A consultant advises the owners of a beauty salon that their profit p each month can be modeled by p(x) = -50x^2 +3500x – 2500, where x is the average cost that a customer is charged. What range of costs will bring in a profit of at least 50 000? *
a. (21.8, 48.2)
b. [21.8, 48.2]
c. (-∞, 21.8) U (48.2, +∞)
d. (-∞, 21.8] U [48.2, +∞)
Answers
Given : profit p each month can be modeled by p(x) = -50x^2 +3500x – 2500
x is the average cost that a customer is charged
To Find : range of costs will bring in a profit of at least 50 000? *
a. (21.8, 48.2)
b. [21.8, 48.2]
c. (-∞, 21.8) U (48.2, +∞)
d. (-∞, 21.8] U [48.2, +∞)
Solution:
p(x) = -50x² +3500x – 2500
profit of at least 50 000
=> -50x² +3500x – 2500 > 50000
=> 0 > 50x² - 3500x + 52500
=> 50x² - 3500x + 52500 < 0
=> x² - 70x + 1050 < 0
=> x² - 21.771x - 48.229x + 1050 < 0
=> (x - 21.771)(x - 48.229) < 0
=> 21.771 < x < 48.229
=> 21.8 ≤ x ≤ 48.2
x ∈ [21.8, 48.2]
range of costs charged x ∈ [21.8, 48.2] will bring in a profit of at least 50 000
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