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 = \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 6 m 8 m 9 m

Explanation:

Step1: Find \( a \) using Pythagorean theorem

The base of the rectangular prism is a rectangle with length 8 m and width 6 m. For a right triangle with legs 6 m and 8 m, \( a^2 = 6^2 + 8^2 \).
\( a^2 = 36 + 64 = 100 \), so \( a = \sqrt{100} = 10 \) m.

Step2: Find \( b \) using Pythagorean theorem

Now, \( b \) is the hypotenuse of a right triangle with legs \( a = 10 \) m and 9 m. So \( b^2 = 10^2 + 9^2 \).
\( b^2 = 100 + 81 = 181 \), so \( b = \sqrt{181} \approx 13.5 \) m (rounded to nearest tenth).

Answer:

(a) \( a = 10 \) m
(b) \( b \approx 13.5 \) m