A room is 12m long, 9m wide and 8m high. Find the cosine of the angle which a diagonal of the room makes with the floor of the room.

1. Analyze the question

We are given a rectangular room with dimensions:

• Length ($L$) = 12m
• Width ($W$) = 9m
• Height ($H$) = 8m

We need to find the cosine of the angle ($\theta$) between the space diagonal of the room and the floor.

2. Explain the concept

• The Floor Diagonal: The floor is a rectangle with length 12m and width 9m. The diagonal of the floor ($d$) forms a right-angled triangle with the length and width. Using Pythagoras' theorem: $d = \sqrt{L^2 + W^2}$.
• The Space Diagonal: The diagonal of the room connects a bottom corner to the opposite top corner. This forms a right-angled triangle where the base is the floor diagonal ($d$), the height is the room height ($H$), and the hypotenuse is the space diagonal ($D$).
• The Angle ($\theta$): The angle $\theta$ is between the floor diagonal ($d$) and the space diagonal ($D$). In this right-angled triangle, $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{d}{D}$.

3. Show the solution

Step 1: Calculate the floor diagonal ($d$)

Step 2: Calculate the space diagonal ($D$) The space diagonal $D$ is the hypotenuse of the triangle with base $d$ (15m) and height $H$ (8m):

Step 3: Calculate $\cos(\theta)$

4. Identify the answer

Comparing our result to the options provided, the correct option is A.

5. Tip

When dealing with 3D geometry problems in JAMB, always visualize the two-step Pythagoras process: first, find the diagonal of the base, then use that result to find the space diagonal. Also, keep an eye out for common Pythagorean triples like (8, 15, 17) to save time during calculations!