The path of water sprayed from a fountain is modeled by h=−4.9t2+58.8t,
where h is the height of the water in meters after t seconds. Determine the maximum height of the water and the amount of time it takes the water to reach the ground.
To find the maximum height, we need to find the vertex of the parabolic equation h=−4.9t2+58.8t. The vertex occurs at t=−b/2a, where a=−4.9 and b=58.8.
t=−58.8/(2(-4.9))=6
So the water reaches its maximum height at 6 seconds.
To find the maximum height, we plug in t=6 into the equation:
h=−4.9(6)2+58.8(6)= 176.4 meters
Therefore, the maximum height of the water is 176.4 meters.
To find the amount of time it takes for the water to reach the ground, we need to find when h=0. Set h=0 in the equation:
0=−4.9t2+58.8t
0=t(−4.9t+58.8)
t=0 or t=12
Since t=0 is the starting point, we ignore that solution. Therefore, the water reaches the ground after 12 seconds.
So the amount of time it takes for the water to reach the ground is 12 seconds.
To determine the maximum height of the water, we need to find the vertex of the parabolic equation h = -4.9t^2 + 58.8t.
The vertex of a parabola can be found using the formula t = -b / (2a), where a and b are the coefficients of the quadratic equation.
In this case, a = -4.9 and b = 58.8.
Using the formula, we have t = -58.8 / (2(-4.9)) = 6 seconds.
To find the maximum height, substitute this value of t into the equation h = -4.9t^2 + 58.8t:
h = -4.9(6)^2 + 58.8(6)
h = -4.9(36) + 352.8
h = -176.4 + 352.8
h = 176.4 meters
Therefore, the maximum height of the water is 176.4 meters.
Now, let's determine the amount of time it takes for the water to reach the ground. Since the ground is at a height of 0 meters, we need to solve the equation -4.9t^2 + 58.8t = 0 for t.
Factoring out t, we have t(-4.9t + 58.8) = 0.
This equation will be true if t = 0 (at the start) or when -4.9t + 58.8 = 0 (reaching the ground).
Solving -4.9t + 58.8 = 0, we get t = 58.8 / 4.9 = 12 seconds.
Therefore, it takes 12 seconds for the water to reach the ground.
To determine the maximum height of the water, we need to find the vertex of the quadratic equation h = -4.9t^2 + 58.8t.
The vertex of a quadratic function in the form h = at^2 + bt + c is given by the coordinates (t, h) where t = -b/2a and h = f(t), where f(t) is the value of h at t.
In this case, a = -4.9 and b = 58.8. Let's substitute these values into the vertex formula:
t = -(-58.8) / (2 * -4.9)
= 58.8 / 9.8
≈ 6
Now, substitute t = 6 into the equation to find the maximum height:
h = -4.9(6)^2 + 58.8(6)
= -4.9(36) + 352.8
= -176.4 + 352.8
≈ 176.4
Therefore, the maximum height of the water is approximately 176.4 meters.
To determine the amount of time it takes for the water to reach the ground, we need to find the time when h = 0.
Set h = 0 in the equation:
-4.9t^2 +58.8t = 0
We can factor out t to solve this equation:
t(-4.9t + 58.8) = 0
This equation is satisfied when either t = 0 or -4.9t + 58.8 = 0.
For t = 0, the water has not yet been sprayed and has not reached the ground.
To find the time it takes for the water to reach the ground, solve -4.9t + 58.8 = 0:
-4.9t = -58.8
Divide both sides by -4.9:
t = -58.8 / -4.9
≈ 12
Therefore, it takes approximately 12 seconds for the water to reach the ground.