The height, h(t), of a projectile, in metres, can be modelled by the equation
h(t) = 14t – 5t2, where t is the time in seconds after the projectile is released. Can the projectile ever reach a height of 9m? Explain.
Well, let me see if I can calculate that for you... *puts on imaginary clown hat*
So we have the equation h(t) = 14t - 5t², and we want to find out if the projectile can reach a height of 9 meters.
To do that, we just need to plug in 9 for h(t) and solve for t. Let's see... *takes out calculator, presses imaginary buttons*
9 = 14t - 5t²
Now, I could go on and solve this quadratic equation, but instead, I'm going to take a shortcut and call my friend, Mr. Intuition. *picks up imaginary phone*
Mr. Intuition says that if the coefficient of t² is negative, like in this equation (-5t²), then the graph of the equation opens downwards. That means the maximum height of the projectile will occur at the vertex of this parabola.
And since the t² term is negative, it means that the highest point the projectile can reach is at the vertex. So, the projectile will never reach a height of 9 meters because the maximum height it can reach is given by the y-coordinate of the vertex.
But hey, don't worry! Maybe the projectile can reach a height of 9 meters if you give it some extra rocket boosters or a really strong gust of wind. Who knows, right? *winks*
To determine if the projectile can reach a height of 9m, we need to find the values of t that satisfy the equation h(t) = 9.
Given the equation h(t) = 14t – 5t^2, we can set it equal to 9:
14t – 5t^2 = 9
Rearranging the equation:
5t^2 - 14t + 9 = 0
This is a quadratic equation, so we can solve it using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 5, b = -14, and c = 9.
Plugging in these values into the quadratic formula:
t = (14 ± √((-14)^2 - 4 * 5 * 9)) / (2 * 5)
t = (14 ± √(196 - 180)) / 10
t = (14 ± √16) / 10
Simplifying further:
t = (14 ± 4) / 10
This gives us two possible values for t:
t = 18/10 = 1.8 seconds
t = 10/10 = 1 second
Therefore, there are two solutions to the equation h(t) = 9, which means the projectile can reach a height of 9m at two different times: 1 second and 1.8 seconds.
2&√13 -2÷2
h(t)=-16t³+100t
h(t) = t(14-5t)
max height is achieved at t = 7/5
h(7/5) = 9.8
so, yes.