Final answer:
The asymptotes of the hyperbola with vertices at (±9, 0) and foci at (±15, 0) are y = ±(4/3)x.
Step-by-step explanation:
The question asks for the asymptotes of a hyperbola with given vertices at (±9, 0) and foci at (±15, 0). To find the asymptotes of a hyperbola, we need to use the formula y = ±(b/a)x, where a is the distance from the center to the vertices along the x-axis, and b is the distance from the center to the co-vertices along the y-axis. In this case, the distance a is 9, since that's the distance from the center to the vertices. The foci have a distance of c from the center, where c is 15 in this case. We use the relationship c2 = a2 + b2 to solve for b.
Plugging into this equation, we get 152 = 92 + b2, which simplifies to 225 = 81 + b2. Solving for b, we find that b2 = 144, so b = 12. Now we have the values of a and b, and we can write down the equations for the asymptotes: y = ±(12/9)x, or simplified, y = ±(4/3)x.