Ainsley’sandCaleb’sapartmentbuildingsareexactlythesameheight. Ainsley measures the distance between the buildings as 51 m and observes that the angle of depression from the roof of her building to the bottom of Caleb’s is about 64°. How tall, to the nearest metre, is each building?

tan 54 = h/51

To solve this problem, we can use trigonometry, specifically the tangent function.

Let's define some variables:
- Let h be the height of each building (which is the same for both Ainsley and Caleb).
- Let d be the horizontal distance between the two buildings, which Ainsley measured as 51 m.

From Ainsley's point of view, if we draw a horizontal line from the bottom of Caleb's building to the line of sight from the roof of Ainsley's building, we can form a right triangle. The angle of depression of 64° is the angle between the line of sight and the horizontal line we just drew.

Now, let's look at the tangent function in relation to this right triangle:
tan(angle) = opposite/adjacent
In this case, the angle is 64°, the opposite side is the height of the buildings (h), and the adjacent side is the horizontal distance between the buildings (d).

So, we have the equation:
tan(64°) = h/d

Now, let's solve for h:
h = d * tan(64°)
h = 51 * tan(64°)
h ≈ 105 meters

Therefore, each building is approximately 105 meters tall.