Problem Setup

We are given a rectangular roof with dimensions $12\text{ m} \times 4\text{ m}$. The roof is supported by $4\text{ m}$ tall poles, meaning the roof is $H = 4\text{ m}$ above the ground.

Sunlight falls on the roof at an angle of $45^\circ$ from the east. We interpret this as the sun's rays having an altitude angle of $\alpha = 45^\circ$ with respect to the horizontal ground, and the sun is located in the East, so the rays travel horizontally towards the West.

Shadow Projection Geometry

The shadow on the ground is the projection of the roof area along the direction of the sun's rays. Let the roof lie in a horizontal plane at height $H$. The sun's rays travel in a direction $\vec{d}$ with components $(d_x, d_y, d_z)$.

Since the sun is from the East and its rays travel West, the horizontal component $d_x$ is negative, and $d_y=0$ (no North-South component). The angle with the horizontal is $\alpha = 45^\circ$, so $d_z = -\sin\alpha$. The horizontal component magnitude is $\cos\alpha$. Thus, $\vec{d}$ is proportional to $(-\cos\alpha, 0, -\sin\alpha)$.

A point $(x, y, H)$ on the roof projects onto the ground ($z=0$) at point $(x', y', 0)$. The projection line is $P(t) = (x, y, H) + t\vec{d}$. Setting $P(t)_z = 0$: $H + t(-\sin\alpha) = 0$ $t = \frac{H}{\sin\alpha}$ Substituting $t$ back into the x and y coordinates:

$x' = x + t(-\cos\alpha) = x - \frac{H \cos\alpha}{\sin\alpha} = x - H \cot\alpha$ $y' = y + t(0) = y$

With $H=4\text{ m}$ and $\alpha=45^\circ$, we have $\cot(45^\circ) = 1$. The projection transformation is:

$(x, y, 4) \rightarrow (x - 4, y, 0)$

Calculating Shadow Area

Let the roof dimensions be $W=4\text{ m}$ (East-West) and $L=12\text{ m}$ (North-South). The roof occupies the region $0 \le x \le 4$ and $0 \le y \le 12$ at height $z=4$. The projection transformation shifts the x-coordinates by $-4$ while leaving the y-coordinates unchanged. The projected shadow on the ground ($z=0$) occupies the region $-4 \le x' \le 0$ and $0 \le y' \le 12$. The dimensions of the shadow rectangle are:

• Width (E-W direction): $0 - (-4) = 4\text{ m}$
• Length (N-S direction): $12 - 0 = 12\text{ m}$

The area of the shadow is $4\text{ m} \times 12\text{ m} = 48\text{ m}^2$.

Alternatively, if the roof dimensions are $L=12\text{ m}$ (East-West) and $W=4\text{ m}$ (North-South), the roof occupies $0 \le x \le 12$ and $0 \le y \le 4$ at $z=4$. The projection transforms this to $-4 \le x' \le 8$ and $0 \le y' \le 4$. The shadow dimensions are $12\text{ m}$ (E-W) and $4\text{ m}$ (N-S). The area remains $12\text{ m} \times 4\text{ m} = 48\text{ m}^2$.

The projection is a translation parallel to the ground, which preserves the area.

Final Answer

The area of the shadow cast on the ground is $48\text{ m}^2$.