Could someone please help me answer this question i am stuck!
A Marketing company provides the following formulas to the owner of an amusement park to help him make decisions about the park.
-The number , N, of people who attend the park is a function of price, p, in dollars:
N(p)=-(p+5)(p-17) assuming the minimum ticket price is $12.
- The revenue, R, is R(p)=N(p)multiplied by p
- The cost,C, of running the amusement park can be modelled by the function C(P)=75+12N(p)
a) write the combined function y=R(p)-C(p) and explain what it represents
b) Determine the interval(s) for p for which R(p)>C(p)
c)Identify the optimum ticket price for the amusement park and determine the maximum profit per ticket.
*for question b) and c) technology can be used to support the answer.
Thank you
(a) revenue-cost = profit
y = p(-(p+5)(p-17)) - (75+12(-(p+5)(p-17)))
= -(p^3-24p^2+59p+1095)
(b) R>C means P>0
The roots are at -5.2, 13.1, 16.1
From what you know about the shape of cubics, P>0 in the interval (13.1,16.1)
(c)
y' = -(3p^2-48p+59)
y' = 0 at p=14.65
y(14.65) = 47.37
is there any other way to find the maximum profit
a graphical or numerical method would do the trick, if you don't have calculus to help.
Sure! Let's break it down step by step.
a) To write the combined function y = R(p) - C(p), we need to substitute the given expressions for R(p) and C(p):
R(p) = N(p) * p
C(p) = 75 + 12 * N(p)
Therefore, the combined function y = R(p) - C(p) becomes:
y = N(p) * p - (75 + 12 * N(p))
This combined function represents the profit (or loss) made by the amusement park at different ticket prices. It takes into account both the revenue generated from ticket sales (N(p) * p) and the cost of running the park (75 + 12 * N(p)).
b) To determine the interval(s) for p where R(p) > C(p), we need to compare the revenue and cost functions. Since R(p) = N(p) * p and C(p) = 75 + 12 * N(p), we can rewrite the inequality as follows:
N(p) * p > 75 + 12 * N(p)
By rearranging the expression, we have:
N(p) * p - 12 * N(p) > 75
Factoring out N(p):
N(p) * (p - 12) > 75
Now we have an inequality involving N(p), which is the number of people who attend the park as a function of price. To determine the interval(s) for p, we need to find the values of p for which the inequality holds true.
To do this, we can use technology to plot the graph of the inequality and identify the x-values where it is above the line y = 75. Graphing calculators or software like Desmos can be used for this purpose.
c) To identify the optimum ticket price for the amusement park and determine the maximum profit per ticket, we can analyze the combined function y = R(p) - C(p). The optimum ticket price corresponds to the maximum value of y.
Again, technology can be used to plot the graph of y = R(p) - C(p) and find the value of p that corresponds to the maximum y. By analyzing the graph or using optimization techniques, we can determine the optimum ticket price and the maximum profit per ticket.
In summary, to answer questions b) and c), we can use technology such as graphing calculators or software to plot and analyze the functions involved, finding the intervals where revenue is greater than cost and determining the optimum ticket price and maximum profit per ticket.