Maximize Revenue: Tubs & Solar Heaters Sales Calculation

by SLV Team 57 views

Hey guys! Today, we're diving into a super practical math problem: figuring out how to maximize revenue when you're selling two different products. In this case, we're talking about tubs and solar heaters. Imagine you're running a business that sells both, and you want to know exactly how many of each you should sell to make the most money. Sounds interesting, right? This is where math, specifically multivariable calculus, comes to the rescue. We'll break down the problem step by step, making it super clear and easy to follow. So, let's jump in and see how we can use math to boost those sales!

Problem Setup: The Revenue Function

So, let's start with the core of our problem: the revenue function. This function, often written as R(x, y), tells us how much money we'll make based on the number of tubs (x) and solar heaters (y) we sell. In our case, the revenue function looks like this:

R(x, y) = 12 + 74x + 85y - 3x² - 5y² - 5xy

Now, this might look a bit intimidating at first glance, but don't worry, we'll break it down. The '12' is a constant, meaning it doesn't change no matter how many tubs or heaters we sell. The '74x' tells us that for each tub we sell, we get $74. Similarly, '85y' means we get $85 for each solar heater. The '-3x²' and '-5y²' terms are where things get a little more interesting. These are quadratic terms, and they tell us that as we sell more tubs or heaters, the revenue from each additional unit decreases slightly. This is pretty realistic, as we might need to lower prices or face increased competition if we sell too much of one product. Finally, the '-5xy' term shows that there's some interaction between the sales of tubs and heaters. Maybe if people buy a tub, they're less likely to buy a heater, or vice versa. Understanding this revenue function is the first and most crucial step in maximizing our income.

Understanding Partial Derivatives

To find the maximum revenue, we're going to use a cool mathematical tool called partial derivatives. Think of partial derivatives as a way to see how the revenue changes when we tweak the number of tubs or heaters we sell, one at a time. It's like having two knobs, one for tubs and one for heaters, and we want to figure out how turning each knob affects our revenue. The partial derivative with respect to 'x' (tubs) tells us how the revenue changes when we sell one more tub, assuming the number of heaters we sell stays the same. Similarly, the partial derivative with respect to 'y' (heaters) tells us how revenue changes when we sell one more heater, keeping the number of tubs constant. By finding these partial derivatives, we can pinpoint the exact combination of tubs and heaters that will give us the highest revenue. It's like finding the peak of a mountain by feeling the slope in different directions. These derivatives are super powerful for optimization problems like this!

Calculating the First Partial Derivatives

Okay, let's get our hands dirty and calculate those partial derivatives! Remember, the revenue function is:

R(x, y) = 12 + 74x + 85y - 3x² - 5y² - 5xy

First, we'll find the partial derivative with respect to x, which we write as ∂R/∂x. This means we treat 'y' as a constant and differentiate with respect to 'x'. So, here's how it goes:

  • The derivative of 12 (a constant) is 0.
  • The derivative of 74x is 74.
  • The derivative of 85y (treating 'y' as a constant) is 0.
  • The derivative of -3x² is -6x.
  • The derivative of -5y² (treating 'y' as a constant) is 0.
  • The derivative of -5xy (treating 'y' as a constant) is -5y.

So, putting it all together, we get:

∂R/∂x = 74 - 6x - 5y

Now, let's find the partial derivative with respect to y, written as ∂R/∂y. This time, we treat 'x' as a constant and differentiate with respect to 'y'.

  • The derivative of 12 (a constant) is 0.
  • The derivative of 74x (treating 'x' as a constant) is 0.
  • The derivative of 85y is 85.
  • The derivative of -3x² (treating 'x' as a constant) is 0.
  • The derivative of -5y² is -10y.
  • The derivative of -5xy (treating 'x' as a constant) is -5x.

So, we get:

∂R/∂y = 85 - 10y - 5x

These two equations, ∂R/∂x and ∂R/∂y, are our key to finding the maximum revenue. They tell us how the revenue changes with respect to small changes in 'x' and 'y'. Next up, we'll use these to find the critical points, which are the potential locations of our maximum revenue!

Finding Critical Points: Where the Magic Happens

Alright, now that we've got our partial derivatives, it's time to find the critical points. These are like the potential treasure spots on our revenue map – the points where the revenue could be at its highest (or lowest). To find them, we need to solve a system of equations. Remember those partial derivatives we calculated?

