The function that describes the height depending on time is a quadratic function:
[tex]h(t)=-16t^2+64t+60[/tex]Write the quadratic function in the vertex form of a parabola:
[tex]f(x)=a(x-m)^2+n[/tex]The vertex has coordinates (m,n).
Factor out -16 from the first two terms:
[tex]h(t)=-16(t^2-4t)+60[/tex]Add 4 - 4 inside the parenthesis to complete the square:
[tex]\begin{gathered} h(t)=-16(t^2-4t+4-4)+60 \\ =-16\lbrack(t-2)^2-4\rbrack+60 \end{gathered}[/tex]Use the distributive property to simplify the expression:
[tex]\begin{gathered} h(t)=-16(t-2)^2-16(-4)+60 \\ =-16(t-2)^2+64+60 \\ =-16(t-2)^2+124 \end{gathered}[/tex]Therefore, the vertex form of the equation is:
[tex]h(t)=-16(t-2)^2+124[/tex]Which means that the vertex of the parabola, is:
[tex](2,124)[/tex]Since the coefficient of (t-2)^2 is negative, then the vertex represents the maximum height.
Since:
[tex]h(2)=124[/tex]Then, it takes 2 seconds to reach the height of 124, which is the maximum height.
The graph can be sketched as a parabola with vertex (2,124) and y-intercept (0,60).
Therefore:
a) It takes 2 seconds to reach the maximum height.
b) The maximum height is 124 feet.
c)
