we have the ordered pairs
(0,0)
(1,76)
(2,120)
(3,132)
(4,112)
Plot the given points
see the attached figure
In this problem we have a vertical parabola open downward
The vertex represents a maximum
The equation is of the form
y=ax^2+bx+c -----> quadratic equation
using a quadratic regression calculator
we have that
a=-16
b=92
c=0
therefore
y=-16x^2+92x
For x=5 sec
substitute
y=-16(5)^2+92(5)
y=60
the answer is
Alternative method (approximate solution)
The quadratic equation in vertex form is equal to
y=a(x-h)^2+k
where
(h,k) is the vertex
I will assume that the vertex in this problem is the point (3,132)
so
(h,k)=(3,132)
substitute
y=a(x-3)^2+132
Find out the value of a
we have the point (0,0)
substitute in the equation
0=a(0-3)^2+132
0=9a+132
a=-132/9
a=-14.67
therefore
y=-14.67(x-3)^2+132
For x=5
y=-14.67(5-3)^2+132
y=73.33 units
Third Method
using the equation
y=ax^2+bx+c
points (0,0), (1,76) and (2,120)
(0,0) --------> 0=a(0)^2+b(0)+c ----------> c=0
y=ax^2+bx
(1.76) ------> 76=a(1)^2+b(1) ----------> a+b=76 ------> equation 1
(2,120) ----> 120=a(2)^2+b(2) ----> 4a+2b=120 ----> equation 2
solve the system of equations
the solution of this system is
a=-16
b=92
therefore
the equation is
