Question

name________________________ date________
tracking a dive
a diver is standing at the end of the diving board. the jump of the diver can be represented by the function ( d(t)=-16 t^{2}+v t+h ) in feet, ( t ) seconds after they jump where ( v ) is the initial velocity and ( h ) is the initial height.
the diver jumped off an 80 -foot diving board with an initial velocity of 64 feet per second.

1. write a function that represents the height of the diver.
2. write an equation that can be used to find the time it takes for the diver to reach the water. solve that equation.
3. write an equation that can be used to find the time it takes for the diver to reach a height of 80 feet. solve that equation.
4. how long does it take for the diver to reach a height of 128 ft? show or explain your thinking.

Explanation:

Step1: Substitute values into the function

Given \(v = 64\) and \(h=80\), substitute into \(d(t)=-16t^{2}+vt + h\).
\(d(t)=-16t^{2}+64t + 80\)

Step2: Set \(d(t)=0\) for water - level

The water - level is \(d(t) = 0\). So the equation is \(-16t^{2}+64t + 80=0\).
Divide through by \(-16\): \(t^{2}-4t - 5=0\)
Factor: \((t - 5)(t+1)=0\)
Set each factor equal to zero: \(t - 5=0\) or \(t + 1=0\)
Solutions: \(t = 5\) or \(t=-1\). Since time \(t\geq0\), \(t = 5\)

Step3: Set \(d(t)=80\)

The equation is \(-16t^{2}+64t + 80=80\)
Subtract \(80\) from both sides: \(-16t^{2}+64t=0\)
Factor out \(-16t\): \(-16t(t - 4)=0\)
Set each factor equal to zero: \(-16t=0\) or \(t - 4=0\)
Solutions: \(t = 0\) (initial time) or \(t = 4\)

Step4: Set \(d(t)=128\)

The equation is \(-16t^{2}+64t + 80=128\)
Rearrange: \(-16t^{2}+64t-48 = 0\)
Divide through by \(-16\): \(t^{2}-4t + 3=0\)
Factor: \((t - 1)(t - 3)=0\)
Set each factor equal to zero: \(t - 1=0\) or \(t - 3=0\)
Solutions: \(t = 1\) or \(t = 3\)

Answer:

1. \(d(t)=-16t^{2}+64t + 80\)
2. Equation: \(-16t^{2}+64t + 80 = 0\), Solution: \(t = 5\)
3. Equation: \(-16t^{2}+64t + 80=80\), Solutions: \(t = 0\) and \(t = 4\)
4. Equation: \(-16t^{2}+64t + 80=128\), Solutions: \(t = 1\) and \(t = 3\)