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The function \( h(t) \) gives the height of a rock thrown off the Golden Gate Bridge in feet above the water at \( t \) seconds after it is thrown. \( h(t)=-16 t^{2}+30 t+225 \) Question: How long until the rock is 89 feet above the water? Enter your final answer rounded to the nearest tenth (one decimal place.) Enter the correct units for your answer in the second box. IF there is no valid answer enter N/A in all blanks. *Remember: When calculating, do not do any rounding until the end of your calculations. Answer:

Ask by Mccarthy Chambers. in the United States
Mar 13,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

It takes 4.0 seconds for the rock to be 89 feet above the water.

Solution

Given the function \( h(t) = -16t^2 + 30t + 225 \) that represents the height of a rock thrown off the Golden Gate Bridge in feet above the water at \( t \) seconds after it is thrown, we need to find how long it takes for the rock to be 89 feet above the water. To find the time when the rock is 89 feet above the water, we set the function \( h(t) \) equal to 89 and solve for \( t \): \[ -16t^2 + 30t + 225 = 89 \] Now, we can solve this quadratic equation to find the value of \( t \) when the rock is 89 feet above the water. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(-16t^{2}+30t+225=89\) - step1: Move the expression to the left side: \(-16t^{2}+30t+136=0\) - step2: Factor the expression: \(-2\left(t-4\right)\left(8t+17\right)=0\) - step3: Divide the terms: \(\left(t-4\right)\left(8t+17\right)=0\) - step4: Separate into possible cases: \(\begin{align}&t-4=0\\&8t+17=0\end{align}\) - step5: Solve the equation: \(\begin{align}&t=4\\&t=-\frac{17}{8}\end{align}\) - step6: Rewrite: \(t_{1}=-\frac{17}{8},t_{2}=4\) The solutions to the equation are \( t_1 = -\frac{17}{8} \) and \( t_2 = 4 \). Since time cannot be negative, the valid solution is \( t = 4 \) seconds. Therefore, it takes 4 seconds for the rock to be 89 feet above the water. The answer is: 4 seconds.

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To find the time \( t \) when the rock is 89 feet above the water, set \( h(t) = 89 \): \[ -16t^2 + 30t + 225 = 89 \] Rearranging gives: \[ -16t^2 + 30t + 136 = 0 \] Next, we can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16 \), \( b = 30 \), and \( c = 136 \). Calculating the discriminant: \[ b^2 - 4ac = 30^2 - 4(-16)(136) = 900 + 8704 = 9604 \] Now, take the square root: \[ \sqrt{9604} = 98 \] Plugging into the quadratic formula: \[ t = \frac{-30 \pm 98}{2(-16)} \] Calculating the two possible values for \( t \): 1. \( t = \frac{-30 + 98}{-32} = \frac{68}{-32} = -2.125 \) (not valid) 2. \( t = \frac{-30 - 98}{-32} = \frac{-128}{-32} = 4 \) The valid solution is \( t = 4 \) seconds. So, the final answer is: Answer: 4 Units: seconds

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