Respuesta :
General Idea:
If we have a quadratic function of the form f(x)=ax^{2} +bx+c , then the function will attain its maximum value only if a < 0 & its maximum value will be at x=-\frac{b}{2a} .
Applying the concept:
The height h is modeled by h = −16t^2 + vt + c, where v is the initial velocity, and c is the beginning height of the firecracker above the ground. The firecracker is placed on the roof of a building of height 15 feet and is fired at an initial velocity of 100 feet per second. Substituting 15 for c and 100 for v, we get the function as [tex] h=-16t^2+100t+15 [/tex].
Comparing the function f(x)=ax^{2} +bx+c with the given function [tex] h=-16t^{2} +100t+15 [/tex], we get [tex] a = -16 [/tex], [tex] b = 100 [/tex] and [tex] c=15 [/tex].
The maximum height of the soccer ball will occur at t=\frac{-b}{2a}=\frac{-100}{2(-16)} = \frac{-100}{-32}=3.125 seconds
The maximum height is found by substituting [tex] t=3.125 [/tex] in the function as below:
[tex] h=-16(3.125)^2+100(3.125)+15=171.25 feet [/tex]
Conclusion:
Yes ! The firecracker reaches a height of 100 feet before it bursts.
