Answer
The magnitude of $\mathscr{E}$ is $~~1.00~mV$
Work Step by Step
$B = (8.00\times 10^{-2}~T/m\cdot s)~y~t~\hat{k}$
Note that the magnetic field changes over time and the magnetic field is in the +z direction.
Also, the magnitude of the magnetic field varies linearly at different values of $y$
Since the y values of the loop vary from $y = 0$ to $y = 0.250~m$, the average magnitude of the magnetic field through the loop is as follows:
$B_{ave} = [(8.00\times 10^{-2}~T/m\cdot s)~t~\hat{k}]~(\frac{0.250~m}{2}) = (1.00\times 10^{-2}~T/s)~t~\hat{k}$
The magnetic flux through the loop also changes over time.
We can find the induced emf in the loop:
$\mathscr{E} = -\frac{d\Phi}{dt}$
$\mathscr{E} = -A~\frac{dB}{dt}$
$\mathscr{E} = -(0.400~m)(0.250~m)~(1.00\times 10^{-2}~T/s)$
$\mathscr{E} = -1.00~mV$
The magnitude of $\mathscr{E}$ is $~~1.00~mV$
