Calculus 1 question need help in answering/solving

Question

Naomi is a business major who invented a remote control that transforms bad professors into

good ones. She is currently selling 2000 remotes per month to students at a price of $10 each.

(a) How much revenue is she currently making per month?

Naomi believes she can make a larger profit by increasing the price on each remote.

(b) If the price per remote becomes $10 + x, how much revenue will Naomi now be making

per month? Write an expression involving x.

As the price per remotes go up, fewer students can afford the cost. Students start to opt out

of buying one.

(c) If the number of students buying remotes per month is 2000 − 20x, how much revenue

is Naomi now making per month? Write an expression involving x.

(d) Naomi decides to use her knowledge of calculus to make more money. How much more

than $10 should she sell each remote for to maximize her revenue?

Answer 1

a) If she is selling 2000 remotes per month and each remote has a price of $10 then obviously her revenue is 2000•10 = $20,000 per month.

b) The problem is not very good at this point because we are not told what the quantity of remotes is that she is selling - we are left to "guess" that she is still selling 2000 remotes per month although (reading on to question c) that is a bad assumption. Nevertheless, I believe the problem wishes you to use 2000 as the quantity. We are told the price is (10 + x) so her revenue is 2000(10 + x) = 20,000 + 2000x

So R(x) = 20,000 + 2000x

c) Now we are told that the quantity of remotes she sells is 2000 − 20x so her revenue is (2000 - 20x)(10 + x) = 20000 + 2000x - 200x - 20x² = 20000 + 1800x - 20x²

So R(x) = 20000 + 1800x - 20x²

d) The maximum revenue occurs at the vertex of the parabola. You can find the vertex without calculus, it is found using x = -b/2a where, in this case, a = -20, and b = 1800 so x = -1800/(2•(-20)) = 1800/40 = 45

Using calculus, if we set the derivative of the revenue function equal to zero, we can find local minimums or maximums.

R(x) = 20000 + 1800x - 20x²

R'(x) = 1800 - 40x

1800 - 40x = 0

40x = 1800

x = 45

So she should sell her remotes for $45 more than $10