Question
Mrs. Parkes plans to increase the price of her lotion bottles. It currently costs $20.00 and she has 1000 people who purchase the lotion. Each time she increases the price by $0.50, twenty fewer persons will buy the lotion bottles. She has stated that an equation that models the revenue
is y = (20 + 0.5x)(1000 - 20x). Determine the maximum amount of money made? If 20+0.5x represents the new price of the lotion, based on the value of x in the vertex, what is the new price that would give the maximum revenue?
is y = (20 + 0.5x)(1000 - 20x). Determine the maximum amount of money made? If 20+0.5x represents the new price of the lotion, based on the value of x in the vertex, what is the new price that would give the maximum revenue?
Answers
GPT 3.5
To find the maximum amount of money made, we need to find the vertex of the revenue equation y = (20 + 0.5x)(1000 - 20x).
First, we need to expand the equation:
y = 20,000 + 500x - 400x - 10x^2
y = -10x^2 + 100x + 20,000
Now, the x-coordinate of the vertex is given by:
x = -b/(2a) = -100/(2*(-10)) = 5
To find the maximum revenue, we substitute x = 5 into the revenue equation:
y = -10(5)^2 + 100(5) + 20,000
y = -250 + 500 + 20,000
y = 20,250
Therefore, the maximum amount of money made is $20,250.
To find the new price that would give the maximum revenue, we substitute x = 5 into the equation for the new price: 20 + 0.5x
new price = 20 + 0.5(5)
new price = 20 + 2.5
new price = $22.50
Therefore, the new price that would give the maximum revenue is $22.50.
First, we need to expand the equation:
y = 20,000 + 500x - 400x - 10x^2
y = -10x^2 + 100x + 20,000
Now, the x-coordinate of the vertex is given by:
x = -b/(2a) = -100/(2*(-10)) = 5
To find the maximum revenue, we substitute x = 5 into the revenue equation:
y = -10(5)^2 + 100(5) + 20,000
y = -250 + 500 + 20,000
y = 20,250
Therefore, the maximum amount of money made is $20,250.
To find the new price that would give the maximum revenue, we substitute x = 5 into the equation for the new price: 20 + 0.5x
new price = 20 + 0.5(5)
new price = 20 + 2.5
new price = $22.50
Therefore, the new price that would give the maximum revenue is $22.50.