Questions: A person has a taco stand. They have found that their daily costs are approximated by C(x) = x^2 - 30x + 455, where C(x) is the cost, in dollars, to sell x units of tacos. Find the number of units of tacos they should sell to minimize costs. What is the minimum cost? The person should sell units of tacos to minimize the costs. The minimum cost is . (Simplify your answers. Type integers or fractions.)

$A person has a taco stand. They have found that their daily costs are approximated by C(x) = x^2 - 30x + 455, where C(x) is the cost, in dollars, to sell x units of tacos. Find the number of units of tacos they should sell to minimize costs. What is the minimum cost? The person should sell units of tacos to minimize the costs. The minimum cost is . (Simplify your answers. Type integers or fractions.)$

Transcript text: A person has a taco stand. They have found that their daily costs are approximated by $C(x)=x^{2}-30 x+455$, where $C(x)$ is the cost, in dollars, to sell $x$ units of tacos. Find the number of units of tacos they should sell to minimize costs. What is the minimum cost? The person should sell $\square$ units of tacos to minimize the costs. The minimum cost is $\$$ $\square$ . (Simplify your answers. Type integers or fractions.)

Solution

Solution Steps

Step 1: Find the x-coordinate of the vertex

To find the number of units $x$ that minimizes the daily costs, we calculate the x-coordinate of the vertex of the quadratic function $C(x) = x^2 - 30x + 455$. The formula to find the x-coordinate of the vertex is $x = -\frac{b}{2a}$, substituting the given values, we get $x = -\frac{-30}{2*1} = 15$.

Step 2: Calculate the minimum cost

Substituting $x = 15$ back into the original cost function to find the minimum cost, we get: $C(15) = 1(15)^2 - 30(15) + 455 = 230$.