Problema Solution
A manufacturer charges $24 for stereo headphones and had been selling 1000 per week. He estimates that for every $1 decrease in price 100 more headphones could be sold per week.
a. Write an expression to represent the price of the headphones.
b. Write and expression for the number of headphones sold.
c. Write a quadratic function to represent the total revenue received in one week.
d. Identify the domain of the function.
e. Identify the range of the function.
f. What price will maximize the total revenue?
Answer provided by our tutors
let x = number of dollar reductions in price
a. Write an expression to represent the price of the headphones.
the reduced price per set of headphones is: Price = (24-x)
b. Write and expression for the number of headphones sold.
x = additional number of 100's of units sold
Units Sold = 1000+100x
c. Write a quadratic function to represent the total revenue received in one week.
Revenue: R = Price * Units Sold
R = (24-x) (1000+100x)
R = -100x^2 + 1400x + 24000
click here to see the step by step simplification:
d. Identify the domain of the function.
24 - x >= 0
24 >= x
the domain of the function is (0, 24]
e. Identify the range of the function.
R max = c - b^2/(4a), where a = -100, b = 1400, c = 24000
R max = 24000 - 1400^2/(-4*100)
R max = $28,900
click here to see the step by step solution:
the range of the function is [0, 28900]
f. What price will maximize the total revenue?
we need to find x such that R = 28900 that is
-100x^2 + 1400x + 24000 = 28900
by solving we find
x = $7
click here to see the step by step solution of the equation:
Price = (24-x)
Price = (24-7)
Price = $17
the price of $17 will maximize the total revenue.