What about in the following case?

Question

Homi · Accepted Answer

To find the shortest distance an ant travels on the surface of a cuboid, we unfold the cuboid's faces into a flat, two-dimensional shape called a net. Then, we calculate the straight-line distance between the ant's starting point and the laddoo's position on this net. We must consider different ways to unfold the cuboid, as the shortest path might cross different faces.

Step 1 — Identify positions

Let us define the dimensions of the cuboid. Let its length (front-to-back) be $L$ , its width (left-to-right) be $W$ , and its height (bottom-to-top) be $H$ . We place the cuboid in a 3D coordinate system with its bottom-left-back corner at the origin $(0,0,0)$ .

The ant is on the right face. From the diagram, we assume it is at the center of this face in terms of its length and height. So, the ant's position is $A = (W, L/2, H/2)$ .

The laddoo is on the top face. It is at the center of the edge between the top face and the front face. This edge runs from $(0,0,H)$ to $(W,0,H)$ . So, the laddoo's position is $L = (W/2, 0, H)$ .

Step 2 — Unfold the cuboid

We will unfold the cuboid's faces into a 2D plane. This allows us to find the shortest path, which is a straight line on the unfolded net. We need to consider different ways the faces can be unfolded to find the true shortest path.

Step 3 — Calculate distances for different paths

We will consider three common unfoldings that connect the ant's starting point to the laddoo.

Path 1: Ant crosses the top-right edge. This path involves the ant moving from the right face directly onto the top face. We unfold these two faces. Let us place the right face in a 2D coordinate system. Its dimensions are $L \times H$ . The ant's position on this unfolded face is $A' = (L/2, H/2)$ . The top face is attached to the top edge of the right face. Its dimensions are $W \times L$ . The laddoo is at $(W/2, 0, H)$ in 3D. On the top face, its coordinates are $(W/2, 0)$ (distance from left edge, distance from front edge). When we unfold, the laddoo's position on the 2D net is $L' = (L/2, H + W/2)$ . The distance $D_1$ is: $D_1 = \sqrt{(L/2 - L/2)^2 + (H/2 - (H+W/2))^2}$ $= \sqrt{0^2 + (-H/2 - W/2)^2}$ $= \sqrt{(H/2 + W/2)^2}$

$\boxed{\frac{H+W}{2}}$

Diagram 1

Path 2: Ant crosses the front-right edge, then the front-top edge. This path involves the ant moving from the right face, across the front face, and then onto the top face. We unfold these three faces. Let us place the front face in a 2D coordinate system. Its dimensions are $W \times H$ . The right face is attached to the right side of the front face. Its dimensions are $L \times H$ . The ant's position on this unfolded net is $A' = (W + L/2, H/2)$ . The top face is attached to the top side of the front face. Its dimensions are $W \times L$ . The laddoo's position on this unfolded net is $L' = (W/2, H)$ . The distance $D_2$ is: $D_2 = \sqrt{((W+L/2) - W/2)^2 + (H/2 - H)^2}$ $= \sqrt{(W/2 + L/2)^2 + (-H/2)^2}$ $= \sqrt{\frac{(W+L)^2}{4} + \frac{H^2}{4}}$

$\boxed{\frac{1}{2} \sqrt{(W+L)^2 + H^2}}$

Diagram 2

Path 3: Ant crosses the back-right edge, then the back-top edge. This path involves the ant moving from the right face, across the back face, and then onto the top face. We unfold these three faces. Let us place the back face in a 2D coordinate system. Its dimensions are $W \times H$ . The right face is attached to the right side of the back face. Its dimensions are $L \times H$ . The ant's position on this unfolded net is $A' = (W + L/2, H/2)$ . The top face is attached to the top side of the back face. Its dimensions are $W \times L$ . The laddoo is at $(W/2, 0, H)$ in 3D. On the top face, its coordinates are $(W/2, L)$ (distance from left edge, distance from back edge). The laddoo's position on this unfolded net is $L' = (W/2, H+L)$ . The distance $D_3$ is: $D_3 = \sqrt{((W+L/2) - W/2)^2 + (H/2 - (H+L))^2}$ $= \sqrt{(L/2)^2 + (-H/2 - L)^2}$ $= \sqrt{\frac{L^2}{4} + \frac{(H+2L)^2}{4}}$

$\boxed{\frac{1}{2} \sqrt{L^2 + (H+2L)^2}}$

Diagram 3

Answer

To find the shortest distance the ant has to travel to reach the laddoo, we must calculate the distances for all possible unfoldings and choose the minimum value. In this case, the shortest distance will be the minimum of the following three expressions:

(i) $\frac{H+W}{2}$ (ii) $\frac{1}{2} \sqrt{(W+L)^2 + H^2}$ (iii) $\frac{1}{2} \sqrt{L^2 + (H+2L)^2}$

Question 22

Answer

More questions in IT