The weekly revenue for a company is r= -4p^2 + 50p +1017 , where p is the price of the company’s product. Use the discriminate to find whether there is a price for which the weekly revenue would be $2000.
To find out if there is a price for which the weekly revenue would be $2000, we need to set up the equation:
-4p^2 + 50p +1017 = 2000
Rearrange the equation to form a quadratic equation in standard form:
-4p^2 + 50p + 1017 - 2000 = 0
Simplify:
-4p^2 + 50p - 983 = 0
Now we have a quadratic equation in the form of:
ap^2 + bp + c = 0
where a = -4, b = 50, and c = -983.
To determine whether there is a price for which the weekly revenue would be $2000, we need to find the discriminant, which is given by the formula:
D = b^2 - 4ac
Substituting the values from our equation, we have:
D = 50^2 - 4(-4)(-983)
D = 2500 - 15728
D = -13228
Since the discriminant is negative (D < 0), there are no real solutions to the equation. Therefore, there is no price for which the weekly revenue would be $2000.
To determine whether there is a price for which the weekly revenue would be $2000, we can use the quadratic equation and the discriminant.
The quadratic equation is given by:
r = -4p^2 + 50p + 1017
To find the price (p) for which the weekly revenue (r) would be $2000, we set r = 2000:
2000 = -4p^2 + 50p + 1017
Rearranging the equation:
4p^2 - 50p - 1017 = 0
Now we can calculate the discriminant, which is given by:
D = b^2 - 4ac
In our equation, a = 4, b = -50, and c = -1017.
Substituting these values:
D = (-50)^2 - 4(4)(-1017)
D = 2500 + 16272
D = 18772
The discriminant (D) is positive because it is greater than zero, which means that there are two distinct real solutions for p. Therefore, there are prices for which the weekly revenue would be $2000.
To determine whether there is a price for which the weekly revenue would be $2000, we need to use the discriminant. The discriminant is a mathematical term used to determine the nature of the roots of a quadratic equation.
The quadratic equation for the given revenue function is:
r = -4p^2 + 50p + 1017
To find the discriminant, we will use the formula:
discriminant = b^2 - 4ac
Where a, b, and c are the coefficients of the quadratic equation.
For our revenue equation, the coefficients are:
a = -4
b = 50
c = 1017
Substituting these values into the formula, we can calculate the discriminant:
discriminant = (50)^2 - 4(-4)(1017)
= 2500 - (-16272)
= 2500 + 16272
= 18772
The discriminant is calculated to be 18772.
Now, let's analyze the discriminant to determine if there is a price for which the weekly revenue would be $2000:
If the discriminant is greater than zero (D > 0), then there are two distinct real roots. In this case, there will be two different prices for which the weekly revenue would be $2000.
If the discriminant is equal to zero (D = 0), then there is one real root (one price) for which the weekly revenue would be $2000.
If the discriminant is less than zero (D < 0), then there are no real roots, and there is no price for which the weekly revenue would be $2000.
Since the discriminant is 18772, which is greater than zero (D > 0), there are two distinct real roots, and therefore, there are two different prices for which the weekly revenue would be $2000.
To find these prices, the next step would be to solve the quadratic equation for r = 2000.