Answer
A) 2 seconds
B) What is the initial height of the diver? 10 m (above the surface)
What is the height after 3 seconds? 20 meters below the surface
Step-by-step explanation
A) The height, h, in meters of a diver, t, seconds after diving off a platform can be modeled by:
[tex]h(t)=-5t^2+5t+10[/tex]When the diver hits the water's surface the height is 0 meters, that is, h(t) = 0.
[tex]0=-5t^2+5t+10[/tex]We can solve this equation with help of the quadratic formula with the coefficients a = -5, b = 5, and c = 10, as follows:
[tex]\begin{gathered} t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ t_{1,2}=\frac{-5\pm\sqrt{5^2-4\cdot(-5)\operatorname{\cdot}10}}{2(-5)} \\ t_{1,2}=\frac{-5\pm\sqrt{225}}{-10} \\ t_1=\frac{-5+15}{-10}=-1 \\ t_2=\frac{-5-15}{-10}=2 \end{gathered}[/tex]Given that variable t measures time, then the negative result has no sense and it is discarded.
Therefore, it takes the diver 2 seconds to hit the water's surface
B) What is the initial height of the diver?
At the initial height, t = 0. Substituting this value into the formula, we get:
[tex]\begin{gathered} h(0)=-5(0)^2+5(0)+10 \\ h(0)=0+0+10 \\ h(0)=10\text{ m} \end{gathered}[/tex]The initial height is 10 meters.
What is the height after 3 seconds?
Substituting t = 3 seconds into height's formula, we get:
[tex]\begin{gathered} h(3)=-5(3)^2+5(3)+10 \\ h(3)=-5(9)+5(3)+10 \\ h(3)=-45+15+10 \\ h(3)=-20\text{ m} \end{gathered}[/tex]The height after 3 seconds is 20 meters below the surface.
