1. While following the directions on a treasure map, a pirate walks 45.0 m north, then turns and walks 7.5 m east. What single straight-line displacement could the pirate have taken to reach the treasure?

2. A hummingbird flies 1.2 m along a straight path at a height of 3.4 m above the ground. Upon spotting a flower below, the hummingbird drops directly downward 1.4 m to hover in front of the flower. What is the hummingbird's total displacement?

On both of these, the paths are right triangles. Use

distance^2=leg1^2 + leg2^2

1. d = (7.5^2 + 45.0^2)^0.5, so d = 45.6207189772 m = 46 m

2. d = (1.2^2 + (-1.4)^2)^0.5, so d = 1.84390889146 m = 1.8

1. Ahoy there, matey! So, our pirate friend walked 45.0 m north and then 7.5 m east. To find the single straight-line displacement, we'll use a bit of pirate trigonometry!

Using the Pythagorean theorem, we can find the hypotenuse (which represents the straight-line displacement):
distance^2 = leg1^2 + leg2^2

In our case, leg1 is the distance walked north (45.0 m) and leg2 is the distance walked east (7.5 m).

So, distance^2 = (45.0 m)^2 + (7.5 m)^2

Solving this equation will give us the answer to where the pirate could have sailed to reach the treasure. Just be careful to avoid any ticklish parrots guarding the loot!

2. Ooooh, we have a hovering hummingbird here with some vertical action! Let's find its total displacement, shall we?

The hummingbird first flies 1.2 m along a straight path at a height of 3.4 m above the ground. Then it drops directly downward by 1.4 m to hover in front of the flower.

Now, to calculate the total displacement, we'll consider both the horizontal and vertical movements.

The horizontal displacement is 1.2 m (since the hummingbird flew in a straight line).

For the vertical displacement, we're dropping 1.4 m downward, but we need to factor in the initial height of 3.4 m.

So, the vertical displacement is (3.4 m - 1.4 m) = 2.0 m.

To find the total displacement, we can use the Pythagorean theorem again:

distance^2 = horizontal displacement^2 + vertical displacement^2

distance^2 = (1.2 m)^2 + (2.0 m)^2

Solve this equation to get the total displacement of our hummingbird. Hopefully, it doesn't get too dizzy with all that up-and-down flying!

1. To find the single straight-line displacement of the pirate, we can use the Pythagorean theorem. With the given information, the distances walked north and east form the legs of a right triangle.

Using the Pythagorean theorem formula: distance^2 = leg1^2 + leg2^2

Let's plug in the values:
distance^2 = (45.0 m)^2 + (7.5 m)^2

Calculating:
distance^2 = 2025 m^2 + 56.25 m^2
distance^2 = 2081.25 m^2

Taking the square root of both sides, we find:
distance = √2081.25 m^2
distance ≈ 45.61 m

Therefore, the single straight-line displacement the pirate could have taken to reach the treasure is approximately 45.61 meters.

2. To find the hummingbird's total displacement, we'll use the same Pythagorean theorem principle again.

Here, we have a right triangle formed by the horizontal distance flown (1.2 m) and the vertical distance dropped (1.4 m). We want to find the total displacement, which will be the hypotenuse of the triangle.

Using the Pythagorean theorem formula: distance^2 = leg1^2 + leg2^2

Let's plug in the values:
distance^2 = (1.2 m)^2 + (1.4 m)^2

Calculating:
distance^2 = 1.44 m^2 + 1.96 m^2
distance^2 = 3.4 m^2

Taking the square root of both sides, we find:
distance = √3.4 m^2
distance ≈ 1.84 m

Therefore, the hummingbird's total displacement is approximately 1.84 meters.

To find the single straight-line displacement in the first question, we can use 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 lengths of the other two sides.

In this case, the northward distance of 45.0 m represents one leg of the right triangle, and the eastward distance of 7.5 m represents the other leg. We can use the Pythagorean theorem to find the length of the hypotenuse, which represents the straight-line displacement.

Using the formula you provided:

distance^2 = leg1^2 + leg2^2

Let's substitute the values from the question:

distance^2 = 45.0^2 + 7.5^2

Simplifying:

distance^2 = 2025 + 56.25
distance^2 = 2081.25

Now, to find the distance, we take the square root of both sides:

distance = √2081.25
distance ≈ 45.60 m

So, the single straight-line displacement the pirate could have taken to reach the treasure is about 45.60 m.

Now, let's move on to the second question.

To find the hummingbird's total displacement, we can again use the Pythagorean theorem. This time, we have two legs of the right triangle representing the hummingbird's horizontal and vertical movements.

The horizontal distance of 1.2 m represents one leg of the triangle, and the vertical distance of 1.4 m represents the other leg. The total displacement can be found by using the Pythagorean theorem.

Using the same formula:

distance^2 = leg1^2 + leg2^2

Substituting the values:

distance^2 = 1.2^2 + 1.4^2

Simplifying:

distance^2 = 1.44 + 1.96
distance^2 = 3.4

Taking the square root of both sides:

distance = √3.4
distance ≈ 1.84 m

Therefore, the hummingbird's total displacement is approximately 1.84 m.