Question

question 23 (mandatory) (1 point) saved
the cost of running a carwash is a function of the number of cars washed per hour. the cost function is ( c(x) = 1.73x^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 \( C(x) = 1.75x^2 - 18.25x + 47.2 \) is a quadratic function in the form \( ax^2 + bx + c \), where \( a = 1.75 \), \( b = -18.25 \), and \( c = 47.2 \). For a quadratic function with \( a>0 \), the parabola opens upwards, and the vertex represents the minimum 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 \( x = -\frac{b}{2a} \). Substitute \( a = 1.75 \) and \( b = -18.25 \) into the formula:

$$ x = -\frac{-18.25}{2\times1.75}=\frac{18.25}{3.5}\approx5.21 $$

This value is approximately 5, which matches option c.

Answer:

c) 5 cars/h