Full solution

Q. Alain throws a stone off a bridge into a river below.

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The stone's height (in meters above the water),

x

seconds after Alain threw it, is modeled by:

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h(x)=-5x^{2}+10x+15

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How many seconds after being thrown will the stone hit the water?

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\text{seconds}

Write Equation for Stone's Height: Write down the equation that models the stone's height above the water. $\newline$ The equation given is $h(x) = -5x^2 + 10x + 15$ , where $h(x)$ is the height in meters and $x$ is the time in seconds after the stone is thrown.
Set Height to $0$ : Set the height $h(x)$ to $0$ to find when the stone will hit the water. $\newline$ $0 = -5x^2 + 10x + 15$
Factor Quadratic Equation: Factor the quadratic equation to solve for $x$ . $\newline$ To factor, we look for two numbers that multiply to $-5 \times 15 = -75$ and add up to $10$ . However, since the quadratic is not easily factorable, 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 $a = -5$ , $b = 10$ , and $c = 15$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ . $\newline$ Discriminant = $10^2 - 4(-5)(15) = 100 - (-300) = 100 + 300 = 400$
Calculate Possible Values for x: Calculate the two possible values for x using the quadratic formula. $\newline$ $x = \frac{{-10 \pm \sqrt{400}}}{{2 \cdot -5}}$ $\newline$ $x = \frac{{-10 \pm 20}}{{-10}}$
Solve for x: Solve for the two possible values of x. $\newline$ $x = \frac{{-10 + 20}}{{-10}} = \frac{{10}}{{-10}} = -1$ (This value is not physically meaningful as time cannot be negative.) $\newline$ $x = \frac{{-10 - 20}}{{-10}} = \frac{{-30}}{{-10}} = 3$ (This is the time in seconds when the stone hits the water.)