A manufacturer has been selling one thousand seven hundred television sets a week at
four hundred eighty dollars each. A market survey indicates that for each fourteen dollar rebate
offered to a buyer, the number of sets sold will increase by one hundred forty per week.
The demand function P of X,
where X is the number of television sets sold per week, P of X would be the price.
Part B: How large of a rebate should the company offer to a buyer to maximize revenue?
Part C: If the weekly cost function is given here,
how should it set the amount of rebate to maximize profit?
We'll begin by determining the demand function P of X. Because we're told that for each
fourteen dollar rebate, the number of television sets sold will increase by one hundred forty per
week, this indicates that P of X will be a linear function. Any point on our function
P of X will have the x-coordinate as the number of televisions sold or X,
and the y-coordinate will be P of X, which should be the price.
To find the two points on the linear function, we know that one thousand seven hundred televisions
are sold when the price is four hundred eighty dollars. Now, to find the second
point, we know that if we decrease the price by fourteen dollars, the number sold will increase
by one hundred forty. So if we drop the price by fourteen dollars, four hundred eighty minus
fourteen would be four hundred sixty-six, and that would increase the number of televisions
sold by one hundred forty, making it one thousand eight hundred forty.
From here, we can find the slope. To keep things organized, let's call these points
one and point two. The slope will be equal to Y sub two minus Y sub one or the change
in the price divided by X sub two minus X sub one or the change in the quantity sold. Here
we have negative fourteen divided by one hundred forty, which simplifies to negative one over ten.
So the slope is negative one over ten. Now, to find the equation of P of X,
we use this slope and one of these two points. Let's use point one.
Using point-slope form of a line where Y is our function P of X,
we would have P of X minus Y sub one, which would be four hundred eighty, equals M, which is
negative one over ten, times the quantity X minus X sub one, or minus one thousand seven hundred.
Let's go ahead and solve this for P of X. So we have P of X
minus four hundred eighty equals negative one over ten times the
quantity X minus one thousand seven hundred. Now, distributing the negative one over ten gives us P
of X minus four hundred eighty equals negative one over ten times X plus one hundred seventy.
Now, adding four hundred eighty to both sides gives us our demand function:
P of X equals negative one over ten X plus six hundred fifty. Again,
where X is the quantity sold and P of X should be the price.
Now, how large of a rebate should the company offer to a buyer to maximize
revenue? Let's begin by determining the revenue function. The revenue function R
of X would be equal to the price times the quantity or in our case P of X times X,
where P of X we just found is negative one over ten X plus six hundred fifty.
Now, multiplying by X gives us the revenue function R of
X equals negative one over ten X squared plus six hundred fifty X.
Now, to maximize the revenue, we'll begin by determining the critical numbers,
which is where the first derivative would be equal to zero or undefined.
For R prime of X, we would have negative two times one over ten, which is negative one over
five, times X plus six hundred fifty. This is never undefined,
so to find the only critical number, we'll set this equal to zero and solve. Subtracting
six hundred fifty on both sides gives us negative one over five X equals negative six hundred fifty.
Now, to solve for X, we multiply both sides by negative five,
which gives us X equals three thousand two hundred fifty.
Now, to verify that this X value does maximize the revenue function and not minimize it,
we can use either the first or second derivative test. Notice if we find the
second derivative R double prime of X, this is equal to just negative one over five. Since the
second derivative is negative, that means the function is concave down, which is good news
because that means at this critical number, we have a high point or a relative maximum.
So the revenue is maximized at this quantity. But the question asks about how large of a rebate we
should have. To determine the amount of rebate, we first need to determine the
price of the televisions at this quantity, which should be P of three thousand two hundred fifty.
Using the demand function, we have P
of X equal negative one over ten times three thousand two hundred fifty plus six hundred fifty.
So we would have negative three hundred twenty-five plus six hundred fifty,
which is two hundred twenty-five. So this is the price that would maximize revenue.
But again, the question asked about the rebate. Remember the original price was
four hundred eighty dollars. So the rebate amount would be equal to the
original price minus the price that maximizes revenue, which is two hundred twenty-five.
This is equal to two hundred fifty-five, which means revenue
is maximized when the rebate is two hundred fifty-five dollars.
This question is a little tricky because we need to ensure we answer the correct question.
Now, for part C, if the weekly cost function is given here,
how should it set the amount of the rebate to maximize profit? So now
the goal is to maximize profit, where profit is equal to revenue minus cost.
We'll begin by determining the profit function P of X equals R of X minus C of X,
where we found the revenue function in part B and we're given the cost
function. Let's go ahead and clear the parentheses and combine like terms.
So we'd have negative one hundred thirty-six thousand and then minus one hundred sixty X.
So we have two like terms here. Our profit function P of X equals negative one over
ten X squared plus four hundred ninety X minus one hundred thirty-six thousand.
Now, to maximize this, we'll begin by determining the critical numbers by
finding where the first derivative is equal to zero or undefined.
So we would have negative one over five X plus four hundred ninety equals zero. This
is never undefined. So we set it equal to zero and solve for X.
Negative one over five X equals negative four hundred ninety.
Now, multiplying both sides by negative five gives us X equals two thousand four hundred fifty.
Again, this is the quantity. To verify that P of X is maximized at this value of X,
we can use the second derivative test. Notice how P prime of X is
equal to negative one over five again, verifying that the function is concave
down, and therefore we have a high point or a maximum function value at this critical number.
But again, the question asked for the rebate amount, not the quantity that
should be produced and sold. So now we'll find the price that maximizes the profit,
which would be P of two thousand four hundred fifty.
So we have negative one over ten times two thousand four
hundred fifty plus six hundred fifty.
This would be negative two hundred forty-five plus six hundred fifty, which is four hundred five.
So a price of four hundred five dollars will maximize the revenue. This means the rebate
amount is equal to the original price of four hundred eighty dollars minus the
price that would maximize revenue of four hundred five dollars.
This difference is equal to seventy-five dollars, which
means profit is maximized when the rebate is seventy-five dollars.
I hope you found this helpful!