A spider and a fly are in a room that has length 8 m, width 4 m, and height 4 m. The spider is on one end wall 1 cm from the floor midway from the two side walls. The fly is caught in the spider’s web on the other end wall 1 cm from the ceiling and also midway from the two side walls. What is the shortest distance the spider can walk to enjoy his meal? (Hint: Draw a two-dimensional picture.)
4m
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To find the shortest distance the spider can walk to enjoy its meal, we can visualize the room and the spider's position. Let's draw a two-dimensional representation of the room:
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|__________P_____________|______F_|
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In the diagram:
- The line represents the floor and the ceiling.
- The two vertical lines represent the side walls.
- The letter "P" represents the spider's position, which is 1 cm above the floor and midway between the side walls.
- The letter "F" represents the fly's position, which is 1 cm below the ceiling and midway between the side walls.
To find the shortest distance, the spider can walk to the fly, we need to determine the diagonal distance from the spider's position (P) to the fly's position (F) in this 2D representation.
Using the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides, we can find the shortest distance.
The formula for the Pythagorean theorem is:
c^2 = a^2 + b^2
In our case, the spider's path represents the hypotenuse (c), and the distance from P to the side wall represents one side (a), while the distance from F to the side wall represents the other side (b).
Now, let's calculate the values:
Side a:
The distance from P to the side wall is half of the room's width, which is 4 m / 2 = 2 m.
Side b:
The distance from F to the side wall is half of the room's width, which is 4 m / 2 = 2 m.
Now we can substitute these values into the Pythagorean theorem formula:
c^2 = a^2 + b^2
c^2 = (2 m)^2 + (2 m)^2
c^2 = 4 m^2 + 4 m^2
c^2 = 8 m^2
To find the shortest distance (c), we need to take the square root of c^2:
c = √(8 m^2)
c = √(4 x 2 m^2)
c = √(4) x √(2 m^2)
c = 2 m x √(2)
c ≈ 2 m x 1.414 (approximating √2 as 1.414)
c ≈ 2.828 m
Therefore, the shortest distance the spider can walk to enjoy its meal is approximately 2.828 meters.