# A cannonball was shot upwards out of a cannon from the top of a castle.  The initial height of the cannonball was 40 m above the ground.Two seconds after being shot, the cannonball was again at...

A cannonball was shot upwards out of a cannon from the top of a castle.  The initial height of the cannonball was 40 m above the ground.

Two seconds after being shot, the cannonball was again at the height of 40 m above the ground.  Three seconds after being shot the cannonball hit the ground.  What is the height of the cannonball 2.5 seconds after being shot?

mrsq | Certified Educator

In this problem we have 2 variables, the height of the cannonball and the time.  The time is independent (x) and the height is dependent (y). Using the information in the problem we can determine two points:

(0, 40)             This means at 0 minutes, the cannonball is at 40 m

(3, 0)     This means that a 3 minutes, the height is 0 meters

When writing an equation, we need to find the slope.  We can do this using the slope formula:

(y2 - y1)/(x2-x1)

Plug your values of x and y above into this formula and you get:

(0 - 40)/(3-0) = -40/3

Now you have the slope (-40/3) as well as 2 points.  Now input what you are given into the point-slope formula: y - y1= m(x - x1).

Remember that m = the slope and x1 and y1 = either of your original points.  In this case, I will you (3, 0), but you could also use (40, 0) as your point if you prefer.

y - y1 =m(x-x1)

y - 0 = -40/3(x - 3) Substitute

y = -40/3x + 40 Distribute

Last, remember how we said x represented seconds and y represented the height?  The original question asked that the height was a 2.5 seconds.  So we plug in 2.5 for x into our equation and solve for y as follows:

y = -40/3(2.5) + 40

y = -33.333 + 40

y = 6.667 feet (approximately)

After 2.5 seconds, the cannonball is approximately 6.667 feet off the ground.   Keep up the good work and keep practicing!