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Precalculus : Building Concepts and Connections

A triangular-shaped playground with vertices at points $P, Q,$ and $R$ is to be enlarged to form a triangle with vertices $P, R,$ and a new point, $S$ (see the figure). Segment $P Q$ is twice as long as segment $R Q$ and the length of segment $P R$ is 45 feet. Let $d(x)$ be the length of segment PS, where $x$ is the measure of angle QPS in radians. Find constants $A$ and $C$ such that $A$ is positive, $0<x<\frac{\pi}{2},$ and $d(x)=A \sec (x+C) .$ Use your function to determine the length of segment PS if $x=\frac{2 \pi}{15}$. (FIGURE CAN'T COPY)