SOLUTION: Hi, please help me answer this question: The president of a company producing water pumps has gathered data suggesting that the average cost in dollars producing x units per hou

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: Hi, please help me answer this question: The president of a company producing water pumps has gathered data suggesting that the average cost in dollars producing x units per hou      Log On


   

Question 107033: Hi, please help me answer this question:
The president of a company producing water pumps has gathered data suggesting that the average cost in dollars producing x units per hour of a new style of pump will be C(x)=x^2-22x+166. Determine the number of pumps per hour to produce to keep the average cost below $75 per pump.
Thanks!
Found 2 solutions by Annabelle1, edjones:
Answer by Annabelle1(69) About Me  (Show Source):
You can put this solution on YOUR website!
you need to solve C(x)=x^2-22x+166<75
s0 x^2-22x+166-75<0
x^2-22x+91<0
this isn't factorisable so put it into the quadratic formula
a=1 b=-22 c=91
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%2822+%2B-+sqrt%28+22%5E2-4%2A1%2A91+%29%29%2F%282%2A1%29+
x=(22+-sqrt(120)/2
x=16.477
or
x=5.523


Answer by edjones(8007) About Me  (Show Source):
You can put this solution on YOUR website!
graph%28500%2C500%2C-2%2C40%2C-10%2C90%2Cx%5E2-22x%2B166%29
Between 6 and 16 pumps per hour should do the trick according to my graphing calculator and the graph above. At 11 pumps/h the cost reaches a minimum of about $45.
.
if you want to do the math:
x^2-22x+166<=75
x^2-22x+91<=0
5.52...<=x<=16.48...
Ed
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-22x%2B91+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-22%29%5E2-4%2A1%2A91=120.

Discriminant d=120 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--22%2B-sqrt%28+120+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-22%29%2Bsqrt%28+120+%29%29%2F2%5C1+=+16.4772255750517
x%5B2%5D+=+%28-%28-22%29-sqrt%28+120+%29%29%2F2%5C1+=+5.52277442494834

Quadratic expression 1x%5E2%2B-22x%2B91 can be factored:
1x%5E2%2B-22x%2B91+=+%28x-16.4772255750517%29%2A%28x-5.52277442494834%29
Again, the answer is: 16.4772255750517, 5.52277442494834. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-22%2Ax%2B91+%29