SOLUTION: Woods Limited manufactures and sells x small patio tables each day. The daily cost of production is modeled as C(x) = 3x2 + 1500, and the revenue produced each day is modeled as R(

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Question 959010: Woods Limited manufactures and sells x small patio tables each day. The daily cost of production is modeled as C(x) = 3x2 + 1500, and the revenue produced each day is modeled as R(x) = 180x. Suppose the company is limited by time to producing at least 30 tables every day. What is the break-even point? If the company sells 45 tables, then what is the profit? How many tables must the company produce in order to get a profit of $1,125? What happen if the company produces 58 tables?

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Woods Limited manufactures and sells x small patio tables each day.
The daily cost of production is modeled as
C%28x%29+=+3x%5E2+%2B+1500, and the revenue produced each day is modeled as R%28x%29+=+180x.
Suppose the company is limited by time to producing at least 30 (at least means ≥30)tables every day.

1. What is the break-even point?
to find it set R%28x%29+=+C%28x%29, then solve the resulting quadratic equation
he break-even point occurs when
+3x%5E2+%2B+1500=180x
+3x%5E2+-180x%2B+1500=0 .......simplify, divide all terms by 3
+x%5E2+-60x%2B+500=0 ........use quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-60%29%2B-+sqrt%28+%28-60%29%5E2-4%2A1%2A500+%29%29%2F%282%2A1%29+
x+=+%2860%2B-+sqrt%28+3600-2000+%29%29%2F2+
x+=+%2860%2B-+sqrt%28+1600+%29%29%2F2+
x+=+%2860%2B-+40%29%2F2+
solutions:
x+=+%2860%2B+40%29%2F2+
x+=+50+
or
x+=+%2860-+40%29%2F2+
x+=+10+
your solution is highlight%28x=50%29 because the company is producing at least 30 tables to have the break-even point ; so, disregard x+=+10+

2. If the company sells 45 tables, then what is the profit?
The profit function P%28x%29+=+R%28x%29+-+C%28x%29,
write P%28x%29 in terms of x, then evaluate P%2845%29
P%28x%29+=+180x+-3x%5E2+%2B+1500
P%2845%29+=+180%2A45+-3%2A45%5E2+%2B+1500
P%2845%29+=+8100+-6075+%2B+1500
P%2845%29+=+3525=> the profit the company earns if sells 45 tables

3.
How many tables must the company produce in order to get a profit of $1125?
Set P%28x%29+=+1125, then solve the resulting quadratic equation
1125+=+180x+-3x%5E2+%2B+1500
3x%5E2-180x%2B1125-1500=0
3x%5E2-180x-375=0
x%5E2-60x-125=0........use quadratic formula
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-%28-60%29+%2B-+sqrt%28+%28-60%29%5E2-4%2A1%2A%28-125%29+%29%29%2F%282%2A1%29+
x+=+%2860+%2B-+sqrt%28+3600%2B500+%29%29%2F2+

x+=+%2860+%2B-+sqrt%28+4100+%29%29%2F2+
x+=+%2860+%2B-+64.03124237432849%29%2F2+
x+=+%2860+%2B-+64.03%29%2F2+
we need only positive value since x represents an order
x+=+%2860+%2B+64.03%29%2F2+
x+=+124.03%2F2+
x+=+62.01562118716424+- exact solution
x+=+62+round it to whole number since x represents tables



4.
What happen if the company produces 58 tables?
evaluate P%2858%29
P%2858%29+=+180%2A58+-3%2A58%5E2+%2B+1500
P%2858%29+=+10440+-10092+%2B+1500
P%2858%29+=+348+%2B+1500
P%2858%29+=+1848 => the profit the company earns if sells 58 tables