A highway is to be built between two towns, one of which lies 49.0 km south and 95.0 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?
To find the shortest length of the highway between the two towns, we can use the Pythagorean theorem and trigonometry.
(a) The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the distance between the two towns forms a right triangle with sides of 49.0 km (south) and 95.0 km (west). Let's call the hypotenuse, or the shortest length of the highway, "d".
Using the Pythagorean theorem, we can calculate "d":
d² = (49.0 km)² + (95.0 km)²
Plugging in the values, we have:
d² = 2401 km² + 9025 km²
d² = 11,426 km²
Taking the square root of both sides, we find:
d ≈ √11,426 km
d ≈ 107.006 km
Therefore, the shortest length of the highway between the two towns is approximately 107.006 km.
(b) To find the angle, we can use trigonometry. We know the lengths of the sides of the right triangle: 49.0 km (south) and 95.0 km (west).
Let's call the angle we are looking for "θ". The tangent of an angle in a right triangle is equal to the ratio of the lengths of the opposite side to the adjacent side.
Using the tangent function, we can calculate the angle "θ" as follows:
tan(θ) = (49.0 km) / (95.0 km)
To find the angle "θ", we can take the inverse tangent (also known as arctan) of both sides:
θ = arctan(49.0 km / 95.0 km)
Using a scientific calculator, we find:
θ ≈ 26.08 degrees
Therefore, the highway would be directed at an angle of approximately 26.08 degrees as a positive angle with respect to due west.
To find the shortest length of highway and the angle of direction, we can use the Pythagorean theorem and trigonometric functions.
(a) The shortest length of highway can be found using the Pythagorean theorem, which states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). In this case, the two sides represent the distances south (a) and west (b) between the towns. Thus, we have:
c^2 = a^2 + b^2
Let's calculate:
a = 49.0 km (distance south)
b = 95.0 km (distance west)
c^2 = (49.0 km)^2 + (95.0 km)^2
c^2 = 2401.0 km^2 + 9025.0 km^2
c^2 = 11426.0 km^2
c = sqrt(11426.0 km^2)
c ≈ 106.96 km
Therefore, the shortest length of the highway is approximately 106.96 km.
(b) To find the angle at which the highway would be directed, we can use trigonometric functions. We have the opposite side (a) as the distance south and the adjacent side (b) as the distance west. We can use the tangent function:
tan(theta) = a/b
Let's calculate:
theta = tan^(-1)(a/b)
theta = tan^(-1)(49.0 km/95.0 km)
theta ≈ 26.40 degrees
Therefore, the highway would be directed at an angle of approximately 26.40 degrees as a positive angle with respect to due west.