Jake dives from a diving board that is 18 feet off the ground. He reaches a maximum height of 26 feet after 2 seconds. What is the standard and vertex formula of this quadratic equation and in how many seconds with Jake hit the water?

The time it will take Jake to hit the water ia approximately 4.46 seconds

Step-by-step explanation:

The given parameters in the question are; feet

The height off the ground, of the diving board from which Jake dives, h₀ = 18 feet

g = The acceleration due to gravity

The maximum height Jake reaches, h = 26 feet

The time Jake takes to reach maximum height = 2 seconds

We have;

h = h₀ + u·t + 1/2·g·t²

v² = u² - 2·g·(h - h₀)

v = u - gt

u = gt

Where;

v = The final velocity

u = The initial velocity

t = The time taken

u = 9.81 × 2 = 19.62 m/s

The vertex form of a quadratic equation is f(x) = a(x - h)² + k², with the vertex = (h, k)

Substituting the values, gives;

26 = 18 + u·t - 1/2 × 9.81 × t²

8 = 19.62·t - 1/2 × 9.81 × t² = 19.62·t - 4.905·t²

19.62·t - 4.905·t² - 8 = 0

Therefore, the standard form is  -4.905·t² + 19.62·t - 8 = 0

The coordinate of the vertex, is (2, 8), the vertex form is therefore;

-4.905(t - 2)² + 8

2) The time it will take Jake to hit the water is given as follows;

The height at which Jake meets the water = -18 feet from initial height

∴ -4.905·t² + 19.62·t - 8 = -18

-4.905·t² + 19.62·t - 8 + 18 = 0

-4.905·t² + 19.62·t + 10 = 0

t² - 4·t - 2.04 = 0

By the quadratic formula, we have;

t = (4 ± √((-4)² - 4 × 1 × -2.04))/(2×1)

t ≈ 4.46 seconds or t ≈ -0.46 seconds

∴ t ≈ 4.46 seconds

The time it will take Jake to hit the water ≈ 4.46 seconds.

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