways to solve this, but let's use the substitution method because it's nice and clear. From the first equation (x + y = 110), we can easily express 'x' in terms of 'y': x = 110 - y. Now, substitute this value of 'x' into the second equation: 11(110 - y) + 66y = 3960. Simplify this to get: 1210 - 11y + 66y = 3960. Combine the 'y' terms: 55y = 3960 - 1210, which simplifies to 55y = 2750. To find 'y', divide both sides by 55: y = 2750 / 55, which gives us y = 50 liters. So, we need 50 liters of hot water. Now that we know 'y', we can plug it back into our first equation (x + y = 110) to find 'x': x + 50 = 110. Therefore, x = 110 - 50, which gives us x = 60 liters. So, we need 60 liters of cold water. Therefore, to get 110 liters of water at 36 degrees Celsius, you'll need 60 liters of cold water (11 degrees Celsius) and 50 liters of hot water (66 degrees Celsius). Congratulations, we solved it! This straightforward process is applicable in countless scenarios beyond bathwater, like calculating the concentration of a solution or understanding the mixing of different grades of fuel. Keep practicing, and you will be good at it!
The Formula
Let's get down to the formula, so you can do this at home! We used this method to find the volume. We can generalize this solution into a formula for any two temperatures and a desired final temperature. Let's define:
T_c= Temperature of the cold waterT_h= Temperature of the hot waterT_f= Final desired temperatureV_f= Total final volumeV_c= Volume of cold waterV_h= Volume of hot water
From our equations, we can derive the formula. The volume of cold water (Vc) can be calculated as: V_c = (V_f * (T_h - T_f)) / (T_h - T_c). Similarly, the volume of hot water (Vh) can be calculated as: V_h = (V_f * (T_f - T_c)) / (T_h - T_c). Using these formulas, you can easily calculate the required volumes of hot and cold water for any given temperatures and desired final temperature. Just plug in the values, and you're good to go! This formula is a direct application of the weighted average principle, which is super useful in various fields, like calculating the final price of a blended product or determining the optimal mix of ingredients. Feel free to use this for your next water mixture! This formula is your new best friend; keep it safe!
Conclusion
Awesome work, guys! We've successfully solved our water-mixing problem. We've figured out that to get 110 liters of water at 36 degrees Celsius, you need 60 liters of cold water at 11 degrees and 50 liters of hot water at 66 degrees. We've also created a handy formula that you can use to calculate this for any temperature. This isn't just a math problem; it's a practical application of algebra and problem-solving skills that can be applied in everyday life. From mixing solutions in the lab to understanding how different components blend in various mixtures, the principles we've used here are broadly applicable. Keep practicing and keep experimenting with the formula; you'll become a pro at these problems in no time. Now you are ready to prepare your perfect bath! You can now adjust your next bathwater to be exactly as you like it!