Answer
Verifying algebraically, the year is $2007$.
Work Step by Step
Verifying algebraically:
$$P=-0.0038t^2+0.272t+5.25$$
Substituting $P=7$ into the equation:
$$7=-0.0038t^2+0.272t+5.25$$
$$-0.0038t^2+0.272t-1.75$$
Using the quadratic formula:
$$t=\frac{-0.272\pm\sqrt{0.272^2-4(-0.0038)(-1.75)}}{2(-0.0038)}$$
$$t_1=\frac{-0.272+\sqrt{0.272^2-4(-0.0038)(-1.75)}}{2(-0.0038)}=7.15$$
$$t_2=\frac{-0.272-\sqrt{0.272^2-4(-0.0038)(-1.75)}}{2(-0.0038)}=64.43$$
Based on the constraint of $1\leq t\leq14$, take $t=7.15\approx7$.
Then, the year is:
$$Year=2000+7=2007$$
