SOLUTION: The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N of items produced in a year, and the formula is as follows: P= -0.2N^2 + 3.6N - 9

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Question 1130474: The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N of items produced in a year, and the formula is as follows:
P= -0.2N^2 + 3.6N - 9
(a) Determine the two break-even points for this manufacturer-that is, the two production levels at which the profit is zero.
(c) How many items should they produce to get this maximum point?
Please show me how you get the answer, thank you.
Found 3 solutions by Boreal, josmiceli, MathTherapy:
Answer by Boreal(15235) About Me  (Show Source):
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P= -0.2N^2 + 3.6N - 9
This is a quadratic with a=-0.2, b=3.6 and c= -9
The n value at the vertex is -b/2a or -3.6/-0.4, which is +9
so when n=9, the profit is maximum
put that back into the equation
P=-0.2*81+32.4-9, or 7.2 thousands or $7200.
setting the equation equal to zero will give the two break even points.
can use the quadratic equation
n=(-1/.4)(-3.6+/- sqrt (12.96-4(-0.2)(-9));that is sqrt of (12.96-7.2) or sqrt (5.76), which is 2.4
n=-2.5(-3.6+2.4) which is 3
n--2.5(-3.6-2.4) which is 15
That makes sense, because both are six units from the vertex/maximum, consistent with symmetry that quadratics have around the vertex.
Can also multiply everything by -5 to make the equation n^2-18n+45=0
That factors into (n-15)(n-3)=0, and n=3, n=15
graph%28300%2C300%2C-10%2C20%2C-10%2C10%2C-.2x%5E2%2B3.6x-9%29

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+P%28N%29+=+0+ ( zero profit )
+-.2N%5E2+%2B+3.6N+-+9+=+0+
+N+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+a+=+-.2+
+b+=+3.6+
+c+=+-9+
--------------------
+N+=+%28-3.6+%2B-+sqrt%28+3.6%5E2-4%2A%28-.2%29%2A%28-9%29+%29%29%2F%282%2A%28-.2%29%29+
+N+=+%28-3.6+%2B-+sqrt%28+12.96+-+7.2+%29%29%2F%28%28-.4%29%29+
+N+=+%28-3.6+%2B-+sqrt%28+5.76+%29%29%2F%28%28-.4%29%29+
+N+=+%28+-3.6+%2B+2.4+%29%2F+%28-.4+%29+
+N+=+%28-1.2%29%2F%28-.4%29+
+N+=+3+
and
+N+=+%28+-3.6+-+2.4+%29+%2F+%28-.4%29+
+N+=+6%2F.4+
+N+=+15+
The two break even points are
+N+=+3+ and +N+=+15+
-------------------------------------
check:
+-.2N%5E2+%2B+3.6N+-+9+=+0+
+-.2%2A15%5E2+%2B+3.6%2A15+-+9+=+0+
+-.2%2A225+%2B+54+-+9+=+0+
+-45+%2B+54+-+9+=+0+
+0+=+0+
OK
You can check other solution

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
The profit P, in thousands of dollars, that a manufacturer makes is a function of the number N of items produced in a year, and the formula is as follows:
P= -0.2N^2 + 3.6N - 9
(a) Determine the two break-even points for this manufacturer-that is, the two production levels at which the profit is zero.
(c) How many items should they produce to get this maximum point?
Please show me how you get the answer, thank you.
(a)  matrix%281%2C3%2C+0%2C+%22=%22%2C+-+.2N%5E2+%2B+3.6N+-+9%29

     matrix%281%2C3%2C+0%2C+%22=%22%2C+N%5E2+-+18N+%2B+45%29 ------- Multiplying by - 5 to clear DECIMALS

     0 = (N - 3)(N - 15)----- Factoring trinomial

     N - 3 = 0             OR                 N - 15 = 0

     N, or 2 break-even points/production levels at which profit is 0 are: highlight_green%28matrix%281%2C3%2C+%223%2C%22%2C+and%2C+15%29%29

(c)  Number of items to be produced in order to achieve maximum profit is where