If a square Mercator map shows 1000 miles at latitude 30°, how many miles does it show at latitude 60°?
As far as I know, in a Mercator map, every horizontal strip is stretched by $\cos x$ so that the distance from the equator to $x$ north is $\int_0^xR\sec x\,dx$. The distance to latitude 30° should be $\int_0^{\pi/6}R\sec x\,dx=R\ln\sqrt3=1000$. Thus, $R=\frac{1000}{\ln\sqrt3}$. At latitude 60°, $\int_0^{\pi/3}R\sec x\,dx=R\ln(2+\sqrt3)=1000\frac{\ln(2+\sqrt3)}{\ln\sqrt3}$. But the answer key says $\frac{1000}{\sqrt3}$. Where was I wrong?
No need to integrate $R\sec{x}$.
All you need to do is find the scale factor $k$.
$$k=\frac{R\sec{30^o}}{R\sec{60^o}}$$ $$k=\frac{\sec{30^o}}{\sec{60^o}}$$ $$k=\frac{\cos{60^o}}{\cos{30^o}}$$ $$k=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$$ $$k=\frac{1}{\sqrt{3}}$$ Now multiply this by your radius $R$. Hence, the Mercator map will show $\frac{1000}{\sqrt{3}}$ miles at latitude $60^o$.