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 = 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
Answer
Explanation:
Step1: Find the length of the diagonal of the base
The base of the rectangular - prism is a rectangle with sides 5m and 12m. We use the Pythagorean theorem (c^{2}=a^{2}+b^{2}) to find the length of the diagonal of the base (a). Here, (a = \sqrt{5^{2}+12^{2}}). [a=\sqrt{25 + 144}=\sqrt{169}=13]
Step2: Find the length of the space - diagonal (b)
The height of the rectangular prism is 6m. Now, we consider a right - triangle with legs (a = 13m) (the diagonal of the base) and the height (h = 6m). Using the Pythagorean theorem again, (b=\sqrt{a^{2}+h^{2}}). Substitute (a = 13) and (h = 6) into the formula: (b=\sqrt{13^{2}+6^{2}}=\sqrt{169 + 36}=\sqrt{205}\approx14.3)
Answer:
(a) (a = 13) (b) (b\approx14.3)