Answer
(a) $A = 0.600~m/s^3$
(b) $\alpha(t) = (2.40~rad/s^3)~t$
(c) $t = 3.54~s$
(d) $\theta = 17.7~rad$
Work Step by Step
(a) $a(t)= A~t$
$a(3)= A~(3.00~s) = 1.80~m/s^2$
$A = 0.600~m/s^3$
(b) $\alpha(t) = \frac{a(t)}{r}$
$\alpha(t) = \frac{(0.600~m/s^3)~t}{0.250~m}$
$\alpha(t) = (2.40~rad/s^3)~t$
(c) $\omega (t) = \int_{0}^{t}\alpha(t)~dt$
$\omega (t) = \int_{0}^{t}(2.40~rad/s^3)~t~dt$
$\omega (t) = (1.20~rad/s^3)~t^2$
We can find $t$ when $\omega = 15.0~rad/s$
$(1.20~rad/s^3)~t^2 = 15.0~rad/s$
$t = \sqrt{\frac{15.0~rad/s}{1.20~rad/s^3}}$
$t = 3.54~s$
(d) $\theta (t) = \int_{0}^{t}~\omega(t)~dt$
$\theta (t) = \int_{0}^{t}~(1.20~rad/s^3)~t^2~dt$
$\theta (t) = (0.400~rad/s^3)~t^3$
We can find $\theta$ when $t = 3.54~s$:
$\theta = (0.400~rad/s^3)(3.54~s)^3$
$\theta = 17.7~rad$
