the path of a football kicked by a field goal kicker can be modeled by the equation $y = -0.03x^2 + 1.53x$, where $x$ is the horizontal distance in yards and $y$ is the corresponding height in yards.

what is the footballs maximum height? round to the nearest tenth.

how far is the football kicked?

Question

the path of a football kicked by a field goal kicker can be modeled by the equation $y = -0.03x^2 + 1.53x$, where $x$ is the horizontal distance in yards and $y$ is the corresponding height in yards.

what is the footballs maximum height? round to the nearest tenth.

how far is the football kicked?

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Vertex of a Quadratic Function", "Zeros of Quadratic Functions" ], "new_concepts": [], "current_concepts": [ "Vertex of a Quadratic Function", "Zeros of Quadratic Functions", "Quadratic Modeling" ] } ## Find the horizontal distance at the vertex $$ x_v = -\frac{b}{2a} = -\frac{1.53}{2(-0.03)} = \frac{1.53}{0.06} = 25.5\text{ yards} $$ ## Calculate the maximum height $$ \begin{aligned} y_{\max} &= -0.03(25.5)^2 + 1.53(25.5) \ &= -0.03(650.25) + 39.015 \ &= -19.5075 + 39.015 \ &= 19.5075 \approx 19.5\text{ yards} \end{aligned} $$ ## Find the total horizontal distance kicked $$ \begin{aligned} -0.03x^2 + 1.53x &= 0 \ x(-0.03x + 1.53) &= 0 \ x = 0 \quad &\text{or} \quad x = \frac{1.53}{0.03} = 51\text{ yards} \end{aligned} $$ ## Question 1 What is the football's maximum height? Round to the nearest tenth. 19.5 yds. ## Question 2 How far is the football kicked? 51 yds { "elements": [ { "type": "functiongraph", "params": [ { "js": "-0.03*x*x + 1.53*x", "latex": "-0.03x^2 + 1.53x" }, 0, 51 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 3, "name": "y = -0.03x^2 + 1.53x", "withLabel": false } }, { "type": "point", "params": [ [25.5, 19.5] ], "properties": { "name": "Vertex (25.5, 19.5)", "size": 4, "color": "#F2557F", "withLabel": true } }, { "type": "point", "params": [ [51, 0] ], "properties": { "name": "Landing (51, 0)", "size": 4, "color": "#5583F2", "withLabel": true } } ], "timestamps": [0.5, 1.0] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Functions" ] }

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QUESTION IMAGE

Question

Explanation:

Find the horizontal distance at the vertex

Calculate the maximum height

Find the total horizontal distance kicked

Answer:

Question 1

Question 2