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 \(a\) using the Pythagorean theorem for the base of the rectangular prism

The base of the rectangular prism has sides \(3m\) and \(4m\). By the Pythagorean theorem \(a^{2}=3^{2}+4^{2}\).

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Step2: Find \(b\) using the Pythagorean theorem for the space - diagonal of the rectangular prism

Now, consider a right - triangle with legs \(a = 5m\) and \(10m\), and hypotenuse \(b\). By the Pythagorean theorem \(b^{2}=a^{2}+10^{2}\). Substitute \(a = 5\) into the equation:

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Answer:

(a) \(a = 5\) m
(b) \(b\approx11.2\) m