Question

handmade necklaces
antonia sells handmade necklaces at a craft fair and needs to assess the costs and revenue.
cost: $c(x) = 6x + 405$
revenue: $r(x) = -x^2 + 60x$
where $x$ represents the number of necklaces sold.
how many necklaces need to be sold to reach the maximum revenue?
30
25
20
45

Explanation:

Step1: Identify revenue function type

The revenue function $R(x) = -x^2 + 60x$ is a quadratic function in the form $ax^2+bx+c$, where $a=-1$, $b=60$, $c=0$. Since $a<0$, the parabola opens downward, so its vertex is the maximum point.

Step2: Calculate vertex x-coordinate

For a quadratic $ax^2+bx+c$, the x-coordinate of the vertex is given by $x = -\frac{b}{2a}$.
Substitute $a=-1$, $b=60$:
$$x = -\frac{60}{2(-1)} = \frac{60}{2} = 30$$

Step3: Verify with graph

The graph shows the revenue parabola peaks at $(30, 900)$, confirming the x-value (number of necklaces) at maximum revenue is 30.

Answer:

30