  • ∂R/∂x = 74 - 6x - 5y
  • ∂R/∂y = 85 - 10y - 5x

We need to set both of these equal to zero and solve for 'x' and 'y'. Why zero? Because at a maximum (or minimum) point, the rate of change in revenue with respect to both 'x' and 'y' will be zero. Imagine being at the top of a hill – if you take a tiny step in any direction, you won't go up any further. So, we have two equations:

  1. 74 - 6x - 5y = 0
  2. 85 - 10y - 5x = 0

Now, we need to solve this system of equations. There are a few ways to do this, like substitution or elimination. Let's use elimination. First, we can rearrange the equations to make them look a bit cleaner:

  1. 6x + 5y = 74
  2. 5x + 10y = 85

To eliminate 'x', we can multiply the first equation by 5 and the second equation by -6:

  1. 30x + 25y = 370
  2. -30x - 60y = -510

Now, add the two equations together. The 'x' terms cancel out, and we're left with:

-35y = -140

Divide both sides by -35, and we get:

y = 4

Great! We've found 'y'. Now, we can plug this value back into either of our original equations to solve for 'x'. Let's use the first one:

6x + 5(4) = 74

6x + 20 = 74

6x = 54

Divide both sides by 6, and we get:

x = 9

So, our critical point is (x, y) = (9, 4). This means that selling 9 tubs and 4 solar heaters is a potential sweet spot for maximizing revenue. But we're not done yet! We need to confirm that this point is actually a maximum and not a minimum or a saddle point. For that, we'll need to use the second partial derivative test, which we'll tackle next.

The Second Partial Derivative Test: Confirming the Maximum

Okay, we've found our critical point (9, 4), which is a potential maximum revenue spot. But how do we know for sure? This is where the second partial derivative test comes in handy. Think of it as a way to check the curvature of our revenue surface at the critical point. Is it curving downwards like a hilltop (a maximum), upwards like a valley (a minimum), or something else entirely (a saddle point)? To perform this test, we need to calculate the second partial derivatives. Remember our first partial derivatives:

  • ∂R/∂x = 74 - 6x - 5y
  • ∂R/∂y = 85 - 10y - 5x

Now, let's find the second partial derivatives:

  • ∂²R/∂x²: This is the partial derivative of ∂R/∂x with respect to x. So, we differentiate 74 - 6x - 5y with respect to x, treating 'y' as a constant. The result is -6.
  • ∂²R/∂y²: This is the partial derivative of ∂R/∂y with respect to y. We differentiate 85 - 10y - 5x with respect to y, treating 'x' as a constant. The result is -10.
  • ∂²R/∂x∂y: This is the partial derivative of ∂R/∂x with respect to y (or, equivalently, the partial derivative of ∂R/∂y with respect to x). We differentiate 74 - 6x - 5y with respect to y, treating 'x' as a constant. The result is -5.

Now, we use these second partial derivatives to calculate something called the discriminant, often denoted as D:

D = (∂²R/∂x²)(∂²R/∂y²) - (∂²R/∂x∂y)²

Plug in the values we just found:

D = (-6)(-10) - (-5)² D = 60 - 25 D = 35

Now, we analyze the results:

  1. If D > 0 and ∂²R/∂x² < 0, we have a local maximum.
  2. If D > 0 and ∂²R/∂x² > 0, we have a local minimum.
  3. If D < 0, we have a saddle point.
  4. If D = 0, the test is inconclusive.

In our case, D = 35 (which is greater than 0) and ∂²R/∂x² = -6 (which is less than 0). So, we have a local maximum! This confirms that selling 9 tubs and 4 solar heaters will indeed maximize our revenue.

Calculating the Maximum Revenue: The Grand Finale

Alright, we've done the hard work! We've found the critical point (9, 4) and confirmed that it corresponds to a maximum revenue. Now, for the grand finale: calculating the actual maximum revenue. To do this, we simply plug our values of x = 9 and y = 4 back into our original revenue function:

R(x, y) = 12 + 74x + 85y - 3x² - 5y² - 5xy

So, let's plug in x = 9 and y = 4:

R(9, 4) = 12 + 74(9) + 85(4) - 3(9)² - 5(4)² - 5(9)(4) R(9, 4) = 12 + 666 + 340 - 243 - 80 - 180 R(9, 4) = 515

So, the maximum revenue we can achieve by selling 9 tubs and 4 solar heaters is $515! That's pretty awesome, right? We've used math to optimize our business and figure out the perfect sales strategy. This is a great example of how calculus can be applied to real-world problems to make smart decisions and maximize profits. Remember, this is just one example, and the same principles can be applied to a wide range of business and economic scenarios. Keep those calculations coming!

Conclusion: Math for the Win!

So, there you have it, guys! We've successfully tackled a real-world optimization problem using the power of calculus. We started with a revenue function, found partial derivatives, identified critical points, used the second partial derivative test to confirm a maximum, and finally, calculated the maximum revenue. It's pretty amazing how math can help us make smart decisions in business and beyond. By selling 9 tubs and 4 solar heaters, we can achieve a maximum revenue of $515. This whole process shows that understanding the math behind the scenes can really give you a competitive edge. Whether you're running a business, managing resources, or just trying to make the best decisions in your daily life, these mathematical tools can be incredibly valuable. So, keep learning, keep exploring, and keep applying math to solve those real-world challenges. You never know what you might discover!