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Alain throws a stone off a bridge into a river below. The stone's height (in meters above the water), xx seconds after Alain threw it, is modeled by: h(x)=−5x2+10x+15h(x)= -5x^2+10x+15. How many seconds after being thrown will the stone hit the water? seconds\text{seconds}

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Full solution

Q. Alain throws a stone off a bridge into a river below. The stone's height (in meters above the water), xx seconds after Alain threw it, is modeled by: h(x)=−5x2+10x+15h(x)= -5x^2+10x+15. How many seconds after being thrown will the stone hit the water? seconds\text{seconds}
  1. Understand the problem: Understand the problem.\newlineWe need to find the time when the stone hits the water, which is when its height is 00 meters.
  2. Set up the equation: Set up the equation to find when the height is 00. We need to solve the equation h(x)=−5x2+10x+15=0h(x) = -5x^2 + 10x + 15 = 0.
  3. Solve the quadratic equation: Solve the quadratic equation.\newlineWe can use the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, where a=−5a = -5, b=10b = 10, and c=15c = 15.
  4. Calculate the discriminant: Calculate the discriminant b2−4acb^2 - 4ac.\newlineDiscriminant = b2−4ac=(10)2−4(−5)(15)=100+300=400b^2 - 4ac = (10)^2 - 4(-5)(15) = 100 + 300 = 400.
  5. Calculate square root: Calculate the square root of the discriminant. 400=20\sqrt{400} = 20.
  6. Apply quadratic formula: Apply the quadratic formula to find the values of xx.x=−10±202⋅−5x = \frac{{-10 \pm 20}}{{2 \cdot -5}}.
  7. Calculate possible values: Calculate the two possible values for xx.x1=($−10x_1 = (\$-10 + 2020) / −10-10 = 1010 / −10-10 = −1-1 (This value is not physically meaningful as time cannot be negative).x_2 = (\(-10 - 2020) / −10-10 = x1=($−10x_1 = (\$-1000 / −10-10 = x1=($−10x_1 = (\$-1022 (This is the time in seconds when the stone hits the water).

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