Maria just moved to a new part of the state, and she decides to take her car for a spin to see the area. Maria’s starting point is Culver City. She drives directly toward River City, located 25 miles East and 60 miles North of Culver City. After driving for 10 miles, she turns and heads due South for 15 miles, then stops to fill up her gas tank. How far away is Maria from River City?

the triangle with sides 25 and 60 is just a scaled-up version of a 5-12-13 right triangle. The hypotenuse is 65.

Draw the diagram. By similar triangles, her distance x East is

x/10 = 25/65 = 5/13
x = 50/13
Her distance y North is
y/15 = 60/65 = 12/13
y = 120/13
So her distance z from Culver City is
z^2 = (50/13)^2 + (15 - 120/13)^2
z = 25/√13 = 6.93 miles

where did 65 come from?

CR = 25+60i = 65mi[22.6o].

V1 = 10mi[22.6o] CW.
V2 = 15mi[180o] CW.

V1+V2 = 10[22.6]+15[180] = (10*sin22.6+15*sin180)+(10*cos22.6+15*cos180)i
V1+V2 = 3.84-5.77i = 6.93mi[-34o] = 6.93mi[326o] CW from Culver City.

D = 65[22.6]+6.93[326-180]. 326-180 = 146o.
D = (65*sin22.6+6.93*sin146)+(65*cos22.6+6.93*cos146)i
D = 28.9+54.3i.
D = sqrt(28.9^2+54.3^2) = ___ miles from River City.

To find out how far Maria is from River City, we can use the concepts of distance traveled and direction.

First, let's visualize Maria's route:

1. Maria starts in Culver City.
2. She drives 10 miles in an unknown direction (which we will determine later).
3. She turns south and drives 15 miles.
4. She stops to fill up her gas tank.

To determine the distance between Maria's final location and River City, we need to find the remaining distance between her final location and River City. We will break it down into two parts:

1. The North-South distance covered from her final location to River City.
2. The East-West distance covered from her final location to River City.

Let's calculate each part:

1. North-South Distance:
- Maria initially drove 60 miles north from Culver City.
- After turning south and driving 15 miles, she is now 60 - 15 = 45 miles north of her starting point.

2. East-West Distance:
- Maria initially drove 25 miles east from Culver City.
- After turning south and driving 15 miles, she is still 25 miles east of her starting point.

Now, we have a right-angled triangle formed by Maria's final location, River City, and the unknown point where Maria turned. To find the distance between Maria's final location and River City, we can use the Pythagorean theorem.

Distance^2 = (North-South Distance)^2 + (East-West Distance)^2

applying this formula, we get:

Distance^2 = (45 miles)^2 + (25 miles)^2
Distance^2 = 2025 + 625
Distance^2 = 2650

Therefore,

Distance = sqrt(2650)
Distance ≈ 51.48 miles

So, Maria is approximately 51.48 miles away from River City.