Problema Solution
Ben and Sarah own lots of ponies and often take them to fairs and other gatherings and sell pony rides. They have found that when they charge $6.00 per ride, they average 310 riders a day. Ben has been perfectly satisfied traveling around the countryside and bringing in that daily average revenue of $1860.00. Sarah, however, wants to bring in more money. She says that if they charge $12.00 per ride, their daily revenue will be $3720.00. Ben doesn't agree. He says they won't be able to attract an average of 310 riders each day at that price, so their daily revenue won't increase that much. Each one sees the point the other is trying to make. They finally agree that for every dollar they increase the price of a ride, they will probably lose 20 customers daily. Based on that assumption, what price per ride should they charge to achieve the maximum amount of revenue? What is the maximum revenue amount?
Answer provided by our tutors
The formula for sales revenue is the selling price multiply by the items sold.
For the price of $6 and average of 310 riders per day the average revenue is 6*310 = $1860
Sarah thinks that:
For the price of $12 and average of 310 riders per day the average revenue will be 12*310 = $3720
After the discussion with Ben they decide:
For $1 increased price the number of riders will go down by 20 then
For $x increased price the number of riders will decrease by x*20
Thus for the revenue we can write the following function
f(x) = (6 + x)*(310 - 20x)
f(x) = -20*x^2 + 190*x + 1830
We need to find the maximum of f(x) - parabolic function
Since f(x) is parabolic function and -20 < 0 the function has maximum in the vertex.
If we calculate the coordinates of the vertex we get the solution
x = 19/4 = 4.75
y = 9125/4 = 2281.25 approximately
That is they need to increase the price to 6 + 4.75 = $10.75 to get maximum revenue amount of $2281.25.