The initial height for the potato launch is 45 meters. The maximum height is 167.5 meters after 5 seconds where the potato is launched, which represents the vertex of a parabola at (5, 167.5).
The formula for finding the vertex of a parabola is t = h = -b/2a and k = f(t). We know t = 5 seconds and f(t) = k = 167.5 meters.
We will use the quadratic equation in vertex form: f(t) = a(t - h)2 + k
1.) Substitute h = 5 and k = 167.5 ⇒ f(t) = a(t - 5)2 + 167.5
2.) We need to find 'a' in order to write the quadratic equation. To do that, we pick out a point in the parabola. Based on the given information, we can use the initial point (0, 45).
a(0 - 5)2 + 167.5 = 45
a(-5)2 + 167.5 = 45
25a + 167.5 = 45
25a = -122.5
a = -4.9
3.) Quadratic Equation: f(t) = -4.9(t - 5)2 + 167.5
4.) Now we can use the quadratic equation to find how long will it take for the potato to hit the water after the launch. In this case, f(t) = 0 and find 't'.
-4.9(t - 5)2 + 167.5 = 0
-4.9(t - 5)2 = -167.5
(t - 5)2 = 34.183673469387755102040816326531
t - 5 = 5.8466805513374642642526455176671
t = 10.846680551337464264252645517667
After the launch, the potato will hit the water at about 10.85 seconds.
