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 length of the base - diagonal (a)

In the base of the rectangular prism with length 8 m and width 6 m, use the Pythagorean theorem $a^{2}=6^{2}+8^{2}$.
$a^{2}=36 + 64$
$a^{2}=100$
$a=\sqrt{100}=10$ m

Step2: Find the length of the space - diagonal (b)

Now consider a right - triangle with legs a = 10 m and the height of the prism 9 m. Use the Pythagorean theorem again: $b^{2}=a^{2}+9^{2}$. Since a = 10, then $b^{2}=10^{2}+9^{2}$.
$b^{2}=100+81$
$b^{2}=181$
$b=\sqrt{181}\approx13.5$ m

Answer:

(a) $a = 10$ m
(b) $b\approx13.5$ m