X = 47 km.
Y = 69 km.
a. d^2 = X^2+Y^2 = 47^2 + 69^2 = 6970
d = 83.49 km.
b. Tan A = Y/X = 69/47 = 1.46809
A = 55.74o W of S = 34.26o S of W.
Y = 69 km.
a. d^2 = X^2+Y^2 = 47^2 + 69^2 = 6970
d = 83.49 km.
b. Tan A = Y/X = 69/47 = 1.46809
A = 55.74o W of S = 34.26o S of W.
In this scenario, we can consider the distance between the two towns as the hypotenuse of a right triangle. The distance north-south can be one of the sides, and the distance east-west can be the other side. Let's call the north-south distance y (47.0 km) and the east-west distance x (69.0 km).
Using the Pythagorean theorem, we can calculate the length of the hypotenuse (the shortest length of highway) as follows:
Length of hypotenuse^2 = x^2 + y^2
Length of hypotenuse = √(x^2 + y^2)
Let's calculate it:
Length of hypotenuse = √(69.0 km)^2 + (47.0 km)^2
Length of hypotenuse = √(4761 km^2 + 2209 km^2)
Length of hypotenuse = √(6970 km^2)
Length of hypotenuse ≈ 83.5 km
So, the shortest length of the highway that can be built between the two towns is approximately 83.5 km.
To find the direction of the highway (the angle with respect to due west), we can use trigonometry. In this case, we can use the inverse tangent function (arctan) to find the angle.
Let's calculate it:
Angle = arctan(y/x)
Angle = arctan(47.0 km/69.0 km)
Angle ≈ 34.24 degrees (rounded to two decimal places)
Therefore, the highway should be directed at an angle of approximately 34.24 degrees (positive angle with respect to due west).