Question

1. a businesss revenue is modeled by $r(x) = -4x^2 + 80x$, where $x$ is the number of units sold. how many units should be sold to maximize revenue?

a. 20 units
b. 22 units
c. 18 units
d. 10 units

1. what are the roots of the quadratic equation $4x^2 - 4x + 1 = 0?$

a. $x = 2$
b. $x = 1$ and $x = 2$
c. $x = 1$ and $x = 3$
d. $x = \frac{1}{2}$

Explanation:

Question 1:

Step1: Identify the function type

The revenue function \( R(x) = -4x^2 + 80x \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -4 \), \( b = 80 \), and \( c = 0 \). For a quadratic function, if \( a < 0 \), the parabola opens downward, and the vertex is the maximum point.

Step2: Find the x-coordinate of the vertex

The formula for the x-coordinate of the vertex of a quadratic function \( f(x) = ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substituting \( a = -4 \) and \( b = 80 \) into the formula:

$$ x = -\frac{80}{2 \times (-4)} = -\frac{80}{-8} = 10 $$

Wait, that seems conflicting with the options? Wait, no, wait, let's recalculate. Wait, \( a = -4 \), \( b = 80 \). So \( x = -b/(2a) = -80/(2(-4)) = -80/(-8) = 10 \)? But option d is 10 units. Wait, but let's check again. Wait, maybe I made a mistake? Wait, the function is \( R(x) = -4x^2 + 80x \). So \( a = -4 \), \( b = 80 \). Then \( x = -b/(2a) = -80/(2(-4)) = -80/-8 = 10 \). So the number of units to maximize revenue is 10? But let's check the options. Option d is 10 units. Wait, but let's verify by expanding or using another method. Alternatively, maybe I misread the function. Wait, the function is \( R(x) = -4x^2 + 80x \). Let's factor it: \( R(x) = -4x(x - 20) \). The roots are at \( x = 0 \) and \( x = 20 \). The vertex is at the midpoint of the roots, so \( (0 + 20)/2 = 10 \). Yes, that's correct. So the x-coordinate of the vertex is 10, so 10 units should be sold to maximize revenue.

Step1: Identify the quadratic equation

The quadratic equation is \( 4x^2 - 4x + 1 = 0 \). We can solve this using factoring, completing the square, or the quadratic formula. Let's try factoring. Notice that \( 4x^2 - 4x + 1 \) is a perfect square trinomial. \( (2x - 1)^2 = 4x^2 - 4x + 1 \). So the equation becomes \( (2x - 1)^2 = 0 \).

Step2: Solve for x

Taking the square root of both sides: \( 2x - 1 = 0 \). Solving for x: \( 2x = 1 \) => \( x = \frac{1}{2} \). So the root is \( x = \frac{1}{2} \) (a repeated root).

Answer:

d. 10 units

Question 2: