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Sagot :
Functions
The total cost of Zacaria's kayaks store is made of a fixed cost and a variable cost. The fixed cost must be paid even if no kayaks are sold, and the variable cost depends on the number of kayaks sold per year (called x in the rest of this answer).
a. We know each kayak is sold for $425, thus for x kayaks, the total income (or revenue) function is:
R(x) = 425x
b. As mentioned above, the total cost is the sum of the fixed and the variable cost. Each kayak cost $315, thus:
C(x) = 315x + 15,600
c. The break-even point occurs when the cost and the revenue are balanced (no loss-no profit). Equating both functions:
315x + 15,600 = 425x
Subtracting 315x:
15,600 = 425x - 315x
Simplifying:
15,600 = 110x
Solving for x:
x = 15,600 / 110
x = 141.8
Thus, 142 kayaks approximately must be sold to be in break-even condition.
d. Since the earnings are proportional to the number of kayaks sold:
R(x) = 425x
There is no limit for the value of x that maximizes this function, i.e., the greater the value of x, the greater the earnings. There is no maximum point in this function.
In case the earnings function is referring to the profit function, then:
E(x) = R(x) - C(x)
E(x)= 425x - (15,600 + 315x)
E(x) = 110x - 15,600
This function doesn't have a maximum point because it's a linear equation, so there is not a theoretical maximum value for x
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