Questions: Question 9 0/1 pt This question is not about solving the stated problem, but about understanding it. A rocket is launched, and its height above sea level t seconds after launch is given by the equation h(t)=-4.9 t^2+1400 t+470. a) From what height was the rocket launched? To answer this question, we'd find: The t coordinate of the vertex ✓ b) What is the maximum height the rocket reaches? To answer this question, we'd find: The h coordinate of the vertex ~ c) If the rocket will splash down in the ocean, when will it splash down? To answer this question, we'd find: The h coordinate of the vertex ✓

Question 9
0/1 pt

This question is not about solving the stated problem, but about understanding it.

A rocket is launched, and its height above sea level t seconds after launch is given by the equation h(t)=-4.9 t^2+1400 t+470.
a) From what height was the rocket launched?

To answer this question, we'd find: The t coordinate of the vertex ✓
b) What is the maximum height the rocket reaches?

To answer this question, we'd find: The h coordinate of the vertex ~
c) If the rocket will splash down in the ocean, when will it splash down?

To answer this question, we'd find: The h coordinate of the vertex ✓

Transcript text: Question 9 0/1 pt This question is not about solving the stated problem, but about understanding it. A rocket is launched, and its height above sea level $t$ seconds after launch is given by the equation $h(t)=-4.9 t^{2}+1400 t+470$. a) From what height was the rocket launched? To answer this question, we'd find: The t coordinate of the vertex $\checkmark$ b) What is the maximum height the rocket reaches? To answer this question, we'd find: The $h$ coordinate of the vertex $\sim$ c) If the rocket will splash down in the ocean, when will it splash down? To answer this question, we'd find: The $h$ coordinate of the vertex $\checkmark$

Solution

Solution Steps

Solution Approach

a) To find the height from which the rocket was launched, evaluate the height function $ h(t) $ at $ t = 0 $.

b) To find the maximum height the rocket reaches, determine the vertex of the parabola represented by the height function. The vertex form of a quadratic function gives the maximum height at the vertex.

c) To find when the rocket will splash down, solve for $ t $ when $ h(t) = 0 $.

Step 1: Height at Launch

To find the height from which the rocket was launched, we evaluate the height function $ h(t) $ at $ t = 0 $: \[ h(0) = -4.9(0)^2 + 1400(0) + 470 = 470 \] Thus, the height at launch is $ 470 $ meters.

Step 2: Maximum Height

To find the maximum height the rocket reaches, we calculate the vertex of the parabola. The time at which the maximum height occurs is given by: \[ t = -\frac{b}{2a} = -\frac{1400}{2 \cdot -4.9} \approx 142.8571 \] Substituting this value back into the height function: \[ h\left(142.8571\right) = -4.9(142.8571)^2 + 1400(142.8571) + 470 \approx 100470 \] Thus, the maximum height is $ 100470 $ meters.

Step 3: Time of Splash Down

To find when the rocket will splash down, we solve for $ t $ when $ h(t) = 0 $: \[ h(t) = -4.9t^2 + 1400t + 470 = 0 \] The solutions to this equation are approximately: \[ t \approx -0.3353 \quad \text{and} \quad t \approx 286.0496 \] Since time cannot be negative, the time when the rocket splashes down is approximately $ 286.0496 $ seconds.