Antoine stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground), $x$ seconds after Antoine threw it, is modeled by: $\newline$ $h(x)=-2x^{2}+4x+16$ $\newline$ How many seconds after being thrown will the ball hit the ground? $\newline$ seconds

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Q. Antoine stands on a balcony and throws a ball to his dog, who is at ground level. The ball's height (in meters above the ground),

x

seconds after Antoine threw it, is modeled by:

\newline

h(x)=-2x^{2}+4x+16

\newline

How many seconds after being thrown will the ball hit the ground?

\newline

seconds

Write height equation: Write down the equation that models the height of the ball. $\newline$ The equation given is $h(x) = -2x^2 + 4x + 16$ , where $h(x)$ represents the height of the ball $x$ seconds after being thrown.
Set equation equal to zero: Set the height equation equal to zero to find when the ball will hit the ground. $\newline$ To find when the ball hits the ground, we need to solve for $x$ when $h(x) = 0$ . $\newline$ $0 = -2x^2 + 4x + 16$
Factor quadratic equation: Factor the quadratic equation to solve for $x$ . $\newline$ We can factor the quadratic equation by finding two numbers that multiply to $-32$ ( $-2 \times 16$ ) and add to $4$ (the coefficient of $x$ ). However, this quadratic does not factor easily, so we will use the quadratic formula instead.
Apply quadratic formula: Apply the quadratic formula to find the values of x. $\newline$ The quadratic formula is $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$ , where $a = -2$ , $b = 4$ , and $c = 16$ . $\newline$ $x = \frac{{-4 \pm \sqrt{{4^2 - 4(-2)(16)}}}}{{2(-2)}}$
Calculate discriminant: Calculate the discriminant $b^2 - 4ac$ . $\newline$ Discriminant = $4^2 - 4(-2)(16)$ $\newline$ Discriminant = $16 + 128$ $\newline$ Discriminant = $144$
Calculate values of x: Calculate the values of x using the quadratic formula. $\newline$ $x = \frac{{-4 \pm \sqrt{144}}}{{-4}}$ $\newline$ $x = \frac{{-4 \pm 12}}{{-4}}$
Solve for possible values of x: Solve for the two possible values of x. $\newline$ $x = \frac{(-4 + 12)}{(-4)} = \frac{8}{(-4)} = -2$ $\newline$ $x = \frac{(-4 - 12)}{(-4)} = \frac{-16}{(-4)} = 4$ $\newline$ Since time cannot be negative, we discard the negative value.
Conclude time of ball hitting ground: Conclude the time when the ball will hit the ground. $\newline$ The ball will hit the ground after $4$ seconds.