Question

the equation $-16x^2 + 10x + 15 = y$ represents the height $y$, in feet, of a flotation device above the water after $x$ seconds. the linear term represents the initial velocity. the constant term represents the initial height.

a. if the initial velocity is 0, when should the flotation device land in the water?

b. if the initial height is 0, when should the flotation device be in the water?

a. the flotation device should land in the water at the time \boxed{} s.
( type an integer or decimal rounded to two decimal places as needed )

Explanation:

Step1: Set up equation for part a

When the device lands in water, height $y=0$, initial velocity term (10x) is removed, so:
$-16x^2 + 15 = 0$

Step2: Rearrange to solve for $x^2$

$16x^2 = 15$
$x^2 = \frac{15}{16}$

Step3: Solve for positive $x$

$x = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4}$
$x \approx 0.97$

Step4: Set up equation for part b

When initial height is 0, constant term (15) is removed, set $y=0$:
$-16x^2 + 10x = 0$

Step5: Factor and solve for $x$

$x(-16x + 10) = 0$
Solutions: $x=0$ (initial time) or $-16x + 10 = 0$
$16x = 10$
$x = \frac{10}{16} = 0.625 \approx 0.63$

Answer:

a. 0.97
b. 0.63