Question

question 23 (mandatory) (1 point)
the cost of running a carwash is a function of the number of cars washed per hour.
the cost function is $c(x) = 1.72x^2 - 18.25x + 47.2$, where $c(x)$ is the cost in dollars, and $x$ is
the number of cars washed per hour. determine the level of approximate number of
cars per hour is most economic.

a) 25 cars/h

b) 3 cars/h

c) 5 cars/h

d) 12 cars/h

Explanation:

Step1: Identify the function type

The cost function \( C(x) = 1.72x^2 - 18.25x + 47.2 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = 1.72 \), \( b = -18.25 \), and \( c = 47.2 \). Since \( a>0 \), the parabola opens upwards, and the vertex represents the minimum point (most economic level).

Step2: Find the x-coordinate of the vertex

The formula for the x-coordinate of the vertex of a quadratic function \( ax^2 + bx + c \) is \( x = -\frac{b}{2a} \). Substitute \( a = 1.72 \) and \( b = -18.25 \) into the formula:

$$ x = -\frac{-18.25}{2 \times 1.72} = \frac{18.25}{3.44} \approx 5.305 $$

This value is approximately 5, so we check the options.

Step3: Compare with options

Among the options, 5 cars/h (option c) is the closest to the calculated approximate value.

Answer:

c) 5 cars/h