a ball is thrown into the air and its position is given by (h(t) = -3t^2 + 78t + 2) where (h) is the height of the ball in meters (t) seconds after it has been thrown. find the maximum height reached by the ball and the time at which that happens.

meters

seconds

Question

a ball is thrown into the air and its position is given by (h(t) = -3t^2 + 78t + 2) where (h) is the height of the ball in meters (t) seconds after it has been thrown. find the maximum height reached by the ball and the time at which that happens.

meters

seconds

meters

seconds

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Critical Points", "Absolute Extrema on Closed Intervals" ], "new_concepts": [ "Quadratic Optimization" ], "current_concepts": [ "Critical Points", "Absolute Extrema on Closed Intervals", "Quadratic Optimization" ] } ## Find the critical point by differentiating the height function $$ \begin{aligned} h(t) &= -3t^2 + 78t + 2 \ h'(t) &= -6t + 78 \ -6t + 78 &= 0 \ t &= 13 \end{aligned} $$ ## Calculate the maximum height at the critical point $$ \begin{aligned} h(13) &= -3(13)^2 + 78(13) + 2 \ h(13) &= -3(169) + 1014 + 2 \ h(13) &= -507 + 1014 + 2 \ h(13) &= 509 \end{aligned} $$ A ball is thrown into the air and its position is given by $h(t) = -3t^2 + 78t + 2$ where $h$ is the height of the ball in meters $t$ seconds after it has been thrown. Find the maximum height reached by the ball and the time at which that happens. 509 meters 13 seconds { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Critical Points" ] }

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QUESTION IMAGE

Question

Explanation:

Find the critical point by differentiating the height function

Calculate the maximum height at the critical point

Answer: