A laptop company has discovered that their cost and revenue functions for each day are 𝐶 of 𝑥 equals three 𝑥 squared minus 10𝑥 plus 200 and 𝑅 of 𝑥 equals negative two 𝑥 squared plus 100𝑥 plus 50. Find the maximum and minimum number of laptops they could produce each day while still making a profit. Know as a hint your answers should be integers.
So, in this question, what we’re gonna need to do is set up an inequality using the functions that we’ve been given. And we’re gonna solve that inequality. And that’s gonna give us our maximum and minimum number of laptops that could be produced while still making a profit.
Well, the key word here is “profit” because if we want to make a profit, then this means that the revenue must be greater than the cost. So therefore, if we know that the revenue has got to be greater than the cost, then we can use this to set up an inequality. Because if 𝑅 of 𝑥 is greater than 𝐶 of 𝑥, then we can say that negative two 𝑥 squared plus 100𝑥 plus 50 is gonna be greater than three 𝑥 squared minus 10𝑥 plus 200.
So, now, what we’re gonna need to do is rearrange; we can make one side equal to zero. So, in order to do this, what we’re gonna do is add two 𝑥 squared, subtract 100𝑥, and subtract 50 from each side of our inequality. And when we do that, what we’re gonna get is zero is greater than five 𝑥 squared, and that’s cause three 𝑥 squared plus two 𝑥 squared is five 𝑥 squared. Minus 110𝑥, that’s cause if we’ve got negative 10𝑥 and we subtract 100𝑥, we get negative 110𝑥. And then, plus 150.
So, now, we could solve this inequality as it is. However, we can see that it’s gonna be easier if we simplify because we can see straightaway that five is a factor of five, 110, and 150. So therefore, if we divide through by five, what we’re gonna get is zero is greater than 𝑥 squared minus 22𝑥 plus 30. So, this is now the inequality we want to solve.
However, to solve this, we need to find our critical values first. And to do that, what we’re gonna do is we’re gonna make this a quadratic equal to zero. So, we’re gonna solve 𝑥 squared minus 22𝑥 plus 30 equals zero to find our critical values. However, we could try and do this by factoring. But if we try to do that, we’ll work out quite quickly that there aren’t a pair of factors that will multiply to give us positive 30 and sum to negative 22.
So therefore, what we’re going to use to solve this is our quadratic equation. And actually, the hint points towards the fact that we might need to use something like the quadratic equation because it says, your answers should be integers. Which suggests that the answer you get for the first part of the question is not going to be an integer. So therefore, we’ll use something like the quadratic equation to help us find them.
So, the first thing we need to do is remind ourselves what the quadratic formula is. Well, the quadratic formula is 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 over two 𝑎. However, what are 𝑎, 𝑏, and 𝑐? Well, 𝑎, 𝑏, and 𝑐 are the coefficient of 𝑥 squared, the coefficient of 𝑥, and our numerical constant at the end, in that order. So therefore, in our situation, they’re going to be one, negative 22, and positive 30.
So therefore, what we’re going to do is substitute them into the quadratic equation to give us the answer we are looking for. So, what you get is 𝑥 is equal to 22 plus or minus the square root of negative 22 squared minus four multiplied by one multiplied by 30. And this is all over two multiplied by one. And we got that, as we said, because we substituted in our 𝑎-, 𝑏-, and 𝑐-values.
The one thing we need to remember about the quadratic formula is that we have this bit here which is positive/negative or plus/minus. What this means is that we’re gonna have two different values. And that’s what we’re looking for because we want our two critical points. So, the first one of those critical points we’re gonna get with 22 plus the square root of 364 over two. And that’s because everything we had inside the square root sign above, if we actually calculate that, so if we squared negative 22 and we subtracted four multiplied by one multiplied by 30, we get 364.
And if we put this into our calculator, what we’re gonna get is 20.5. And that’s to one decimal place, which is a suitable degree of accuracy for what we’re looking to do. And then, what we’re gonna do is we’re gonna work out our second possible value of 𝑥. We get this with 22 minus root 364 over two. So, what this is gonna give us is 1.5 to one decimal place.
So, great, we’ve now got both of our critical values. But what are they used for? How are they useful? Well, what we can do is use a sketch to show how we’re gonna use them to help solve the problem. So, if we do a rough sketch of our quadratic, which was 𝑥 squared minus 22𝑥 plus 30 equals zero, we’ll have something that looks a bit like this. The main points because — like we said, it is a rough sketch — are that it crosses the 𝑥-axis at 1.5 and 20.5 as these are our critical values.
And it’s also worth remembering that because it’s positive 𝑥 squared, then it’s gonna be a U-shaped parabola. If it was negative 𝑥 squared, then it’d be an n-shaped parabola. So, what we can do now is use the original inequality, our sketch, and our critical values to solve our inequality.
Because if we’ve got 𝑥 squared minus 22𝑥 plus 30 is less than zero. So if we’re looking for the region where the function is less than zero, then we want the part of the function that’s below the 𝑥-axis. So, we want the area that is shaded here in pink. So therefore, the solution to our inequality would be 𝑥 is greater than 1.5 but less than 20.5, remembering that we rounded those values.
So, have we solved the problem because we’ve solved the inequality? Well, no, because we haven’t mentioned how many laptops need to be sold. And also, our answer is not an integer. Well, therefore, the answer to the problem is going to be the minimum number of laptops that could be sold each day while still making a profit is going to be two laptops. And that’s cause 𝑥 has to be greater than 1.5. Well, the smallest integer that’s greater than 1.5 is two. And the maximum number of laptops they could produce each day while still making a profit is going to be 20 laptops. And that’s because we found out that 𝑥 had to be less than 20.5. And the next integer that’s less than 20.5 is 20. So, there are two answers.