Question

as shown below, this box is 4 cm long, 2 cm wide, and 1 cm high. an ant crawls on the surface of the box from vertex a to vertex b. the length of the shortest route is ____ cm.
a 5
b √29
c 7
d √37

Explanation:

Step1: Unfold the box

There are three ways to unfold the box.
Case 1: Unfold the box such that the path is on the front - right face. The length of the two - dimensional path is \(l_1=\sqrt{(4 + 2)^2+1^2}=\sqrt{36 + 1}=\sqrt{37}\).
Case 2: Unfold the box such that the path is on the front - top face. The length of the two - dimensional path is \(l_2=\sqrt{(4+1)^2 + 2^2}=\sqrt{25+4}=\sqrt{29}\).
Case 3: Unfold the box such that the path is on the left - top face. The length of the two - dimensional path is \(l_3=\sqrt{(2 + 1)^2+4^2}=\sqrt{9 + 16}=5\).

Step2: Compare the lengths

Compare \(l_1=\sqrt{37}\approx6.08\), \(l_2=\sqrt{29}\approx5.39\), \(l_3 = 5\).

Answer:

A. 5