Use the answer key to determine that you accurately solved each problem. ● If necessary, revise your work. 1. The cost of new band uniforms is modeled by C(x) = 0.1x squared + 2.7x + 40 where x is the number of uniforms purchased. How many new uniforms can be purchased for $600? Round your answer to nearest whole number.

1 answer

To determine how many new band uniforms can be purchased for $600 using the cost function \( C(x) = 0.1x^2 + 2.7x + 40 \), we set up the equation with \( C(x) = 600 \):

\[ 0.1x^2 + 2.7x + 40 = 600 \]

Next, we subtract 600 from both sides:

\[ 0.1x^2 + 2.7x + 40 - 600 = 0 \]

This simplifies to:

\[ 0.1x^2 + 2.7x - 560 = 0 \]

To eliminate the decimal, we can multiply the entire equation by 10:

\[ x^2 + 27x - 5600 = 0 \]

Next, we can solve this quadratic equation using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = 27 \), and \( c = -5600 \).

Calculating the discriminant \( b^2 - 4ac \):

\[ b^2 = 27^2 = 729 \] \[ 4ac = 4 \cdot 1 \cdot (-5600) = -22400 \] Thus,

\[ b^2 - 4ac = 729 + 22400 = 23129 \]

Now we apply the quadratic formula:

\[ x = \frac{-27 \pm \sqrt{23129}}{2 \cdot 1} \] \[ x = \frac{-27 \pm 152}{2} \]

Calculating the two potential values for \( x \):

  1. \( x = \frac{-27 + 152}{2} = \frac{125}{2} = 62.5 \)
  2. \( x = \frac{-27 - 152}{2} = \frac{-179}{2} = -89.5 \) (not feasible since we cannot purchase a negative number of uniforms)

Since we need a whole number, we round \( 62.5 \) to the nearest whole number:

\[ x \approx 63 \]

Thus, the number of new uniforms that can be purchased for $600 is approximately 63.