Answer:
a)C(x)=1500+35x
b)R(x)=60 x
c)P(x)=25x-1500
d)To achieve a break-even point, 60 lessons must be conducted.
e)If the studio conducts 82 lessons, it will earn 550.
Step-by-step explanation:
Monthly expenses including rent, utilities, insurance, and advertising total $1500.
The studio charges $60 per private lesson but incurs a $35 variable cost per lesson paid to the instructor.
a) To express the cost function C(x) for x private lessons in a month:
Fixed costs = $1500
Instructor's variable cost for each lesson = $35
The variable cost for x lessons = 35x
Thus, C(x)=1500+35x
b) To express the revenue function R(x) for x private lessons:
Earning from one lesson = $60
For x lessons = 60 x
Hence, R(x)=60 x
c) To formulate the profit function P(x) for x lessons:
Profit = Revenue - Cost
P(x)=R(x)-C(x)
P(x)=60x-1500-35x
P(x)=25x-1500
d) Finding the break-even number of lessons:
Setting R(x)=C(x)
60x=1500+35x
25x=1500
x=60
So, the studio must conduct 60 lessons to break even.
e) If 82 lessons are held, computing profit:
P(x)=25x-1500
P(82)=25(82)-1500
P(82)=550
Thus, the studio will net 550 for 82 lessons.
