Math problem

Question

The model for the amount of production produced by a certain number of employees is given by :

y=-.051x^2+2.33x+20.15

Where y is production in thousands and x is the number of employees.

 

a. How many units of production will 5 employees produce?

b. How many employees will you need to produce 45 units of production?

Answer 1

Part A) The problem states that x represents the number of employees, so replace X in the original equation with 5.

So now you have      

 

  y=-0.051(5)²+2.33(5)+20.15      

 

 now the equation only has one unknown variable, y, which represents the units of production so just solve for y.

 

y=-1.275+11.65+20.15

y=30.52 units of production.

 

Part B) This part tells us that the employees need to produce 45 units, which means y=45 and solve for x. When this is set up it should look like

 

45=-0.051X²+2.33X+20.15

 

Solving for X with numbers on both sides of the equation is a little difficult so let's subtract 45 from both sides. 

 

0=-0.051X²+2.33X-24.85  (let's call this equation 2)

 

Now it is a lot easier to solve for X. By looking at the decimals we can guess that this equation cannot be factored, so we will use the quadratic formula.

 

recall the quadratic formula is (-b±√(b²-4ac))/2a

 

Now using the a,b, and c values from equation 2 the quadratic formula should work to 

 

(-2.33±0.599583)/(-0.102)

 

which produces the X values of

 

X=16.96  and X=28.72

 

Now we cannot have .96 or .72 of an employee so lets round up to X=17 and X=29. The final step is to take both X values that we found and put them back into the original equation because we want y=45.

When this is done,

 

X=17   →    y=-0.051(17)2+2.33(17)+20.15 = 45.021 units of production

and

X=29  →   y=-0.051(29)2+2.33(29)+20.15 = 44.829 units of production

 

The answer then is X=17. 17 employees produce 45 units. 

We do not select X=29 because just like we cannot have .96 of an employee, we cannot have .829 of a unit and 44 whole units. X=17 produce 45 whole units so we do not need to worry about the .021 unit.

 

Good Luck!