To get to her friend's house, Sandra walked along the sidewalk in her neighborhood. If she walked 100 yards at a direction of 45° northeast and then turned to walk 400 yards straight east, how many yards and in what direction would she need to walk if she wanted to walk straight home without having to make any turns?

Did you make your sketch?

On mine, I have a triangle with sides 100 and 400 and the contained angle of 135°
Using the cosine law:
x^2 = 100^2 + 400^2 -2(100)(400)cos135°
x = 475.992..

To get the angle opposite the side of 400:
sinØ/400 = sin135/475.992..
sinØ = .5942..
Ø = 36.46°

so the angle should be 45-36.46 = appr 8.54° north of east

Thanks. All of this was right except it was southwest instead of northeast.

To walk straight home without making any turns, Sandra needs to cancel out her previous steps by walking in the opposite direction.

First, we need to break down Sandra's initial walk into its north and east components.

To do that, we calculate the distance traveled in the north direction by multiplying the total distance by the sine of the angle:
North distance = 100 yards * sin(45°)

North distance = 100 yards * 0.707106781 ≈ 70.71 yards

Next, we calculate the distance traveled in the east direction by multiplying the total distance by the cosine of the angle:
East distance = 100 yards * cos(45°)

East distance = 100 yards * 0.707106781 ≈ 70.71 yards

So Sandra walked approximately 70.71 yards northeast.

To walk straight home, Sandra needs to cancel out the north and east components by walking in the opposite direction.

North distance to walk home = -70.71 yards (negative indicates the opposite direction)

East distance to walk home = -70.71 yards (negative indicates the opposite direction)

Therefore, Sandra needs to walk approximately 70.71 yards in the south direction and 70.71 yards in the west direction to walk straight home without making any turns.

To figure out the number of yards Sandra would need to walk straight home, we can use basic trigonometry. Let's break down her movements step by step:

1. Sandra walks 100 yards at a direction of 45° northeast. This can be visualized as a right-angled triangle, with 45° being the angle between the direction of Sandra's initial walk and the horizontal axis.

2. As the angle is 45° and the distance is 100 yards, we can use trigonometric ratios to find the horizontal and vertical components of her walk. In this case, we are interested in the horizontal component.

The horizontal component can be found using the cosine function:

cos(45°) = Adjacent / Hypotenuse

In this case, the adjacent side is the horizontal component, and the hypotenuse is the distance she walked (100 yards). So:

cos(45°) = (horizontal) / 100

Solving for the horizontal component:

horizontal = cos(45°) * 100

3. Sandra then turns and walks 400 yards straight east. Since she walks east, this portion of her walk is entirely in the horizontal direction.

4. Finally, to calculate how far and in what direction Sandra needs to walk to go straight home, we need to add up the horizontal components of her two walks. The vertical components cancel out each other.

horizontal_total = horizontal (from step 2) + 400

To find the direction of her walk, we can find the angle between the horizontal_total and the horizontal axis using the inverse tangent function:

angle = tan^(-1) (horizontal_total / 0)

Since dividing any number by 0 is undefined, the direction of her walk is simply "straight east".

Hence, Sandra would need to walk a total of horizontal_total yards straight east to walk straight home without making any turns.