find the time that the rocket will hit the ground, to the nearest 100th of second. y=-16x^2+149x+108

Question

A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation,

Answer 1

Hi! In order to solve this problem you will need a calculator.

First, you begin by setting y = 0. This is because y represents the height of the rocket at x seconds after launch. When the rocket hits the ground, its height would be 0 feet, which is why we set y = 0.

Then, you would take your graphing calculator and plot the equation y = -16(x^2) + 149x + 108. If you have a TI-84 plus, you will click "Y=" button which is on the top row of the calculator. Make sure everything is clear by clicking the "clear" button. Then carefully type in the above equation into "Y1 ="

From here, you will click the button that says "Graph," which is located all the way to the right in the row of gray buttons. Once it graphs (depending on the window of your graph), you will see that the graph crosses the x-axis once to the left of the y axis and once to the right. We care about where the graph crosses the x-axis because where the graph touches the x-axis, is where y is equal to 0 and that is what we are trying to find - we are trying to find what x is when y is 0.

We are going to focus on the right side of the graph. This is because where the graph crosses the x-axis to the left of the y-axis, x is negative and we cannot have negative time. This is not physically possible. So we focus on the right side of the graph. I am going to zoom into the right side of the graph by adjusting the Window by going to the button that says "Window." I am going to adjust the Xmin and Xmax to 0 and 15, respectively. This will allow me to better see where the graph crosses the x-axis on the right side. Click "Graph" again and you should see where the graph is crossing the x-axis better now.

To find what the value of x is when y is 0, I will click the blue button that's called "2nd" and then the button that says "Trace," which will bring me to the calc function. From here, I will click "2: zero"

This function calculates the value of x when y is 0. Now, if you look at the bottom left of your screen it will say "left bound?" and we want to set this to the first x value that has a y value greater than 0. You can adjust this by using the up and down buttons. For this graph, the first value of x that has a y value greater than 0 that you can select is x = 9.89 and y = 16.01. Click Enter.

Then, the bottom left of your screen will say "right bound?" and we want to set this to the first x value that has a y value LESS than 0. You can adjust this by using the up and down buttons. For this graph, the first value of x that has a y value less than 0 that you can select is x = 10.05 and y = -11.14. Click Enter.

What this process allowed you to do was tell the calculator "hey my graph crosses the x-axis between these two points."

Then click "Enter" again.

Now if you look at the bottom left of your screen it will say "Zero x= 9.9882913, y=0"

This is the answer (because y is 0 ft at this time of x seconds). To the nearest hundredth, this answer will be 9.99 seconds.

So your rocket will hit the ground 9.99 seconds after launch.

Please let me know if anything is unclear. Hope this helps!