Answer:
3 seconds
Step-by-step explanation:
Our function is
h(t) = -16t²+24t+72
To find how long it takes the roll to hit the ground, we must find the roots of this equation. We first set the function equal to 0:
0 = -16t²+24t+72
Next we factor out the GCF. The GCF of -16, 24 and 72 is 8:
0 = 8(-2t²+3t+9)
To factor the remaining trinomial, we find factors of -2(9) = -18 that sum to 3:
-18 = -1(18) or 1(-18); these do not sum to 3.
-18 = -2(9) or 2(-9); these do not sum to 3.
-18 = -3(6) or 3(-6); -3 and 6 sum to 3. We will split 3t into -3t and 6t:
0 = 8(-2t²-3t+6t+9)
Group together the first two terms and the second two terms:
0 = 8((-2t²-3t)+(6t+9))
Factor out the GCF of each group. The GCF of the first group is -t:
0 = 8(-t(2t+3)+(6t+9))
The GCF of the second group is 3:
0 = 8(-t(2t+3)+3(2t+3))
The common factor is now 2t+3:
0 = 8(2t+3)(-t+3)
Using the zero product property, either 2t+3 = 0 or -t+3 = 0:
2t+3 = 0
Subtract 3 from each side:
2t+3-3 = 0-3
2t = -3
Divide both sides by 2:
2t/2 = -3/2
t = -1.5
-t+3 = 0
Subtract 3 from each side:
-t+3-3 = 0-3
-t = -3
Divide both sides by -1:-
-t/-1 = -3/-1
t = 3
This means the roots are at t = -1.5 and t = 3.
This means the toilet paper hits the ground at -1.5 seconds and 3 seconds. Since we do not have negative time, the correct answer is 3 seconds.
