SOLUTION: I can't pay for it but I need this rather quickly because I have tutoring today. I have been out of school due to medical reasons and the school provided me with a rather unhelpfu

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Question 611230: I can't pay for it but I need this rather quickly because I have tutoring today.
I have been out of school due to medical reasons and the school provided me with a rather unhelpful tutor who rather than teaching the material pushed me straight off into the book work. The problem is I am a year behind in math and need to study for ACT's and SAT's. This is the problem that has been giving me so much trouble:
In making business plans for a pizza sale fund raiser the Band Boosters at Roosevelt High figured out how both sales income I(n) and selling expenses E(n) would probably depend on number of pizzas sold n. They predicted that I(n)= -0.05n2+20n and E(n)= 5n+250.
a. Estimate value(s) of n for which I(n)= E(n) and explain what the solution(s) of that equation tell about prospects of the pizza sale fund raiser. Illustrate your answer with a sketch of the graphs of the two functions involved labeling key points with their coordinates.
b. Write a rule that gives predicted profit P(n) as a function of number of pizzas sold and use that function to estimate the number of pizza sales necessary for the fund raiser to break even. Illustrate yaddad yadda.
c. Use the profit function to estimate the maximum profit possible from this fund raiser. Then find number of pizzas sold, income, and expenses associated with the maximum profit situation.
Answer by ankor@dixie-net.com(22740) (Show Source):
You can put this solution on YOUR website!
In making business plans for a pizza sale fund raiser the Band Boosters at Roosevelt High figured out how both sales income I(n) and selling expenses E(n) would probably depend on number of pizzas sold n.
They predicted that I(n)= -0.05n2+20n and E(n)= 5n+250.
:
a. Estimate value(s) of n for which I(n)= E(n) and explain what the solution(s) of that equation tell about prospects of the pizza sale fund raiser.
;
n = number of items sold
Income = Expenses
-.05n^2 + 20n = 5n + 250
;
arrange as a single quadratic equation
-.05n^2 + 20n - 5n - 250 = 0
-.05n^2 + 15n - 250 = 0
:
Using the quadratic formula, two solutions, round to integers
n ~ 282 items
n ~ 18 items
:
" Illustrate your answer with a sketch of the graphs of the two functions" involved labeling key points with their coordinates.
:
graph two equation y = -.05x^2+20x and y = 5x+250

Note the two points of intersection illustrates the two break even points, about 18 and 282 items
:
:
b. Write a rule that gives predicted profit P(n) as a function of number of pizzas sold and use that function to estimate the number of pizza sales necessary for the fund raiser to break even.
:
The above quadratic equation f(n) = -.05n^2 + 15n - 250 represents the profit
profit is 0 at break even point remember.
:
Max occurs at the axis of symmetry, find that using x - -b/(2a)
n =
n = 150 items for max profit
;
Find that profit substitute 150 for n
f(150) = -.05(150^2) + 15(150) - 250
f(150) = $875 is max profit occurring when n = 150
:
A graph of this

:
c. Use the profit function to estimate the maximum profit possible from this fund raiser. Then find number of pizzas sold, income, and expenses associated with the maximum profit situation.
Using the previous information we know
max profit occurs when 150 pizza are sold
max profit is $850
:
Find expenses replace 150 for in in the expense equation
E = 5(150)+250
E = $1000 expenses
:
Find the total income, replace n with 150 again in the Income equation
I = -.05(150^2) + 20(150)
I = -1125 + 3000
I = $1875 total income
then
1875 - 1000 = $500 profit as we saw before
:
I know this is a lot to grasp, but try to understand each step, you can ask me a question in the comment if you wish. C