Question on: JAMB Mathematics - 2024

To find the cosine of the angle a diagonal of the room makes with the floor, we first need to determine the length of the room's diagonal and the diagonal on the floor.

Let's denote the length, width, and height of the room as l = 12m, w = 9m, and h = 8m respectively.

1. The diagonal of the floor (d_f) can be found using the Pythagorean theorem:

d_f = \[\sqrt{l^2 + w^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15m\]

1. The diagonal of the room (d_r) can be found using the Pythagorean theorem, with the floor diagonal and height:

d_r = \[\sqrt{d_f^2 + h^2} = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17m\]

Now, let \[\theta\] be the angle between the room's diagonal and the floor. The cosine of this angle can be found using the adjacent side (floor diagonal) and the hypotenuse (room diagonal):

cos(\[\theta\]) = \(rac{d_f}{d_r}\) = \[\frac{15}{17}\]