Rebels launch a cannonball from a 58.8ft tall cliff toward their enemy's ship. The cannonball travels 19.6ft per second (ft/s). The equation for the cannonball's height, h, at time, t, after launch is h(t) = -4.9t^2 + 19.6t + 58.8, where h is in feet and t is in seconds.

After how many seconds will the cannonball strike the enemy's ship?

What is tue maximum height the cannonball reaches?

ship impact ... h = 0 ... solve the quadratic for t

time for max height is on the axis of symmetry ... t = -b / 2 a = -19.6 / - 9.8
... plug t into the height equation to find max height

To find the time it takes for the cannonball to strike the enemy's ship, we need to set the height of the cannonball, h(t), equal to zero and solve for t.

The equation for the cannonball's height, h(t) = -4.9t^2 + 19.6t + 58.8, where h is in feet and t is in seconds.

Setting h(t) equal to zero:
0 = -4.9t^2 + 19.6t + 58.8

This is a quadratic equation, we can solve it using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = -4.9, b = 19.6, and c = 58.8.

Plugging these values into the quadratic formula, we get:

t = (-19.6 ± √(19.6^2 - 4(-4.9)(58.8))) / (2(-4.9))

Simplifying further:

t = (-19.6 ± √(384.16 + 1152.48)) / (-9.8)

t = (-19.6 ± √1536.64) / (-9.8)

t = (-19.6 ± 39.2) / (-9.8)

Now we have two possible solutions:

1. t = (-19.6 + 39.2) / (-9.8)
2. t = (-19.6 - 39.2) / (-9.8)

Calculating these solutions:

1. t = 19.6 / (-9.8) = -2
2. t = -59.8 / (-9.8) = 6.1

Since time cannot be negative in this context, we discard the -2 second solution. Therefore, the cannonball will strike the enemy's ship after approximately 6.1 seconds.

To find the maximum height the cannonball reaches, we need to determine the vertex of the parabolic equation h(t) = -4.9t^2 + 19.6t + 58.8.

The vertex of a parabolic equation in the form y = ax^2 + bx + c is given by (-b/2a, f(-b/2a)), where f(x) is the value of the function evaluated at x.

In this case:
a = -4.9
b = 19.6

The vertex is then found at (-b/2a, f(-b/2a)) = (-19.6 / (2*(-4.9)), f(-19.6 / (2*(-4.9))))

Calculating further:

(-19.6 / (2 * (-4.9)), f(-19.6 / (2 * (-4.9))))
(-19.6 / -9.8, f(-19.6 / -9.8))
(2, f(2))

To determine the value of f(2), we plug in t = 2 into the equation h(t) = -4.9t^2 + 19.6t + 58.8:

h(2) = -4.9(2)^2 + 19.6 * 2 + 58.8
h(2) = -19.6 + 39.2 + 58.8
h(2) = 78.4

So, the maximum height the cannonball reaches is 78.4 feet.