A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.
A) $( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )$
B) $( x - 800 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 2400 )$
C) $( x - 800 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 2400 )$
D) $( x - 2400 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 800 )$
E) none of these

Question

A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 2400 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon. 
A) $( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )$ 
B) $( x - 800 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 2400 )$ 
C) $( x - 800 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 2400 )$ 
D) $( x - 2400 ) ^ { 2 } = - \frac { 800 } { 3 } ( y - 800 )$ 
E) none of these

Quizplus · Accepted Answer

We can use the standard form of a parabola: $y = a(x-h)^2+k$, where $(h,k)$ is the vertex. Since the highest point of the path is the vertex (0,2400), we have $h=0$ and $k=2400$. Now we need to find the value of $a$.

Since the cannonball lands 1600 ft from the cannon, we know that the vertex is halfway between the cannon and the landing point. That means the vertex is at $(800, 2400)$ and the landing point is at $(1600,0)$. Using these two points, we can solve for $a$:

$$0 = a(1600-800)^2+2400$$
$$-2400 = 3a(800)^2$$
$$a = -\frac{800}{3}$$

Substituting $h=0$, $k=2400$, and $a=-800/3$ into the standard form of the parabola, we get:

$$(x-0)^2 = -\frac{800}{3}(y-2400)$$

Simplifying, we get:

$$(x-800)^2 = -\frac{800}{3}(y-2400)$$

So the correct answer is (C).

A Cannon Fires a Cannonball as Shown in the Figure (x−1200)2=−4003(y−800)( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )(x−1200)2=−3400​(y−800)

A Cannon Fires a Cannonball as Shown in the Figure $( x - 1200 ) ^ { 2 } = - \frac { 400 } { 3 } ( y - 800 )$