a newspaper company currently charges $11 a week to its subscribers, but is considering raising their price. the company found that the weekly revenue y can be modeled by the function
y= -750(x-40)(x+20) where x is the number $.50 increases in the weekly price.
a. explain how to find the price that will maximize weekly revenue. what is the price?
b. what is the maximum weekly revenue?
20.59
a. To find the price that will maximize weekly revenue, we need to find the value of x that corresponds to the maximum value of the function y.
First, let's rewrite the given function:
y = -750(x - 40)(x + 20)
To find the maximum value of y, we need to find the vertex of the parabolic function. The vertex of a quadratic function in the form of ax^2 + bx + c can be found using the formula: x = -b / (2a).
In this case, a = -750, b = 40, and c = -750 * 20:
x = -40 / (2 * (-750))
x = -40 / (-1500)
x = 2/75
Now, we need to convert x back to the increase in the weekly price. Given that x represents the number of $.50 increases in the weekly price, we can calculate the price by multiplying x by $0.50:
Price = x * $0.50
Price = (2/75) * $0.50
Price = $0.0133
Therefore, the price that will maximize weekly revenue is approximately $0.0133 (or rounded to $0.01).
b. To find the maximum weekly revenue, we substitute the value of x = 2/75 into the given function:
y = -750(x - 40)(x + 20)
y = -750(2/75 - 40)(2/75 + 20)
y = -750((-77 + 2/75)(80/75))
Simplifying further, we get:
y = -750(-6218/75)(80/75)
y ≈ $83.09
Therefore, the maximum weekly revenue is approximately $83.09.
To find the price that will maximize the weekly revenue, we need to understand that the weekly revenue is represented by the function y = -750(x-40)(x+20), where x represents the number of $0.50 increases in the weekly price.
a. To find the price that maximizes the weekly revenue, we need to find the value of x that gives the maximum value for y. Here's how you can do it:
1. Rewrite the revenue function as y = -750x^2 + 11000x - 60000.
2. This function represents a quadratic equation, which has a maximum or minimum value at its vertex.
3. The vertex of a quadratic equation in the form y = ax^2 + bx + c can be found using the formula x = -b/(2a).
In this case, a = -750 and b = 11000.
4. Substitute the values of a and b into the formula: x = -11000/(2*-750).
5. Simplify: x = -11000/-1500 = 7.3333.
6. Since x represents the number of $0.50 increases, multiply x by $0.50 to find the price increase: 7.3333 * $0.50 = $3.6667.
Therefore, the price that will maximize weekly revenue is a $3.67 increase in the weekly price.
b. Now, let's find the maximum weekly revenue associated with the price increase. Substitute the value of x that we found above into the original revenue function:
y = -750(7.3333 - 40)(7.3333 + 20).
Using a calculator or solving manually, we get:
y = -750(-32.6667)(27.3333) = 697333.33
Therefore, the maximum weekly revenue is $697,333.33.