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?
(type an integer or decimal rounded to two decimal places as needed.)
b. the flotation device should be in the water at the time(s) (square) s.
(type an integer or decimal rounded to two decimal places as needed. use a comma to separate answers as needed.)

Explanation:

Step1: Set up equation for part a

Initial velocity = 0, so linear term $10x$ is removed. We need $y=0$ (device lands in water):
$$-16x^2 + 15 = 0$$

Step2: Solve for $x$ (part a)

Rearrange to isolate $x^2$:
$$16x^2 = 15$$
$$x^2 = \frac{15}{16}$$
$$x = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4} \approx 0.97$$
(We discard the negative root since time cannot be negative.)

Step3: Set up equation for part b

Initial height = 0, so constant term $15$ is removed. We need $y=0$:
$$-16x^2 + 10x = 0$$

Step4: Solve for $x$ (part b)

Factor out $x$:
$$x(-16x + 10) = 0$$
Set each factor equal to 0:

1. $x = 0$
2. $-16x + 10 = 0 \implies 16x = 10 \implies x = \frac{10}{16} = 0.625 \approx 0.63$

Answer:

a. 0.97 seconds
b. 0 seconds, 0.63 seconds