You take measurements of the distance traveled by an object that is increasing its speed at a constant rate. the distance traveled as a function of time can be modeled by a quadratic function.
a. write a quadratic function that models distances of 10 ft at 1 sec, 30 ft at 2 sec, and 100 ft at 4 sec.
b. find the zeros of the function.
c. describe what each zero represents for this real-world situation.
if anyone can with even part of this, it would be greatly appreciated!

A)

Quadratic function:   s = at² + bt + c

where s = distance, t = time

when s = 10, t = 1. when s = 30, t = 2, when s = 100, t = 4

Substituting these:

s = at² + bt + c

10 = a(1)² + b*1 + c

10 = a + b + c                       a + b + c = 10 (a)

30 = a(2)² + b*2 + c

30 = 4a + 2b + c                  4a + 2b + c = 30(b)
 
 
100 = a(4)² + b*4 + c

100 = 16a + 4b + c              16a + 4b + c = 100(c)  

 

So we have three unknown equations:

a + b + c = 10 (a)

4a + 2b + c = 30(b)

16a + 4b + c = 100(c) 

So you can input these in a function calculator.

a = 5, b = 5, c = 0

Therefore s = at² + bt + c,   s = 5t² + 5t 

b)

The zeros of  s = 5t² + 5t, is when s = 0. 

 5t² + 5t = 0

5t(t + 1) = 0              

5t = 0           or   t + 1 = 0

t = 0/5                  t = 0 - 1

t = 0                     t = -1

 
c)

Therefore at time = 0, time = -1.

In real life time = 0 is just before timing.

At t = -1, is 1 second before timing.