= Solution
For an ideal <electrical transformer>, let $n=N_p/N_s$. The reflected secondary load is $R_{\mathrm{in}}=n^2R_L$. Matching a $4\,\Omega$ load to an $8\,\Omega$ source therefore requires
$$
\boxed{n=\sqrt{\frac84}=\sqrt2.}
$$
With the generator replaced by its internal resistance, its resistance referred to the secondary is
$$
\boxed{R_{\mathrm{out}}=\left(\frac{N_s}{N_p}\right)^2(8\,\Omega)=4\,\Omega.}
$$
This is the source output resistance relevant to matching and maximum <power> transfer.
To find the windings, use the <inductance of a toroidal winding>. The <mean> magnetic path is $\ell=2\pi(0.10)\,\mathrm m$, the core <area> is $A=30\times10^{-4}=0.003\,\mathrm{m^2}$ and $\mu=500\mu_0$. Thus
$$
L_p=\frac{\mu_0\mu_rN_p^2A}{\ell},
\qquad N_p=\sqrt{\frac{(0.75)2\pi(0.10)}{\mu_0(500)(0.003)}}=500.
$$
The ideal secondary turn count is $N_s=500/\sqrt2=353.55$, giving the practical nominal design
$$
\boxed{N_p=500,\qquad N_s\simeq354.}
$$
With integer turns the irrational ratio cannot be exact: $500:354$ reflects the $4\,\Omega$ load as $7.98\,\Omega$, an approximately $0.25\%$ resistance mismatch. These turn counts use the stated $0.75\,\mathrm H$ self-inductance; no operating <frequency> was specified from which to set a separate minimum <inductance>.
The $200\,\Omega$ condition must refer to the unloaded magnetizing reactance, not to the matched loaded input resistance. With $L_p=0.75\,\mathrm H$,
$$
|X_p|=2\pi fL_p\geq200\,\Omega
\quad\Longrightarrow\quad f\geq42.4\,\mathrm{Hz}.
$$
The secondary self-inductance is $L_s=L_p/n^2=0.375\,\mathrm H$, so its corresponding unloaded coil impedance satisfies
$$
\boxed{Z_{s,\mathrm{coil}}=i\omega L_s,\qquad |Z_{s,\mathrm{coil}}|\geq100\,\Omega.}
$$
If output impedance means this transformed coil reactance, $100\,\Omega$ is its minimum magnitude. It is distinct from the $4\,\Omega$ transformed source resistance. This is the <magnetizing reactance and reflected resistance in a transformer> distinction. A finite magnetizing <inductance> adds a parallel branch, $Z_{\mathrm{in}}=(8\,\Omega)\parallel(i\omega L_p)$, so exact purely resistive matching is an ideal large-reactance approximation; interpreting the loaded input itself as both $8\,\Omega$ and at least $200\,\Omega$ would be impossible.
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