Answer:
6. 50 ft/s
7. y = -16x² + 50x + 16
8. 3.42 seconds
Explanation:
The equation that models the height in terms of the time since launch is a parabola, so in vertex form, it is
y = a(x - h)² + k
Where (h, k) is the vertex and a is a constant. x is the time and y is the height.
In this case, the vertex is the point (1.56, 55) because its maximum is at 1.56 seconds where it reached 55 ft. So, replacing (h, k) = (1.56, 55), we get:
y = a(x - 1.56)² + 55
To find the value of a, we will use the fact that it starts at 16ft, so when the time x = 0, the height y = 16, then
16 = a(0 - 1.56)² + 55
16 = a(-1.56)² + 55
Solving for a, we get:
16 - 55 = a(2.43) + 55 - 55
-39 = a(2.43)
-39/2.43 = a(2.43)/2.43
-16 = a
Therefore, the equation that represents the height based on the time since lunch is:
y = -16(x -1.56)² + 55
y = -16(x² - 2x(1.56) + 1.56²) + 55
y = -16x² + 50x - 39 + 55
y = -16x² + 50x + 16
On an equation that models the height and the time, the initial velocity is the coefficient of the variable x, so in this case, the initial velocity is 50 ft/s because it is the number beside the x.
Finally, the daredevil will hit the ground when the height is equal to 0, so we need to solve the following equation:
y = -16x² + 50x + 16 = 0
-16x² + 50x + 16 = 0
Using the quadratic equation, we get that the solutions are:
[tex]\begin{gathered} x=\frac{-50\pm\sqrt[]{50^2-4(-16)(16)}}{2(-16)}=\frac{-50\pm59.36}{-32} \\ Then \\ x=\frac{-50+59.36}{-32}=-0.29 \\ or \\ x=\frac{-50-59.36}{-32}=3.42 \end{gathered}[/tex]Since the negative time x has no sense, the daredevil hits the ground after 3.42 seconds.
Therefore, the answers are:
6. 50 ft/s
7. y = -16x² + 50x + 16
8. 3.42 seconds