2) suppose the function h(t)= - 5t² + 10t + 3 models the height of a diver above the water (in meters), t…

Explanation:

Step1: Find spring - board height

When the diver leaves the springboard, (t = 0). Substitute (t = 0) into (h(t)=-5t^{2}+10t + 3). [h(0)=-5(0)^{2}+10(0)+3=3]

Step2: Find time to reach maximum height

The function (h(t)=-5t^{2}+10t + 3) is a quadratic function in the form (y = ax^{2}+bx + c) where (a=-5), (b = 10), (c = 3). The time (t) at which the maximum occurs is given by (t=-\frac{b}{2a}). [t=-\frac{10}{2\times(-5)} = 1]

Step3: Find maximum height

Substitute (t = 1) into (h(t)). [h(1)=-5(1)^{2}+10(1)+3=-5 + 10+3=8]

Step4: Find time to enter water

When the diver enters the water, (h(t)=0). So we solve the quadratic equation (-5t^{2}+10t + 3=0). Using the quadratic formula (t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}), where (a=-5), (b = 10), (c = 3). [t=\frac{-10\pm\sqrt{10^{2}-4\times(-5)\times3}}{2\times(-5)}=\frac{-10\pm\sqrt{100 + 60}}{-10}=\frac{-10\pm\sqrt{160}}{-10}=\frac{-10\pm4\sqrt{10}}{-10}] We take the positive root since time cannot be negative. [t=\frac{-10 + 4\sqrt{10}}{-10}\approx\frac{-10+4\times3.1623}{-10}=\frac{-10 + 12.6492}{-10}=\frac{2.6492}{-10}\text{(wrong sign)}] [t=\frac{-10-4\sqrt{10}}{-10}\approx\frac{-10 - 12.6492}{-10}=\frac{-22.6492}{-10}=2.26]

Answer:

a) 3 meters b) 1 second c) 8 meters d) 2.26 seconds