Answer:
A) 144 feet
[tex]\textsf{B)} \quad 16t(6-t)[/tex]
Step-by-step explanation:
Part A
Given polynomial:
[tex]H(t)=96t-16t^2[/tex]
where:
- H(t) is the height of the debris (in feet)
- t is the time (in seconds) after the explosion
To find the height of the debris 3 seconds after the explosion, substitute t = 3 into the polynomial and solve:
[tex]\begin{aligned}\implies \sf Height & = 96(3)-16(3)^2\\ & = 288 - 16(9)\\ & =288-144\\ & =144 \sf \:\: ft\end{aligned}[/tex]
Part B
To factor the polynomial, rewrite 96 as 6 × 16:
[tex]\implies h(t)=6 \cdot 16t-16t^2[/tex]
Rewrite t² as t × t:
[tex]\implies h(t)=6 \cdot 16t-16t \cdot t[/tex]
Factor out the common term 16t:
[tex]\implies h(t)=16t(6-t)[/tex]
Check
Plug t = 3 into the factored expression:
[tex]\begin{aligned}h(3) & = 16(3)(6-3)\\& = 16(3)(3)\\& = 48(3)\\& = 144 \end{aligned}[/tex]
