can you solve this pease?

Question

a certain company the cost function for poducing x items is c(x)=50x+150 and the revenue for selling x items is r(x)= -0.5(x-130)^2+8450. the maximum capacity of the company is 160 items

what is the domain of P(x)?

does calculating P(x) make sense when x=-10 or x=1000?

the company can choose to produce each case and which level of production should they choose?

profit when producing 80 items?=

profit when producing 90 items=

can you explain from our model why the company makes less profit when producing 10 more items?

Answer 1

domain is 0<x<160 in practical economics

or in interval notation [0,160]

but for theoretical math, domain could be considered -infinity to +infinity

although it's financially meaningless below 0 or above 160

still it's useful for considering expansion of the manufacturing facilities to go above 160

but there are finite resources in the universe, so a finite upper boundary is reality

less an zero output is strange, how do you produce a negative output? Maybe

they buy back previously produced output that was previously purchased? even if so,

there's some finite limite to what was previously produced, not -infinity.

P(x) = R(x)-C(x) = -.5(x-130)^2 +8450- 50x -150

P(x) = -.5(x^2 -260x+130^2) +8450 -50x -150

P(x) =-.5x^2 +.5(260x) - .5(16900) +8300 -50x

P(x) = -.5x^2 + 130x - 8450 +8450-50x -150

P(x) = -.5x^2 +80x -150

max profit is when P'(x)=0

P'(x) =-x +80 = 0

x = 80 is the profit maximizing output level

P(80) = -.5(80)^2 +80(80) -150

= -3200+6400 -150

=$3050

P(90) = -.5(90)^2 +80(90)-150

= -8100/2 + 7200 -150

= -4050+7050

= $3000

a loss of $50 profit by increasing output by 10

P'(x) = R'(x) - C'(x) = MR-MC = 0

max Profit is when Marginal Revenue = Marginal Cost

at higher output levels,

such as going from 80 to 90 MC>MR,

adding more cost than revenue for each increase in output

above the profit maximizing output level

P(x) =-.5x^2 +180x -150 = 0 at zero profits

x^2 -360x =-300

x^2 -360x + 180^2 = -300 + 180^2

(x-180)^2 = 32400-300

x-180 =+/- sqr32100

x= 180 +/-10sqr321

x = about 180-10(17.9165)

x =about 180-179.165

x = about 0.835

or about 359.9165

for zero profit, output is about 1 or 360

financially the domain could also be considered about [1, 360]