To determine the width of the deck Mario can install around his swimming pool given that he has [tex]$4080 to spend, we follow these steps:
1. Identify the cost function provided:
The cost function given is \( C(w) = 120w^2 + 1800w \), where \( C(w) \) is the cost in dollars and \( w \) is the width in meters.
2. Set the given total cost equal to the cost function:
Since Mario has $[/tex]4080 to spend, we equate the cost function to this amount:
[tex]\[
120w^2 + 1800w = 4080
\][/tex]
3. Formulate the quadratic equation:
Rearrange the equation to standard form:
[tex]\[
120w^2 + 1800w - 4080 = 0
\][/tex]
4. Simplify the quadratic equation:
We can simplify this equation by dividing every term by 60 to make it easier to solve:
[tex]\[
2w^2 + 30w - 68 = 0
\][/tex]
5. Solve the simplified quadratic equation using the quadratic formula:
The quadratic formula is given by:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\(2w^2 + 30w - 68 = 0\)[/tex], the coefficients are:
[tex]\[
a = 2, \quad b = 30, \quad c = -68
\][/tex]
Plug these values into the quadratic formula:
[tex]\[
w = \frac{-30 \pm \sqrt{30^2 - 4 \cdot 2 \cdot (-68)}}{2 \cdot 2}
\][/tex]
[tex]\[
w = \frac{-30 \pm \sqrt{900 + 544}}{4}
\][/tex]
[tex]\[
w = \frac{-30 \pm \sqrt{1444}}{4}
\][/tex]
[tex]\[
w = \frac{-30 \pm 38}{4}
\][/tex]
6. Calculate the roots:
Solve for the two possible values of [tex]\( w \)[/tex]:
[tex]\[
w_1 = \frac{-30 + 38}{4} = \frac{8}{4} = 2
\][/tex]
[tex]\[
w_2 = \frac{-30 - 38}{4} = \frac{-68}{4} = -17
\][/tex]
7. Select the valid solution:
Since a width cannot be negative, the valid solution is:
[tex]\[
w = 2
\][/tex]
Thus, the width of the deck that Mario can install around his swimming pool, given he has \$4080 to spend, is 2 meters.
