If Jane walks North for 3 miles, turns 45 degrees to the right, and then walks another 4 miles, how many miles will Jane be from her starting point? Give your answer as a decimal rounded to the nearest hundredth.

Plz help!!!!

use the law of cosines. The distance z is found using

z^2 = 3^2+4^2 - 2*3*4*cos(135)

or, you can figure the x- and y-displacements, then use the distance formula as usual.

3N = (0,3)
4NE = (2.828,2.828)
-----------------------
result: (2.828,5.828)

To solve this problem, we can use the Pythagorean theorem to find the distance between Jane's starting point and her ending point.

First, let's break down Jane's movements:

1. Jane walks North for 3 miles.
2. Jane turns 45 degrees to the right.
3. Jane walks another 4 miles.

Since Jane walks North for 3 miles, she goes directly up on the y-axis. Let's call this point A.

When Jane turns 45 degrees to the right, she is changing her direction from North to Northeast. This forms an isosceles right triangle. Let's call the point where Jane turns B.

Jane walks another 4 miles from point B. Let's call this point C.

To find the distance between Jane's starting point (A) and ending point (C), we need to find the length of the hypotenuse of the right triangle formed by points A, B, and C.

Using the Pythagorean theorem, we can calculate the distance:

AC² = AB² + BC²

The length of AB (a leg of the right triangle) can be calculated using the sine function:
sin(45 degrees) = AB / 4 miles
AB = sin(45 degrees) * 4 miles

The length of BC (the other leg of the right triangle) is equal to the distance Jane walks after turning, which is 4 miles.

Now, let's calculate the values:

AB = sin(45 degrees) * 4 miles
AB ≈ 2.83 miles

AC² = AB² + BC²
AC² = (2.83 miles)² + (4 miles)²
AC² = 7.9989 + 16
AC² ≈ 23.9989

Taking the square root of both sides, we get:

AC ≈ √(23.9989)
AC ≈ 4.9 miles (rounded to the nearest hundredth)

Therefore, Jane will be approximately 4.9 miles from her starting point.

To solve this problem, we can use trigonometry. Let's break down the steps:

1. Jane walks north for 3 miles. This means she moves directly upwards.

2. Jane turns 45 degrees to the right. This means she changes her direction by rotating clockwise by 45 degrees from her original north direction.

3. Jane walks another 4 miles. She moves in the new direction after the 45-degree turn.

To find how far Jane is from her starting point, we can use the Pythagorean theorem. This theorem states that for any 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, Jane's path forms a right triangle. The 3-mile segment is one side of the triangle, the 4-mile segment is the other side, and the hypotenuse is the distance from her starting point.

Using the Pythagorean theorem, we can calculate the distance from Jane's starting point as follows:
(distance from starting point)^2 = (3 miles)^2 + (4 miles)^2

Simplifying this equation:
(distance from starting point)^2 = 9 + 16
(distance from starting point)^2 = 25

Taking the square root of both sides to solve for the distance from the starting point:
distance from starting point = √25
distance from starting point = 5

Therefore, Jane will be 5 miles from her starting point.

To give the answer rounded to the nearest hundredth, we know that the answer is already a whole number, so it will be something.00. Since the hundredths digit is 0 in this case, the rounded answer will still be 5.00.

So, Jane will be approximately 5.00 miles from her starting point.

Well, Jane sounds like she's going on quite the adventure! Let's see if our clown math skills are up to the task.

If Jane walks north for 3 miles, we can think of that as her moving "up" on a coordinate plane. So, let's say she starts at point (0,0).

When she turns 45 degrees to the right, it means she's turning clockwise. So, she'll end up facing northeast, at a 45-degree angle to the vertical axis.

Since she walked 3 miles north, she would be at (0,3) at this point.

Now, she walks another 4 miles. Since she's still facing northeast, she'll move diagonally, 45 degrees to the right of vertical. This will effectively be a 45-degree angle from the positive y-axis.

To find out where she ends up, we can use the Pythagorean theorem: a^2 + b^2 = c^2. In this case, a = 3 and b = 4.

Using the formula, we get:
3^2 + 4^2 = c^2
9 + 16 = c^2
25 = c^2

Taking the square root of both sides, we find c = 5.

So Jane will be 5 miles from her starting point. Rounded to the nearest hundredth, that would be 5.00 miles.

Now, if only Jane had brought some clown shoes for the journey, she could have had even more fun! 🤡