Answer :
Since it takes 80 minutes to walk along the sides of the square, it would take 80 minutes for the dog to walk along the diagonal as well, provided that its speed remains constant. So, the dog would take 80 minutes to walk along the diagonal of the square.
The time it takes for the dog to walk along the diagonal of the square can be determined using the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the diagonal of the square is the hypotenuse of a right triangle, and the sides of the square are the other two sides. Let's assume that each side of the square has a length of "s". So, the length of the diagonal, which is also the hypotenuse, can be calculated using the Pythagorean theorem as follows:
diagonal² = s² + s²
diagonal² = 2s²
Now, we know that the dog takes 80 minutes to walk along the sides of the square. Let's denote the speed at which the dog walks along the sides as "v_s" (in distance per minute). Therefore, the total distance covered by the dog along the sides is 4s (since there are four sides in a square), and the time taken is 80 minutes. We can use the formula:
distance = speed × time
4s = v_s × 80
From this equation, we can solve for "v_s" as:
v_s = (4s) / 80
v_s = s / 20
Now, let's calculate the time it would take for the dog to walk along the diagonal. The speed at which the dog walks along the diagonal can be denoted as "v_d" (in distance per minute). The distance along the diagonal is the length of the diagonal, which we found earlier to be √(2s²). Therefore, the time taken can be calculated as:
time = distance / speed
time = √(2s²) / v_d
To find the relationship between v_s and v_d, we can consider that the distances covered along the sides and the diagonal are the same. So, we have:
v_s × 80 = v_d × √(2s²)
Substituting v_s = s / 20, we get:
(s / 20) × 80 = v_d × √(2s²)
Simplifying the equation:
4s = v_d × √(2s²)
To find v_d, we can isolate it:
v_d = (4s) / √(2s²)
v_d = 2s√2 / 2s
v_d = √2
So, the speed at which the dog walks along the diagonal is √2 times the speed at which it walks along the sides. Now, let's calculate the time it would take for the dog to walk along the diagonal:
time = √(2s²) / √2
time = √(s²)
time = s
Therefore, the time it would take for the dog to walk along the diagonal is equal to the length of one side of the square.
In this case, since it takes 80 minutes to walk along the sides of the square, it would take 80 minutes for the dog to walk along the diagonal as well, provided that its speed remains constant.
So, the dog would take 80 minutes to walk along the diagonal of the square.
Learn more about Pythagorean theorem
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