Calculus Help - Optimization?

Question

Hey, could someone help me with this problem? It reads:

The price of a day long van tour of a famous national park starts at 100 dollars per person with a required minimum of 10 people. For each extra rider beyond the required 10, the group gets a 5 dollar per person discount. Set up and solve an optimization problem to determine the number of people that would maximize the amount paid per van load.

The answer is: "Max Per Van Load is @ 15 = 10+5 Passengers"

I can't figure out how they got that, though. Any help?

Answer 1

Edited:

Cost = (x+10)(100-5x)

when x=0, cost =1,00

when x=1, cost =95 etc.

C=100x-5x^2+1,000-50x

C =-5x^2+50x+1,000

dC/dx = -10x+50=0

-10x=--50

x=5 people, 10+5 =15

d^2C/dx^2 = -10 < 0, shows C is maximized.

Answer 2

EDIT:

Apparently I misread the question.



It seems that the question lacks required information.



If the van ride costed 100 dollars regardless of the number of people then that would be a more reasonable optimization question. Furthermore, the maximum number of people that can fit inside the van would be required at the start(not as an answer? odd).



Lets assume that it does cost a fixed amount for the ride of 100 dollars regardless of the number of people up to 10. Obviously through inspection the highest benefit is obtained when as many people as possible are inside the van giving the max discount.



More realistically, the van ride should cost a fixed amount for the first x number of people(we can say 10) then a discounted additional amount is added for each extra person. Lets say, 100 dollars for fixed price up to 10 people meaning that each person pays 10 dollars per seat if the max 10 is reached. For each additional person it costs $5.

If the goal is to pay the lowest average price per person then the optimization would be(again) to fit the maximum number of people into the van.

Answer 3

a million) that's elementary; ==> First choose the objective of the problem; right here that's paper dimensions, so as that its area will be minimum; then next comes lower than what constraints this must be finished? ==> nicely, right here on the selected paper, you're leaving some margins throughout and arriving on the realm for printing, that is given some fastened fee. 2) Now, we are sparkling what we want; the thanks to proceed for construction mathematical equations, so as that it may nicely be solved for paying for the end answer: 3) right here we are given the printing area as 50 squarewherein is consistent; consequently enable us start up from this knowledge; ==> enable the measurement of the printing area be x in (top sensible) by using y in (width sensible) ==> Printing area = xy = 50; == y = 50/x -------(a million) 4) there's a margin of four in each and each at good and bottom are presented; consequently undemanding top of the paper is "x + 8" in; further a margin of two in is presented on both aspects; ==> undemanding width = y + 4 in 5) consequently the realm of the paper is = (x+8)(y+4) 6) Substituting for y from equation (a million), A (x+8)(50/x + 4) = 50 + 400/x + 4x + 32 7) consequently the functionality to be minimized is: A = 80 2 + 4x + 400/x 8) Differentiating this, A' = 0 + 4 - 2 hundred/x^2 = 4 - 400/x^2 9) Equating A' = 0, x^2 = one hundred; ==> x = +/- 10 in 10) yet a measurement can't be unfavourable, consequently we evaluate in common words + 10; So x = 10 in and y = 5 in 11) although we prefer to verify,even if that's minimum or maximum; for which we are able to practice second spinoff try; So again differentiating, A'' = 800/x^3; at x = 10, A'' is > 0; consequently that's minimum consequently we finish for the printing the realm to be 50 squarein, lower than the given constraints of margins, the outer length of the paper should be 18 in by using 9 in; So outer area is = 162 squarein. desire you're defined; have a severe-high quality time.