23) A rocket is launched from a 4.8 -meter tall platform at 20.5 -meters per second. What is the maximum height of the rocket? $$ h(t)=-4.9 t^{2}+v_{0} t+h_{0} $$

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To find the maximum height of the rocket, we need to determine the time at which the rocket reaches its maximum height.

Given:
- Initial height of the platform, $$ h_{0} = 4.8 $$ meters
- Initial velocity of the rocket, $$ v_{0} = 20.5 $$ meters per second
- Acceleration due to gravity, $$ g = 4.9 $$ meters per second squared

The equation for the height of the rocket at time $$ t $$ is given by:
$$ h(t) = -4.9t^{2} + v_{0}t + h_{0} $$

To find the maximum height, we need to find the time at which the velocity of the rocket is zero. The velocity of the rocket is given by the derivative of the height equation with respect to time:
$$ v(t) = \frac{dh}{dt} = -9.8t + v_{0} $$

Setting the velocity to zero to find the time at which the rocket reaches its maximum height:
$$ -9.8t + v_{0} = 0 $$

Now, we can solve for $$ t $$ to find the time at which the rocket reaches its maximum height.
Solve the equation by following steps:
- step0: Solve for $$t$$:
  $$-9.8t+20.5=0$$
- step1: Move the constant to the right side:
  $$-9.8t=0-20.5$$
- step2: Remove 0:
  $$-9.8t=-20.5$$
- step3: Change the signs:
  $$9.8t=20.5$$
- step4: Divide both sides:
  $$\frac{9.8t}{9.8}=\frac{20.5}{9.8}$$
- step5: Divide the numbers:
  $$t=\frac{205}{98}$$
The time at which the rocket reaches its maximum height is $$ t = \frac{205}{98} $$ seconds.

Now, we can substitute this value of $$ t $$ back into the height equation to find the maximum height of the rocket.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$-4.9t^{2}+20.5t+4.8$$
- step1: Substitute:
  $$-4.9\left(\frac{205}{98}\right)^{2}+20.5\times \frac{205}{98}+4.8$$
- step2: Multiply the numbers:
  $$-\frac{8405}{392}+20.5\times \frac{205}{98}+4.8$$
- step3: Multiply the numbers:
  $$-\frac{8405}{392}+\frac{8405}{196}+4.8$$
- step4: Rewrite the expression:
  $$-\frac{8405}{392}+\frac{8405}{196}+\frac{24}{5}$$
- step5: Reduce fractions to a common denominator:
  $$-\frac{8405\times 5}{392\times 5}+\frac{8405\times 2\times 5}{196\times 2\times 5}+\frac{24\times 196\times 2}{5\times 196\times 2}$$
- step6: Multiply the numbers:
  $$-\frac{8405\times 5}{1960}+\frac{8405\times 2\times 5}{196\times 2\times 5}+\frac{24\times 196\times 2}{5\times 196\times 2}$$
- step7: Multiply the terms:
  $$-\frac{8405\times 5}{1960}+\frac{8405\times 2\times 5}{1960}+\frac{24\times 196\times 2}{5\times 196\times 2}$$
- step8: Multiply the terms:
  $$-\frac{8405\times 5}{1960}+\frac{8405\times 2\times 5}{1960}+\frac{24\times 196\times 2}{1960}$$
- step9: Transform the expression:
  $$\frac{-8405\times 5+8405\times 2\times 5+24\times 196\times 2}{1960}$$
- step10: Multiply the numbers:
  $$\frac{-42025+8405\times 2\times 5+24\times 196\times 2}{1960}$$
- step11: Multiply the terms:
  $$\frac{-42025+84050+24\times 196\times 2}{1960}$$
- step12: Multiply the terms:
  $$\frac{-42025+84050+9408}{1960}$$
- step13: Add the numbers:
  $$\frac{51433}{1960}$$
The maximum height of the rocket is $$ 26\frac{473}{1960} $$ meters.
The maximum height of the rocket is approximately 26.24 meters.

A rocket is launched from a 4.8 -meter tall
platform at 20.5 -meters per second. What
is the maximum height of the rocket?

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