Maximum revenue

Question

A company sells bolts for $0.95, selling approx. 10,000 bolts per month. For every $0.15 increase in price, the company will sell 500 fewer bolts. determine the price of the bolt that will maximize the revenue. Complete the square to find the maximum revenue.

Answer 1

How many $0.15 increases will maximize the profit ?

($0.95+$0.15x)(10,000-500x)=0 where x is the number of $0.15 increases

9500+1500x-475x-75x²=0

-75x²+1025x+9500=0

the maximum occurs at the vertex, (h,k), of the parabola

x=-b/2a

x=-1025/(-150)

x=6.8333...

seeing that you have to have whole number increases, you need 7 $0.15 increases

$0.95+7*$0.15=$0.95+$1.05=$2.00

$2.00*(10,000-7*500)

$2.00*(10,000-3500)

$2.00*6500=$13,000 is the maximum revenue

if there were 6 increases...

$1.85*7000=$12,950

if there were 8 increases...

$2.15*6000=$12,900

$2.00 is the price of the bolt and the maximum revenue is $13,000 (7 increases of $0.15)

 

Answer 2

( 0.95, 10000)

 

M = -500/0.15 = -3333.33

 

 

The equation of the line is y = m x +  b

  So b = y  - mx

 

  b = 10000 - (-3333.33)(0.95)

 

     = 13166.66

 

The demand function is D(x) = (-3333.33)X + 13166.66

 

The revenue function is R(x) = x*D(x) = (-3333.33)X^2 + 13166.66*x

 

So yes, in order to maximize the revenue, we must complete the square.

(There is actually an easier way using calculus, which is will address afterwards)

 

Factoring out the leading coefficient.....

(-3333.33){X^2 + 3.95*x                } =

 

Completes the square

(-3333.33){X^2 + 3.95*x   +  3.900625 } + 13002.083    <--leading coefficient is negative,compensates by adding

 

(-3333.33) { X   +  1.975 }^2 + 13002.083

 

 

So the max revenue occurs at $1.975    (Remember.... the leading coefficient is negative....)

 

Secondly, notice the zeros of the revenue function are 0 and 13166.66/3333.33 = 3.95

The axis of symmetry of the parabola where the max/min occurs at the AVERAGE of the zeros.

(0+3.95)/2 = 1.975

 

Finally, calculus fans.....

The derivative of the revenue function is -6666.66X + 13166.66

The max occurs when the derivative is zero.

That is -13166.66/-6666.66  = 1.975

 

Same problem done 3 different ways, same result.   

1.975