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?

a. To find the price that will maximize weekly revenue, we need to find the maximum point of the revenue function. This can be done by finding the x-value at which the revenue function reaches its maximum.

The revenue function is given by y = -750(x-40)(x+20), where x is the number $.50 increases in the weekly price.

To find the x-value that maximizes the revenue, we need to find the vertex of the parabola defined by this function. The vertex of a quadratic function in the form f(x) = ax^2 + bx + c can be found using the formula x = -b / (2a).

Comparing this with the revenue function y = -750(x-40)(x+20), we can see that a = -750, b = 0, and c = 0.

Using the formula for the x-coordinate of the vertex, we can find:
x = -0 / (2(-750))
x = 0

Hence, the price that will maximize weekly revenue is $0.50.

b. To find the maximum weekly revenue, we substitute the x-value of 0.50 into the revenue function:
y = -750(0.50-40)(0.50+20)

Simplifying the expression, we get:
y = -750(-39.50)(20.50)
y = 30,737.50

Therefore, the maximum weekly revenue is $30,737.50.

To find the price that will maximize weekly revenue, we need to determine the value of x that will result in the maximum value of the function y=-750(x-40)(x+20).

a. To find the price that maximizes weekly revenue:
Step 1: Expand the equation:
y = -750(x^2 - 20x + 800)
= -750x^2 + 15000x - 60000

Step 2: We can see that the equation is a quadratic function in the form of ax^2 + bx + c. The maximum value occurs at the vertex of a quadratic function. The x-coordinate of the vertex can be found using the formula: x = -b / (2a).

In our equation, a = -750, and b = 15000. Plugging these values into the formula, we get:
x = -15000 / (2 * (-750))
= -15000 / (-1500)
= 10

Step 3: The price can be found by substituting the value of x back into the equation. Since x represents the number of $0.50 increases, the original price is $11 + ($0.50 * x).
Substituting x = 10:
Price = $11 + ($0.50 * 10)
= $11 + $5
= $16

Therefore, the price that will maximize weekly revenue is $16.

b. To find the maximum weekly revenue:
Step 1: After obtaining the value of x that maximizes weekly revenue (x = 10), substitute this value back into the equation y=-750(x-40)(x+20).
y = -750(10-40)(10+20)
= -750(-30)(30)
= -750 * 900
= -675,000

Hence, the maximum weekly revenue is $675,000.

since y is a parabola, with roots at -20 and 40, the max is achieved halfway between the roots, at x=10.

price = 11.00 + 10*0.50 = 16.00
revenue is y(10) = 675000