SOLUTION: You run a paddle-boat rental business on the Swan River. You currently charge $24.00 per paddle-boat and average 36 rentals a day. An industry journal says that, for every $1.50 in

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: You run a paddle-boat rental business on the Swan River. You currently charge $24.00 per paddle-boat and average 36 rentals a day. An industry journal says that, for every $1.50 in      Log On


   

Question 898589: You run a paddle-boat rental business on the Swan River. You currently charge $24.00 per paddle-boat and average 36 rentals a day. An industry journal says that, for every $1.50 increase in rental price, the business can expect to lose two rentals a day. Use this information to attempt to maximise your daily income.
Using both graphical and algebraic methods, determine the price should you charge to maximise daily income and what is this maximum daily income?
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Rentals__________Price
36_______________24
33_______________25.5
31_______________27

x=price
y=how many rentals
m = slope -%281.5%2F2%29
y=-%281.5%2F2%29x%2Bb-------- use the point, (24, 36) for $24 with 36 rentals.
Find b according to this data point.
b=y%2B%281.5%2F2%29x
b=36%2B%281.5%2F2%2924
b=36%283%2F4%2924
b=36%2B18
b=54
-
highlight%28y=-%283%2F4%29x%2B54%29

INCOME for price of x:
Call this revenue, R.
R=y%2Ax
R=%28-%283%2F4%29x%2B54%29x
highlight%28R=-%283%2F4%29x%5E2%2B54x%29

The function R is a parabola with vertex as a maximum.
The easier way to find this maximum is to find the zeros of R, through factoring, which is seen from the previous step; and then the maximum R occurs at the midpoint of these two zeros.

THE ZEROS OF R
x%2854-%283%2F4%29x%29=0
highlight%28x=0%29 OR 54-%283%2F4%29x=0;
54=%283%2F4%29x
x=54%2A4%2F3
x=%2827%2A2%2A4%29%2F3
x=9%2A2%2A4
highlight%28x=72%29
The zeros of R are 0 and 72. The exact middle of these is 36.
-
This maximum, vertex is R=54x-%283%2F4%29x%5E2----evaluate at x=36.
R=972 dollars, maximum.


graph%28450%2C450%2C-5%2C45%2C-50%2C1000%2C54x-%283%2F4%29x%5E2%29
THE VERTICAL SCALE ON THIS GRAPH IS NOT SHOWING PROPERLY.

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
You run a paddle-boat rental business on the Swan River. You currently charge $24.00 per paddle-boat and average 36 rentals a day. An industry journal says that, for every $1.50 increase in rental price, the business can expect to lose two rentals a day. Use this information to attempt to maximise your daily income.
Using both graphical and algebraic methods, determine the price should you charge to maximise daily income and what is this maximum daily income?


Parabolic equation to graph: highlight_green%28highlight_green%28f%28x%29+=+-+3x%5E2+%2B+6x+%2B+864%29%29, where x represents the NUMBER of price increases

Price to charge in order to maximize daily income: $highlight_green%28highlight_green%2825.50%29%29

Maximum daily income derived from increased price of $25.50: $highlight_green%28highlight_green%28867%29%29

You can do the check!! 



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