Dividing by a fraction is equivalent to multiply by its reciprocal, then:
Now, we need to express the quadratic polynomials using their roots, as follows:
where y1 and y2 are the roots.
Applying the quadratic formula to the first polynomial:
Applying the quadratic formula to the second polynomial:
Applying the quadratic formula to the third polynomial:
Applying the quadratic formula to the fourth polynomial:
Substituting into the rational expression and simplifying:
Answer:
When do you hit the water?
1.075 seconds after you jump.
What is your maximum height?
the maximum height is 12.5626 ft
Step-by-step explanation:
The equation:
h(t) = -16*t^2 + 6*t + 12
Is the height as a function of time.
We know that the initial height is the height when t = 0s
h(0s) = 12
and we know that the diving board is 12 foot tall.
Then the zero in h(t)
h(t) = 0
Represents the surface of the water.
When do you hit the water?
Here we just need to find the value of t such that:
h(t) = 0 = -16*t^2 + 6*t + 12
Using the Bhaskara's formula, we get:
Then we have two solutions, and we only care for the positive solution (because the negative time happens before the jump, so that solution can be discarded)
The positive solution is:
t = (-6 - 28.4)/-32 = 1.075
So you hit the water 1.075 seconds after you jump.
What is your maximum height?
The height equation is a quadratic equation with a negative leading coefficient, then the maximum of this parabola is at the vertex.
We know that the vertex of a general quadratic:
a*x^2 + b*x + c
is at
x = -b/2a
Then in the case of our equation:
h(t) = -16*t^2 + 6*t + 12
The vertex is at:
t = -6/(2*-16) = 6/32 = 0.1875
Evaluating the height equation in that time will give us the maximum height, which is:
h(0.1875) = -16*(0.1875 )^2 + 6*(0.1875) + 12 = 12.5626
And the height is in feet, then the maximum height is 12.5626 ft