Question

a large birdcage has the shape of a rectangular prism. a straight branch of length b (in meters) is placed in the cage going from one corner to the opposite corner as shown in the figure. (the figure is not drawn to scale.) (a) find a. a = m (b) use your answer to part (a) to find b, the length of the branch. round your answer to the nearest tenth of a meter. b = m

Explanation:

Step1: Find the diagonal of the base

In the base of the rectangular - prism (a rectangle with sides 5 m and 12 m), we use the Pythagorean theorem $c^{2}=x^{2}+y^{2}$, where $x = 5$ and $y = 12$.
$a=\sqrt{5^{2}+12^{2}}$
$a=\sqrt{25 + 144}$
$a=\sqrt{169}$
$a = 13$ m

Step2: Find the length of the branch

Now, consider a right - triangle with sides $a = 13$ m and the height of the prism $h=10$ m, and the hypotenuse $b$ (the length of the branch). Using the Pythagorean theorem again, $b=\sqrt{a^{2}+h^{2}}$.
$b=\sqrt{13^{2}+10^{2}}$
$b=\sqrt{169 + 100}$
$b=\sqrt{269}$
$b\approx16.4$ m

Answer:

(a) $a = 13$ m
(b) $b\approx16.4$ m