Final answer:
To find if a weekly revenue of $1800 can be achieved for the given quadratic revenue function, we set the revenue function equal to $1800 and solved for price using the discriminant. The discriminant, obtained from the quadratic equation, turned out to be negative, which indicates that there is no real price that would result in a weekly revenue of $1800.
Step-by-step explanation:
The student wants to determine if a certain weekly revenue goal can be achieved based on the price of the product, using a given quadratic revenue function r and the discriminant of a quadratic equation. The discriminant is part of the quadratic formula that helps to determine the nature of the roots of a quadratic equation.
To find whether a price exists for which the weekly revenue would be $1800 using the revenue function r = -2p² + 40p + 1293, we need to set the revenue r to $1800 and solve for p. This will result in a quadratic equation which we can then analyze using its discriminant.
First, we set the revenue to $1800:
-2p² + 40p + 1293 = 1800.
By subtracting 1800 from both sides, we obtain the quadratic equation:
-2p² + 40p - 507 = 0.
The coefficients of this equation are: a = -2, b = 40, and c = -507. The discriminant of a quadratic equation is given by b² - 4ac. Substituting the coefficients:
Discriminant = 40² - 4(-2)(-507) = 1600 - 4(1014) = 1600 - 4056 = -2456.
A negative discriminant indicates that there are no real solutions to the equation, which means there is no real price p that would result in a weekly revenue of $1800.