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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: h(x)=-2x^(2)+4x+16 How many seconds after being thrown will the ball hit the ground? seconds

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), xx seconds after Antoine threw it, is modeled by:\newlineh(x)=−2x2+4x+16h(x)=-2x^{2}+4x+16\newlineHow many seconds after being thrown will the ball hit the ground?\newlineseconds

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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), xx seconds after Antoine threw it, is modeled by:\newlineh(x)=−2x2+4x+16h(x)=-2x^{2}+4x+16\newlineHow many seconds after being thrown will the ball hit the ground?\newlineseconds
  1. Write height equation: Write down the equation that models the height of the ball.\newlineThe equation given is h(x)=−2x2+4x+16h(x) = -2x^2 + 4x + 16, where h(x)h(x) represents the height of the ball xx seconds after being thrown.
  2. Set equation equal to zero: Set the height equation equal to zero to find when the ball will hit the ground.\newlineTo find when the ball hits the ground, we need to solve for xx when h(x)=0h(x) = 0.\newline0=−2x2+4x+160 = -2x^2 + 4x + 16
  3. Factor quadratic equation: Factor the quadratic equation to solve for xx.\newlineWe can factor the quadratic equation by finding two numbers that multiply to −32-32 (−2×16-2 \times 16) and add to 44 (the coefficient of xx). However, this quadratic does not factor easily, so we will use the quadratic formula instead.
  4. Apply quadratic formula: Apply the quadratic formula to find the values of x.\newlineThe quadratic formula is x=−b±b2−4ac2ax = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}, where a=−2a = -2, b=4b = 4, and c=16c = 16.\newlinex=−4±42−4(−2)(16)2(−2)x = \frac{{-4 \pm \sqrt{{4^2 - 4(-2)(16)}}}}{{2(-2)}}
  5. Calculate discriminant: Calculate the discriminant b2−4acb^2 - 4ac.\newlineDiscriminant = 42−4(−2)(16)4^2 - 4(-2)(16)\newlineDiscriminant = 16+12816 + 128\newlineDiscriminant = 144144
  6. Calculate values of x: Calculate the values of x using the quadratic formula.\newlinex=−4±144−4x = \frac{{-4 \pm \sqrt{144}}}{{-4}}\newlinex=−4±12−4x = \frac{{-4 \pm 12}}{{-4}}
  7. Solve for possible values of x: Solve for the two possible values of x.\newlinex=(−4+12)(−4)=8(−4)=−2x = \frac{(-4 + 12)}{(-4)} = \frac{8}{(-4)} = -2\newlinex=(−4−12)(−4)=−16(−4)=4x = \frac{(-4 - 12)}{(-4)} = \frac{-16}{(-4)} = 4\newlineSince time cannot be negative, we discard the negative value.
  8. Conclude time of ball hitting ground: Conclude the time when the ball will hit the ground.\newlineThe ball will hit the ground after 44 seconds.

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