A line passes through the points (2, 4) and (3, 1). What is its equation in point-slope form? Use one of the specified points in your equation. Write your answer using integers, proper
fractions, and improper fractions. Simplify all fractions.

a. 45 ft

b. 3 seconds

Step-by-step explanation:

a.

At time equals zero, the cannon is fired. The cannonball at that time is just exiting the cannon, so the cannonball at time zero is at the same height as the cannon.

We let t = 0, and evaluate function h(t).

h(t) = -16(t - 3)^2 + 189

h(0) = -16(0 - 3)^2 + 189

h(0) = -16(9) + 189

h(0) = -144 + 189

h(0) = 45

The height of the cannon is 45 ft.

b.

The cannonball is fired. It starts at 45 ft height which is where the cannon is. It goes up until it reaches a maximum height. Then it starts falling to the ground. At some point after it reaches maximum height, it reaches a height of 45 ft as it falls to the ground. We can find the time it takes the cannonball to be at 45 ft height on the way down.

We set the function equal to 45 and solve for t. One answer will be 0 since we already know the height at time zero is 45 ft. We need to find the other time.

-16(t - 3)^2 + 189 = 45

-16(t - 3)^2 = -144

(t - 3)^2 = 9

t - 3 = 3 or t - 3 = -3

t = 6 or t = 0

The other time that the height of the cannonball is zero is 6 seconds.

The graph is symmetric from t = 0 to t = 6. The maximum height happens exactly in the middle of the time period of 0 to 6 seconds. The maximum height happens at t = 3 seconds.