A highway is to be built between two towns, one of which lies 24.0 km south and 76.0 km west of the other. What is the shortest length of highway that can be built between the two towns, and at what angle would this highway be directed?
R^2=X^2+Y^2
R^2=(24.0)^2+(76.0)^2
R^2=6352
R =79.7
To find the shortest length of the highway between the two towns, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled 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, we can consider the distance between the two towns as the hypotenuse of a right-angled triangle. The distance south is the vertical side, and the distance west is the horizontal side. We can label the distance south as y and the distance west as x.
Using the Pythagorean theorem, we have:
hypotenuse^2 = vertical side^2 + horizontal side^2
Substituting the given distances, we have:
hypotenuse^2 = 24.0 km^2 + 76.0 km^2
hypotenuse^2 = 576.0 km^2 + 5776.0 km^2
hypotenuse^2 = 6352.0 km^2
To find the length of the hypotenuse (the shortest length of the highway), we take the square root of both sides:
hypotenuse = √(6352.0 km^2)
hypotenuse = 79.7 km (rounded to one decimal place)
So, the shortest length of the highway that can be built between the two towns is approximately 79.7 km.
To determine the angle at which the highway is directed, we can use trigonometry. We can calculate the angle by taking the inverse tangent (or arctan) of the ratio of the vertical side (y) to the horizontal side (x) of the triangle formed.
angle = arctan(vertical side / horizontal side)
angle = arctan(24.0 km / 76.0 km)
angle ≈ 17.3 degrees (rounded to one decimal place)
Therefore, the highway would be directed at an angle of approximately 17.3 degrees.