Question

you are interested in retrieving a souvenir from a shipwreck located 115 feet below the water. you do not have diving equipment, so your dive is limited by the equation $y = 0.05x^2 - 4x - 38$ assuming you are able to retrieve your souvenir, how long do you have to hold your breath until you resurface? your answer this is a required question

Explanation:

Step1: Set y = -115 (depth of wreck)

$-115 = 0.05x^2 - 4x - 38$

Step2: Rearrange to standard quadratic form

$0.05x^2 - 4x - 38 + 115 = 0$
$0.05x^2 - 4x + 77 = 0$
Multiply all terms by 20 to eliminate decimals:
$x^2 - 80x + 1540 = 0$

Step3: Apply quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here, $a=1$, $b=-80$, $c=1540$
$\Delta = (-80)^2 - 4(1)(1540) = 6400 - 6160 = 240$
$\sqrt{\Delta} = \sqrt{240} = 4\sqrt{15} \approx 15.49$
$x = \frac{80\pm15.49}{2}$

Step4: Calculate valid positive roots

$x_1 = \frac{80 + 15.49}{2} = 47.745$
$x_2 = \frac{80 - 15.49}{2} = 32.255$
We take the larger root, as it represents the total time to resurface.

Answer:

Approximately 47.75 seconds (or exactly $40 + 2\sqrt{15}$ seconds)