QuestionJuly 22, 2025

If a rock is dropped from a height of 87 ft, its poistion t seconds after it is dropped until it hits the ground is given by the function s(t)=-16t^2+87 Round values below to 3 decimal places. How long does it take the rock to hit the ground? square seconds Find the average velocity of the rock from when it is released until when it hits the ground. square 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) square seconds after it is thrown

If a rock is dropped from a height of 87 ft, its poistion t seconds after it is dropped until it hits the ground is given by the function s(t)=-16t^2+87 Round values below to 3 decimal places. How long does it take the rock to hit the ground? square seconds Find the average velocity of the rock from when it is released until when it hits the ground. square 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) square seconds after it is thrown
If a rock is dropped from a height of 87 ft, its poistion t seconds after it is dropped until it hits the ground is given by the function s(t)=-16t^2+87 Round values below to 3 decimal places. How long does it take the rock to hit the ground? square seconds Find the average velocity of the rock from when it is released until when it hits the ground. square 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) square seconds after it is thrown

Solution
3.9(245 votes)

Answer

2.336 seconds ### -37.222 feet per second ### 1.861 seconds after it is thrown Explanation 1. Determine time to hit the ground Set s(t) = 0: -16t^2 + 87 = 0. Solve for t: t = \sqrt{\frac{87}{16}}. 2. Calculate average velocity Average velocity is \frac{s(\text{end}) - s(\text{start})}{\text{time}}. Here, s(\text{end}) = 0, s(\text{start}) = 87, and time is from Step 1. 3. Find instantaneous velocity Instantaneous velocity v(t) = \frac{ds}{dt} = -32t. 4. Apply Mean Value Theorem Set v(c) = \text{average velocity} and solve -32c = \text{average velocity} for c.

Explanation

1. Determine time to hit the ground<br /> Set $s(t) = 0$: $-16t^2 + 87 = 0$. Solve for $t$: $t = \sqrt{\frac{87}{16}}$.<br /><br />2. Calculate average velocity<br /> Average velocity is $\frac{s(\text{end}) - s(\text{start})}{\text{time}}$. Here, $s(\text{end}) = 0$, $s(\text{start}) = 87$, and time is from Step 1.<br /><br />3. Find instantaneous velocity<br /> Instantaneous velocity $v(t) = \frac{ds}{dt} = -32t$.<br /><br />4. Apply Mean Value Theorem<br /> Set $v(c) = \text{average velocity}$ and solve $-32c = \text{average velocity}$ for $c$.
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