s(t) = -2t2 +12t +32
a) Determine the initial height of the stones above the ground.
b) Determine the time taken to reach the maximum height.
c) Calculate the maximum height.
d) How fast was the stone travelling when it hit the ground?
To answer these questions, we'll need to use the equation for height in terms of time, s(t) = -2t^2 + 12t + 32. Let's go through each question step by step:
a) Determine the initial height of the stones above the ground:
To find the initial height, we need to determine the value of s(t) when t = 0. Substitute t = 0 into the equation:
s(0) = -2(0)^2 + 12(0) + 32
s(0) = 32
So, the initial height of the stone above the ground is 32 units.
b) Determine the time taken to reach the maximum height:
The maximum height is reached when the stone reaches its peak. In this case, it's the maximum point on the graph of s(t). To find this time, we can use the formula for the axis of symmetry, which is given by t = -b / (2a), where a and b are the coefficients of the quadratic equation.
In our case, a = -2 and b = 12. Plugging in these values:
t = -(12) / (2(-2))
t = -12 / -4
t = 3
Therefore, the time taken to reach the maximum height is 3 units of time.
c) Calculate the maximum height:
To determine the maximum height, we substitute the time obtained in the previous step, t = 3, into the equation:
s(3) = -2(3)^2 + 12(3) + 32
s(3) = -18 + 36 + 32
s(3) = 50
Hence, the maximum height of the stones is 50 units.
d) How fast was the stone traveling when it hit the ground?
To determine the speed of the stone when it hits the ground, we need to find the time at which the height is equal to zero. In other words, we need to solve the equation s(t) = 0.
-2t^2 + 12t + 32 = 0
We can either use the quadratic formula or factorize the equation. In this case, factoring is more convenient:
-2(t^2 - 6t - 16) = 0
-2(t - 8)(t + 2) = 0
This equation has two solutions: t = 8 and t = -2. Since time cannot be negative in this context, we can ignore t = -2. Therefore, the stone hits the ground at t = 8.
To find the speed of the stone at this time, we need to find the derivative of the height function with respect to time, s'(t), and evaluate it at t = 8. The derivative can be found by differentiating the equation s(t) = -2t^2 + 12t + 32 with respect to t.
s'(t) = -4t + 12
Substituting t = 8 into the derivative:
s'(8) = -4(8) + 12
s'(8) = -32 + 12
s'(8) = -20
Therefore, the stone was traveling at a speed of 20 units per time when it hit the ground.