a large birdcage has the shape of a rectangular prism. a straight branch of length (b) (in meters) is placed…
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=square 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 = square m)
Answer
Explanation:
Step1: Find $a$ using Pythagorean theorem on base
The base of the rectangular - prism has sides of lengths 5 m and 12 m. By the Pythagorean theorem $a^{2}=5^{2}+12^{2}$. $a^{2}=25 + 144=169$. Taking the square - root of both sides, $a=\sqrt{169}=13$ m.
Step2: Find $b$ using Pythagorean theorem on the right - triangle with sides $a$ and height of prism
We know that $a = 13$ m and the height of the prism is 6 m. By the Pythagorean theorem $b^{2}=a^{2}+6^{2}$. Substituting $a = 13$ into the equation, we get $b^{2}=13^{2}+6^{2}=169 + 36=205$. Taking the square - root of both sides, $b=\sqrt{205}\approx14.3$ m.
Answer:
(a) $a = 13$ m (b) $b\approx14.3$ m