if a rock is dropped from a height of 175 ft, its position $t$ seconds after it is dropped until it hits the ground is given by the function $s(t) = -16t^2 + 175$.

round values below to 3 decimal places.

how long does it take the rock to hit the ground?

seconds

find the average velocity of the rock from when it is released until when it hits the ground.

feet per second

what time after the rock is thrown will its instantaneous velocity be equal to its average velocity? (apply the mean value theorem)

seconds after it is thrown

Question

if a rock is dropped from a height of 175 ft, its position $t$ seconds after it is dropped until it hits the ground is given by the function $s(t) = -16t^2 + 175$.

round values below to 3 decimal places.

how long does it take the rock to hit the ground?

seconds

find the average velocity of the rock from when it is released until when it hits the ground.

feet per second

what time after the rock is thrown will its instantaneous velocity be equal to its average velocity? (apply the mean value theorem)

seconds after it is thrown

Accepted Answer

{ "quality": "clear", "question_count": 3, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Position Velocity Acceleration Relations", "Average Rate of Change", "Mean Value Theorem" ], "new_concepts": [], "current_concepts": [ "Position Velocity Acceleration Relations", "Average Rate of Change", "Mean Value Theorem" ] } ## Find the time when the rock hits the ground $$ \begin{aligned} s(t) &= -16t^2 + 175 = 0 \ 16t^2 &= 175 \ t^2 &= \frac{175}{16} = 10.9375 \ t &= \sqrt{10.9375} \approx 3.307189 \approx 3.307 \text{ seconds} \end{aligned} $$ ## Calculate the average velocity $$ \begin{aligned} v_{\text{avg}} &= \frac{s(t_{\text{ground}}) - s(0)}{t_{\text{ground}} - 0} \ v_{\text{avg}} &= \frac{0 - 175}{3.307189 - 0} \approx -52.91502 \approx -52.915 \text{ feet per second} \end{aligned} $$ ## Apply the Mean Value Theorem to find the time of equal instantaneous velocity $$ \begin{aligned} v(t) &= s'(t) = -32t \ -32t &= v_{\text{avg}} \ -32t &= -52.91502 \ t &= \frac{-52.91502}{-32} \approx 1.65359 \approx 1.654 \text{ seconds} \end{aligned} $$ ## Question 1 How long does it take the rock to hit the ground? 3.307 seconds ## Question 2 Find the average velocity of the rock from when it is released until when it hits the ground. -52.915 feet per second ## Question 3 What time after the rock is thrown will its instantaneous velocity be equal to its average velocity? (Apply the Mean Value Theorem) 1.654 seconds after it is thrown { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Calculus", "Mean Value Theorem" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Find the time when the rock hits the ground

Calculate the average velocity

Apply the Mean Value Theorem to find the time of equal instantaneous velocity

Answer:

Question 1

Question 2

Question 3