Using the graph at the left, it shows the height h in feet of a small rocket t seconds after it is launched. The path of the rocket is given by the equation: h(t)= - 16t^2+128t
The graph at the left displays a parabolic curve that opens downwards. This is indicative of the rocket being launched upwards and then falling back to the ground.
The equation h(t) = -16t^2 + 128t represents the height of the rocket at time t seconds after launch. The coefficient of t^2, -16, determines how quickly the rocket is going up or coming down. The negative coefficient indicates that the rocket is moving downwards. The coefficient of t, 128, represents the initial velocity of the rocket.
The vertex of the parabola, the highest point the rocket reaches before falling back to the ground, can be found by evaluating the formula t = -b / 2a, where a = -16 and b = 128 in the equation h(t) = -16t^2 + 128t.
t = - 128 / 2(-16) = 4 seconds
Therefore, the rocket reaches its highest point at 4 seconds. After that, it begins to fall back to the ground.
whats the maximum height the rocket can fly?
To find the maximum height the rocket can fly, we need to substitute the value of t = 4 seconds into the equation h(t) = -16t^2 + 128t:
h(4) = -16(4)^2 + 128(4)
h(4) = -16(16) + 512
h(4) = -256 + 512
h(4) = 256 feet
Therefore, the maximum height the rocket can fly is 256 feet.
what would that be on the graph
On the graph, you would look for the point where the curve reaches its highest point, which corresponds to the maximum height of the rocket.
At t = 4 seconds, as we calculated earlier, the rocket reaches its maximum height of 256 feet. This point on the graph would be the vertex of the parabolic curve. The coordinates of the vertex would be (4, 256) on the graph.
So, on the graph, you would see the curve reaching its peak at the point (4, 256) representing the maximum height the rocket can fly before falling back to the ground.
can you do a graphing point
I'm currently unable to directly create graphs. However, you can easily plot this point on a graphing tool or software such as Desmos or GeoGebra by entering the equation of the rocket's path:
h(t) = -16t^2 + 128t
Then, plot the point (4, 256) on the graph, and you should see the vertex at that point representing the maximum height of the rocket.
If you have any more questions or need further assistance with interpreting the graph or any mathematical concept, feel free to ask!