A highway is to be built between two towns, one of which lies 44.0 km south and 74.0 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?

What does km stand for (I'm only in 4th grade)?

50km

An motorbike traveling down the road at a constant speed emits sound waves from its siren.a boy stand on the side of the road with a detector which register sound waves at a frequency of 460 hz as the motorbike approaches him.after passing him and moving away at the same constant speed,sound waves of frequency 390 hz registered.calculate the speed at which the motorbike is moving?

To find the shortest length of the highway and the angle at which it should be directed, we can use the Pythagorean theorem and trigonometric functions.

Let's assume that the two towns are point A (44.0 km south) and point B (74.0 km west). To find the shortest length of the highway, we need to find the length of the hypotenuse of the right triangle formed by the two towns.

Using the Pythagorean theorem (a^2 + b^2 = c^2), where a and b are the two legs of the triangle, and c is the length of the hypotenuse, we can calculate as follows:

a = 44.0 km
b = 74.0 km
c^2 = a^2 + b^2

c^2 = (44.0 km)^2 + (74.0 km)^2
c^2 = 1936 km^2 + 5476 km^2
c^2 = 7412 km^2

Taking the square root of both sides, we get:

c = √(7412 km^2)
c ≈ 86.06 km

So, the shortest length of the highway that can be built between the two towns is approximately 86.06 km.

To determine the angle at which the highway should be directed, we can use trigonometric functions. In this case, we can use the tangent function (tan):

tan(θ) = a/b

θ = tan^(-1)(a/b)
θ = tan^(-1)(44.0 km / 74.0 km)
θ ≈ 30.96 degrees

So, the highway should be directed at an angle of approximately 30.96 degrees.