Find the break-even point for the firm whose cost function C and revenue function R are given.
C(x)=0.3x+140; R(x)=0.5x
a. P(680)=340
b. P(700)=350
c. P(690)=345
d. P(720)=360
e. P(710)=355
thank you
To find the break-even point for the firm, we need to determine the value of x where the revenue equals the cost.
Given that the cost function is C(x) = 0.3x + 140 and the revenue function is R(x) = 0.5x, we can set up the equation R(x) = C(x) and solve for x.
0.5x = 0.3x + 140
To isolate x, we subtract 0.3x from both sides:
0.5x - 0.3x = 140
0.2x = 140
Next, we divide both sides by 0.2:
x = 140 / 0.2
x = 700
The break-even point occurs when x is equal to 700.
Now, let's check the given options:
a. P(680) = 0.5(680) = 340 (Not a break-even point)
b. P(700) = 0.5(700) = 350 (Break-even point, as revenue equals cost)
c. P(690) = 0.5(690) = 345 (Not a break-even point)
d. P(720) = 0.5(720) = 360 (Not a break-even point)
e. P(710) = 0.5(710) = 355 (Not a break-even point)
Therefore, the only break-even point among the given options is b. P(700) = 350.
To find the break-even point, we need to determine the quantity at which the revenue equals the cost.
The revenue function R(x) represents the total revenue generated by selling x units, and the cost function C(x) represents the total cost incurred in producing x units.
The break-even point occurs when the revenue equals the cost, so we need to solve the equation R(x) = C(x) to find the value of x.
Given:
C(x) = 0.3x + 140
R(x) = 0.5x
Setting R(x) equal to C(x), we have:
0.5x = 0.3x + 140
Subtracting 0.3x from both sides of the equation, we get:
0.2x = 140
To solve for x, we divide both sides of the equation by 0.2:
x = 140 / 0.2
x = 700
Therefore, the break-even point occurs at x = 700.
Now, let's check the options one by one:
a. P(680) = 340
Plugging x = 680 into the revenue function R(x), we get:
R(680) = 0.5 * 680 = 340
Since P(680) equals the revenue at that quantity, this option is correct.
b. P(700) = 350
Plugging x = 700 into the revenue function R(x), we get:
R(700) = 0.5 * 700 = 350
Since P(700) equals the revenue at that quantity, this option is correct.
c. P(690) = 345
Plugging x = 690 into the revenue function R(x), we get:
R(690) = 0.5 * 690 = 345
Since P(690) equals the revenue at that quantity, this option is correct.
d. P(720) = 360
Plugging x = 720 into the revenue function R(x), we get:
R(720) = 0.5 * 720 = 360
Since P(720) equals the revenue at that quantity, this option is correct.
e. P(710) = 355
Plugging x = 710 into the revenue function R(x), we get:
R(710) = 0.5 * 710 = 355
Since P(710) equals the revenue at that quantity, this option is correct.
Therefore, the answer is:
a. P(680) = 340
b. P(700) = 350
c. P(690) = 345
d. P(720) = 360
e. P(710) = 355
just solve C(x) = R(x)
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