A highway is to be built between two towns, one of which lies 98.0 km south and 24.0 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 trigonometric functions.
(a) Shortest Length of the Highway:
We know that the towns are located 98.0 km south and 24.0 km west of each other. This forms a right-angled triangle.
Using the Pythagorean theorem, the shortest length of the highway (the hypotenuse) is given by:
(highway length)^2 = (distance south)^2 + (distance west)^2
Therefore, the shortest length of the highway is:
(highway length) = sqrt((distance south)^2 + (distance west)^2)
= sqrt((98.0 km)^2 + (24.0 km)^2)
Calculating this expression, we find that the shortest length of the highway is approximately 102.55 km.
(b) Angle with Respect to Due West:
To find the angle of the highway with respect to due west, we can use the tangent function.
The tangent of an angle in a right-angled triangle is defined as the ratio of the opposite side to the adjacent side. In this case, the tangent of the required angle is equal to the ratio of the distance south to the distance west.
Therefore, we can calculate the required angle using the tangent function:
(angle) = arctan((distance south) / (distance west))
= arctan(98.0 km / 24.0 km)
Calculating this expression, we find that the angle is approximately 74.54 degrees. Since the angle is measured as a positive angle with respect to due west, the answer is 74.54 degrees.
To find the shortest length of the highway, we can use the Pythagorean theorem and trigonometry.
(a) Using the Pythagorean theorem, we can find the shortest length of the highway (H) as follows:
H² = (98.0 km)² + (24.0 km)²
H² = 9604 km² + 576 km²
H² = 10180 km²
Taking the square root of both sides gives us:
H = √10180 km
H ≈ 100.89 km
Therefore, the shortest length of the highway is approximately 100.89 km.
(b) To find the angle, we can use trigonometry. The angle (θ) can be found using the tangent function:
tan(θ) = opposite / adjacent
tan(θ) = (98.0 km) / (24.0 km)
θ = tan⁻¹(98.0/24.0)
Using a calculator, we find:
θ ≈ 75.96°
Since the angle is with respect to due west, the positive angle would be 75.96°.
Therefore, the highway would be directed at an angle of approximately 75.96° (positive) with respect to due west.