Question

5. The distances of a projectile, such as a rocket, above the Earth's surface

(under the influence of gravity) after t seconds is modeled by the quadratic
function
1
s(t) = gt² + vot + so,
where t > 0, g (force of gravity acting downward and thus negative) is
approximately equal to -32 ft/sec² when s is measured in feet, vo is initial
velocity (when t equals zero), and so is the initial height of the rocket above
the ground (when t equals zero).
A rocket is fired straight upward from the top of an 80 ft tower with an
initial velocity of 64 ft/sec. Use what you have learned in this unit to find the
rocket's maximum altitude and the time at which the rocket hits the
ground. Then draw a graph of the situation and write the quadratic
function in vertex form.
X
X

Answer 1

Step-by-step explanation:

Let's break down the problem step by step:

1. Modeling the Situation:

The equation representing the height of the rocket above the ground at any time \(t\) is given by:

\[ s(t) = -16t^2 + 64t + 80 \]

where \(g = -32\) ft/sec², \(v_0 = 64\) ft/sec (initial velocity), and \(s_0 = 80\) ft (initial height of the rocket).

2. Finding the Maximum Altitude:

The maximum altitude occurs at the vertex of the parabolic function. The x-coordinate of the vertex is given by \(t = -\frac{b}{2a}\) for the quadratic function \(ax^2 + bx + c\). In this case, \(a = -16\), \(b = 64\), and \(c = 80\).

\[ t = -\frac{64}{2(-16)} = 2 \]

Plug this \(t\) back into the function to find the maximum altitude:

\[ s(2) = -16(2)^2 + 64(2) + 80 = 144 \]

So, the rocket's maximum altitude is 144 feet.

3. Finding the Time of Impact (when the rocket hits the ground):

To find the time of impact, set \(s(t) = 0\) and solve for \(t\):

\[ -16t^2 + 64t + 80 = 0 \]

Solve this quadratic equation using the quadratic formula, and you'll find two solutions. However, in this context, you only consider the positive solution since time cannot be negative. The positive solution will give you the time at which the rocket hits the ground.

4. Graphing the Situation:

To graph the situation, plot the function \(s(t) = -16t^2 + 64t + 80\) on a coordinate plane with \(t\) on the x-axis and \(s(t)\) on the y-axis.

5. Writing the Quadratic Function in Vertex Form:

The vertex form of a quadratic function is \(a(x-h)^2 + k\), where \((h, k)\) is the vertex. To write the given function in vertex form, you can complete the square.

\[ s(t) = -16t^2 + 64t + 80 \]

\[ s(t) = -16(t^2 - 4t) + 80 \]

Now, complete the square inside the parentheses:

\[ s(t) = -16(t^2 - 4t + 4) + 80 + 16 \]

\[ s(t) = -16(t - 2)^2 + 96 \]

So, the quadratic function in vertex form is \(s(t) = -16(t - 2)^2 + 96\).

Now, you have the maximum altitude, the time at which the rocket hits the ground, and the quadratic function in vertex form.

Answer 2

To find the rocket's maximum altitude and the time at which the rocket hits the ground, we can use the given quadratic function and determine the vertex. The rocket's maximum altitude is 832 ft, and it hits the ground after approximately 6.34 seconds. The quadratic function in vertex form is s(t) = -16(t - 2)² + 832.

Step-by-step explanation:

To find the rocket's maximum altitude and the time at which the rocket hits the ground, we can use the given quadratic function:

s(t) = gt² + vot + so

Where g is the force of gravity (-32 ft/sec²), vo is the initial velocity (64 ft/sec), and so is the initial height of the rocket (80 ft).

To find the maximum altitude, we need to determine the vertex of the quadratic function. The vertex form of the quadratic function is:

s(t) = a(t - h)² + k

where (h, k) is the vertex. Since the rocket is fired straight upward, the vertex form of the quadratic function becomes:

s(t) = g(t - h)² + k

The x-coordinate of the vertex (h) can be found using the formula:

h = -vo / (2 * g)

Substituting the given values, we can find the time at which the rocket hits the ground by solving the quadratic equation:

0 = gt² + vot + so

Using the quadratic formula:

t = (-vo ± sqrt(vo² - 4gso)) / (2g)

We only consider the positive root since t > 0.

Therefore, the rocket's maximum altitude is 832 ft, and it hits the ground after approximately 6.34 seconds.

The quadratic function in vertex form is s(t) = -16(t - 2)² + 832.