Question

4 compound
as shown below, a box has a length of 4 cm, a width of 3 cm, and a height of 12 cm.

1. ( bd=) cm.
2. a stick has a length of ( sqrt{107} ) cm. can it be shipped in the box?

a yes
b no

Explanation:

Step1: Find the length of the diagonal of the base

The base of the box is a rectangle with length \(l = 4\) cm and width \(w=3\) cm. Using the Pythagorean theorem \(d_{base}=\sqrt{l^{2}+w^{2}}\), where \(d_{base}\) is the diagonal of the base.

$$d_{base}=\sqrt{4^{2}+3^{2}}=\sqrt{16 + 9}=\sqrt{25}=5$$

Step2: Find the length of \(BD'\)

The box has height \(h = 12\) cm. Now, consider a right - triangle with one leg as the diagonal of the base (\(d_{base}=5\) cm) and the other leg as the height of the box (\(h = 12\) cm). Using the Pythagorean theorem for the space - diagonal \(BD'\):

$$BD'=\sqrt{d_{base}^{2}+h^{2}}=\sqrt{5^{2}+12^{2}}=\sqrt{25+144}=\sqrt{169}=13$$

Step3: Compare the length of the stick with \(BD'\)

The length of the stick is \(\sqrt{107}\approx10.34\) cm (since \(\sqrt{100}<\sqrt{107}<\sqrt{121}\), i.e., \(10 < \sqrt{107}<11\)). And \(BD'=13\) cm. Since \(\sqrt{107}<13\)

Answer:

1. \(13\)
2. A. Yes