A highway is to be built between two towns, one of which lies 47.0 km south and 69.0 km west of the other. (a) What is the shortest length of highway that can be built between the two towns, and (b) at what angle would this highway be directed, as a positive angle with respect to due west?
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.
To find the shortest length of highway between the two towns, we can use the Pythagorean theorem. The theorem states that in a 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 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).