A rake is leaning against a wall. The top of the rake hits the wall $63$ inches above the ground and the bottom of the rake is $16$ inches away from the base of the wall. A beetle crawls from the bottom of the rake to the top of the rake. Then the beetle climbs down to the base of the wall and back to the bottom of the rake. How far does the beetle travel in all? $\newline$ $\_\_\_\_\_$ inches

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Pythagorean Theorem and its converse

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Q. A rake is leaning against a wall. The top of the rake hits the wall

63

inches above the ground and the bottom of the rake is

16

inches away from the base of the wall. A beetle crawls from the bottom of the rake to the top of the rake. Then the beetle climbs down to the base of the wall and back to the bottom of the rake. How far does the beetle travel in all?

\newline

\_\_\_\_\_

inches

Question Prompt: question_prompt: How far does the beetle travel in total?
Identify Triangle Components: Identify the legs and hypotenuse of the right triangle formed by the rake and the wall. $\newline$ Legs: $63$ inches (height), $16$ inches (base) $\newline$ Hypotenuse: $x$ (length of the rake)
Use Pythagorean Theorem: Use the Pythagorean Theorem to find the length of the rake. $a^2 + b^2 = c^2$ $63^2 + 16^2 = x^2$
Calculate Squares: Calculate the squares of the legs. $\newline$ $63^2 = 3969$ $\newline$ $16^2 = 256$
Add Squares: Add the squares of the legs to find the square of the hypotenuse. $\newline$ $3969 + 256 = x^2$ $\newline$ $x^2 = 4225$
Find Rake Length: Take the square root of both sides to find the length of the rake. $\sqrt{x^2} = \sqrt{4225}$ $x = 65$
Calculate Total Distance: Calculate the total distance the beetle travels. $\newline$ Distance up the rake: $65$ inches $\newline$ Distance down the wall: $63$ inches $\newline$ Distance along the ground: $16$ inches $\newline$ Total distance = $65 + 63 + 16$
Add Distances: Add up the distances to get the total distance traveled by the beetle. $\newline$ Total distance = $65 + 63 + 16 = 144$ inches