help for math please!!

Question

Answer 1

part a) has been solved.

Part b) can be found using the quadratic equation to solve 0 = -4.9x² + 6x + 5

The quadratic equation is [-b +/- sqrt(b² - 4ac)] / 2a, where a = -4.9, b = 6 and c = 5

Plugging the values a, b and c into the equation yields

[-6 +/- sqrt(6² -4(-4.9)(5))] / 2(-4.9)

= [-6 +/- sqrt(36 + 98)] / -9.8

= [-6 +/- sqrt(134)] / -9.8

There are 2 solutions:

1. [-6 + 11.5758] / -9.8 = 5.5758/-9.8 = -0.5690 and
2. [-6 - 11.5758] / -9.8 = -17.5758 / -9.8 = +1.7935

Clearly, a negative time is not the answer so the ball will reach h(x) = 0 meters in 1.7935 seconds

c) The path of the ball is a parabola and that means the maximum point (in this case) is usually 1/2 the time from initial point to final point. HOWEVER, in this problem the initial height is 5 meters, so the maximum height is attained at 1/2 time from 5 meters back to 5 meters.

The time required to travel from 5 meters to 0 meters using the function h(x) is 1.7935 seconds. There was another solution to the quadratic equation, -0.5690 seconds. This is the time required from 0 meters to 5 meters. The sum of the 2 solutions is 1.2245 seconds, which is the time from 5 meters back to 5 meters when following the parabolic function. One half of this time is 0.61225 seconds and this is the time to reach maximum height.

d) Plugging in 0.61225 seconds into the function for h(x) yields the maximum height = 6.8367 meters.

FYI:

Another way to estimate c) and d) is to plug in different values of x into h(x) and record the results. For example, when x=0, h(x) = 5, when x=1, h(x) = -4.9*1² + 6*1 + 5 = -4.9+6+5 = 6.1 meters. When x=2, h(x) = -4.9*2² +6*2 + 5 = -4.9*4 +12 + 5 = -2.6 meters, so x=2 is too large.

Continuing in this way you'll eventually find that the solution is between x = 0.61 and x = 0.62. This is very time consuming and labor intensive and is not the recommended way to solve the problem.

Answer 2

When x=0 (no time has elapsed):

h(0) = -4.9(0²) + 6(0) +5

= 0 + 5

= 5

When time elapsed = 0, the height of the ball is 5.

So, 5 represents the intitial height of the ball.