First, let us notice that since $APB$ and $CDR$ are equilateral, we have $AB = AP$ and $CD = DR$. Hence, the difference between the perimeters of $APQRD$ and $ABCD$ equals
$$(AD + DR + QR + PQ + AP) - (AD + CD + BC + AB) = PQ + QR - BC.$$
By hypothesis we know that $BC = 24$ and $\text{per}(APQRD) - \text{per}(ABCD) = 32$. Therefore, using the previous equality, we get $PQ + QR = 56$.

On the other hand, we have that $P\hat{B}Q = 360^\circ - A\hat{B}C - C\hat{B}Q - A\hat{B}P$, which gives $P\hat{B}Q = 90^\circ$ since $A\hat{B}C = 150^\circ$ and $A\hat{B}P = C\hat{B}Q = 60^\circ$. Analogously, we can obtain that $Q\hat{C}R = 90^\circ$.
Using the Pythagorean theorem on the triangle $\triangle PBQ$ we find that $PQ = \sqrt{18^2 + 24^2} = 30$ and so, using that $PQ + QR = 56$, it follows that $QR = 26$.
Finally, since $Q\hat{C}R = 90^\circ$, we can apply the Pythagorean theorem again but now in the triangle $\triangle CQR$ in order to get $QR^2 = CQ^2 + CR^2$. From the fact that $QC = BC = 24$ and $QR = 26$ it follows that $CR = 10$. Since $CR = CD$ because $\triangle CDR$ is equilateral, it is $CD = 10$.