the height of an object after it is released can be modeled by the function $f(t) = -16t^2 + vt + s$, where $t$ is the number of seconds after the object is released, $v$ is the upward speed in feet per second at release, and $s$ is the starting height in feet. if a quarterback throws a ball from his hand 6 feet in the air at an upward speed of 25 feet per second, how much time does his teammate have to catch the ball?

about 1.45 seconds
about 1.77 seconds
about 3 seconds
about 6 seconds

Question

the height of an object after it is released can be modeled by the function \$f(t) = -16t^2 + vt + s\$, where \$t\$ is the number of seconds after the object is released, \$v\$ is the upward speed in feet per second at release, and \$s\$ is the starting height in feet. if a quarterback throws a ball from his hand 6 feet in the air at an upward speed of 25 feet per second, how much time does his teammate have to catch the ball?

about 1.45 seconds
about 1.77 seconds
about 3 seconds
about 6 seconds

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Projectile Motion Modeling" ], "new_concepts": [], "current_concepts": [ "Projectile Motion Modeling", "Quadratic Equations" ] } ## Set up the quadratic equation for height $$ \begin{aligned} &f(t) = -16t^2 + vt + s \ &v = 25, \quad s = 6 \ &-16t^2 + 25t + 6 = 0 \end{aligned} $$ ## Solve for t using the quadratic formula $$ \begin{aligned} &t = \frac{-25 \pm \sqrt{25^2 - 4(-16)(6)}}{2(-16)} \ &t = \frac{-25 \pm \sqrt{625 + 384}}{-32} \ &t = \frac{-25 \pm \sqrt{1009}}{-32} \end{aligned} $$ ## Calculate the positive time value $$ \begin{aligned} &\sqrt{1009} \approx 31.76 \ &t = \frac{-25 - 31.76}{-32} \approx 1.77 \text{ seconds} \end{aligned} $$

about 1.45 seconds

(B) about 1.77 seconds

about 3 seconds

about 6 seconds

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Projectile Motion Modeling" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Set up the quadratic equation for height

Solve for t using the quadratic formula

Calculate the positive time value

Answer: