h(t)=-4.9t^(2)+9.8 t+39.2 Kaia throws a stone vertically upward from a bridge. The height, in meters, of the stone above the water t seconds after the throw can be modeled by the quadratic function given. How many seconds after the throw does the stone hit the water? Choose 1 answer: (A) 1 (B) 2 (c) 4 (D) 8

Question

h(t)=-4.9t^(2)+9.8 t+39.2 Kaia throws a stone vertically upward from a bridge. The height, in meters, of the stone above the water  t seconds after the throw can be modeled by the quadratic function given. How many seconds after the throw does the stone hit the water? Choose 1 answer: (A) 1 (B) 2 (c) 4 (D) 8

Bytelearn · Accepted Answer

Step 1: Set height function equal to zero: The height function h(t) = -4.9t^2 + 9.8t + 39.2 models the stone's height above the water in meters at time t seconds after the throw. To find out when the stone hits the water, we need to solve for t when h(t) = 0.Step 2: Solve quadratic equation: Set the height function equal to zero and solve for t: 0 = -4.9t^2 + 9.8t + 39.2 This is a quadratic equation in the standard form of at^2 + bt + c = 0.Step 3: Calculate discriminant: To solve the quadratic equation, we can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a) where a = -4.9, b = 9.8, and c = 39.2.Step 4: Determine number of solutions: First, calculate the discriminant (b^2 - 4ac): Discriminant = (9.8)^2 - 4(-4.9)(39.2) Discriminant = 96.04 - (-4)(4.9)(39.2) Discriminant = 96.04 + 768.32 Discriminant = 864.36Step 5: Calculate square root of discriminant: Since the discriminant is positive, there are two real solutions. Now, calculate the two possible values for t using the quadratic formula: t = (-9.8 ± √864.36) / (2 * -4.9)Step 6: Plug square root into quadratic formula: Calculate the square root of the discriminant: √864.36 = 29.4Step 7: Calculate possible solutions for t: Now, plug the square root back into the quadratic formula to find the two values of t: t = (-9.8 ± 29.4) / (-9.8)Step 8: Determine physically meaningful solution: Calculate the two possible solutions for t: t1 = (-9.8 + 29.4) / (-9.8) t1 = 19.6 / (-9.8) t1 = -2 (This solution is not physically meaningful because time cannot be negative.)  t2 = (-9.8 - 29.4) / (-9.8) t2 = -39.2 / (-9.8) t2 = 4Step 9: Final answer: The positive value of t is the time in seconds after the throw when the stone hits the water. Therefore, the stone hits the water 4 seconds after the throw.

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