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Hi, everyone. Welcome back to integralcalc.com. Today, we’re going to talk about profit
and the marginal profit function. And in this particular case we’ve been given the problem
which tells us that the marginal profit function for producing and selling x units of a certain
product is given by marginal profit of x, negative 0.6x + 420. And they tell us that
the company is currently producing 500 units and at that level of production, they’re
earning $150 000 in profit. So they’ve asked us a couple questions. The first one is, is
it profitable for the company to increase production? And the second question is, if
it is profitable, to find the production level, in other words how many times do they have
to produce to maximize profit and find the maximum profit at that level of production.
So we’ve got a couple of things on the table here. the first thing we want to know is,
is it profitable for the company to increase production? So that’s basically asking us
to find the point at which the company makes its maximum profit. Remember that the profit
function p of x, the derivative of the profit function is marginal profit. And that’s
what we’ve been given. Marginal profit of x. In order to find the maximum profit, we
need to go ahead and set the marginal profit function equal to zero. So we will go ahead
and set the marginal profit, negative 0.6x + 420. We’re going to go ahead and set that
equal to zero and now we need to solve for x. So we’ll add 0.6x to both sides and then
we’ll divide both sides by 0.6 and we’ll find that x is going to be equal to 700. So
what this tells us is that the company is going to make its maximum profit when they’re
making 700 units. We know that they’re currently making 500 units so we already know that it
is going to be profitable for them to increase production. So we’ve gone ahead and answered
this first question here. Yes it will be profitable for them to increase production and they should
increase it to 700 units of production because at that point, they’ll make their maximum
profit. So now, they’ve said if it is profitable and we know that it is to increase production,
find the level of production that maximizes profit. We’ve already done that. We know
that the level of production is 700 units. And then they say find the maximum profit.
So now we need to find the maximum profit and in order to find maximum profit, we need
to have the profit function. Right now, we already have the marginal profit function.
The way to get from marginal profit to profit is by taking the integral of the marginal
profit function since it’s the derivative of the profit function. So if we go ahead
and integrate the marginal profit function, -0.6x + 420, we’ll go ahead and integrate
this and this will be the profit function. So doing so, we’ll get -0.3x^2 + 420x +
c and that’s going to be equal to the profit function p(x). Now that we’ve done this,
we’re going to go ahead and solve for c. we don’t have a complete profit function.
We have this constant of integration in here as a place holder. Luckily for us in our problem,
we were given essentially an initial condition. This has turned in right now to an initial
value problem. We’ve got this initial condition where they’ve told us that the company is
producing 500 units and earning a profit of $150 000 at that level. So we know that profit
when the company is producing 500 units is equal to $150 000. So we can go ahead and
plug in that initial condition and we’ll be able to find c, the constant of integration.
So -0.3 times, and we’re plugging in 500 for x so 500^2 + 420(500) + c and that’s
going to be equal to $150 000. And when we go ahead and solve this for c, we’ll multiply
everything out and then subtract everything that’s on the left side, leaving c by itself.
We’ll find that c = 15 000. So now we can go ahead and write our full profit function
p of x. We’ll find that p of x is going to be equal to -0.3x^2 + 420x + 15 000. We
went ahead and solved for c. So we’ve now got our full profit function
and we can use it to find the maximum profit. We already know that maximum profit will be
attained when x is equal to 700 so we can go ahead and plug 700 into our profit function
for x and whatever we get when we solve that will be our maximum profit. And when we solve this for profit, we’ll fund that
the maximum profit is going to be $162 000. So that answers this last question maximum
profit. The maximum profit is achieved when the company produces 700 units and when they
do that, they make $162, 000. And that is the level of production that’s going to
maximize their profit. If they start making more than 700 units, their profit’s actually
going to decrease. So that’s the maximum right there. So anyway, that’s it. I hope
this video helped you guys and I will see you in the next one. Bye!