quadratic applications

Question

When cannonballs are shot out of a cannon, their flight through the air depends on both the angle at which the cannon is set and the amount of gunpowder that is loaded into the cannon, which affects the initial velocity of the cannonball.

The equation y = 05r - 0.01r represents the parabola flight of a certain cannonball shot at an angle of 26° with the horizon and at an initial velocity of 25 meters per second. In this equation, y is the height of the cannonball, in meters, and a is the horizontal distance traveled, in meters The graph of the equation is shown to the right.

1.  Given that the points (10, 4) and (40, 4) lie on the parabola, at what x coordinate must the vertex Be?
2.  Use the equation and your answer to question 1 to find the maximum height of the cannonball.
3.  Use the point (0, 0) and the location of the vertex to find the total horizontal distance that the cannonball will travel.

When the angle of the cannon is decreased, the cannonball will travel in a different flight. The parabolic flight of the cannonball is shown to the right, with the vertex labeled.

4. What is the total horizontal distance that this cannonball will travel?

Using the same angle, the initial velocity of the cannonball is increased to produce the graph of the flight shown to the right. The point shown represents the total horizontal distance the cannonball will travel.

5. How far will the cannonball travel horizontally before it reaches its maximum height?

Answer 1

1. To find the x-coordinate of the vertex of the parabola, we can use the formula x = -b/2a, where a and b are the coefficients of the quadratic equation in standard form y = ax^2 + bx + c. In this case, a = 0.05 and b = -0.01.

x = -(-0.01) / 2(0.05) = 0.1

So the vertex is at (0.1, h), where h is the maximum height of the cannonball.

1. To find the maximum height of the cannonball, we can substitute the x-coordinate of the vertex into the equation for y.

y = 0.05(0.1) - 0.01(0.1) = 0.004

So the maximum height of the cannonball is 0.004 meters.

1. To find the total horizontal distance that the cannonball will travel, we can use the fact that the vertex of the parabola is the midpoint of the two given points (10, 4) and (40, 4).

The x-coordinate of the midpoint is (10 + 40)/2 = 25. So the vertex is at (25, 4).

The total horizontal distance traveled is twice the horizontal distance from the vertex to either of the given points. So the total distance is:

2(25-10) = 30 meters.

1. From the graph, we can see that the horizontal distance traveled by the cannonball before it hits the ground is approximately 45 meters.
2. To find the horizontal distance traveled by the cannonball before it reaches its maximum height, we can find the x-coordinate of the vertex and divide by 2.

The x-coordinate of the vertex is (5 + 25)/2 = 15. So the cannonball travels 15/2 = 7.5 meters before it reaches its maximum height.