A family vacationed at a beach 120 mi east and 30 mi south of their home. Find the distance and the direction from their home to the beach.
To get the distance, use the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Assuming the point where the home lies is at the origin (0,0), thus the beach lies at (120, -30). Substituting,
d = sqrt((120 - 0)^2 + (-30 - 0)^2)
d = sqrt(14400 + 900)
d = sqrt(15300)
d = 123.69 mi
For the direction, we can just use trigonometry. Let X be the angle:
tan X = opposite/adjacent
tan X = -30/120
X = arctan -0.25
X = -14.04 degrees, or 14.04 degrees below the horizontal, or 14.04 degrees south of east
Therefore the beach is located 123.69 miles, 14.04 degrees south of east from home.
Hope this helps~ :3
To find the distance and direction from their home to the beach, we can use the Pythagorean theorem to calculate the distance and trigonometry to determine the direction.
Step 1: Calculate the distance using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square 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 distance from the home to the beach forms a right triangle, with one side measuring 120 miles and the other side measuring 30 miles. Let's call the distance from the home to the beach "d".
Using the Pythagorean theorem, we can write the equation:
d^2 = 120^2 + 30^2
Step 2: Solve for "d".
Plugging the values into the equation:
d^2 = 14400 + 900
d^2 = 15300
Taking the square root of both sides:
d = √15300
d ≈ 123.6 miles
So, the distance from their home to the beach is approximately 123.6 miles.
Step 3: Determine the direction using trigonometry.
To determine the direction, we need to find the angle that the line connecting the home and the beach makes with the eastward direction.
Using trigonometry, we can find this angle by taking the inverse tangent (arctan) of the ratio of the opposite side (30 miles) to the adjacent side (120 miles).
Let's call this angle "θ".
Using the arctan function:
θ = arctan(30/120)
θ = arctan(0.25)
θ ≈ 14.04 degrees
So, the direction from their home to the beach is approximately 14.04 degrees south of east.