a highway is to be built between two towns, one of which lies 35.0 km south and 72.0 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 with respect due to west? can some one solve this problem step by step?
I see the angle S of W at an angle arctan(35/72).
The length? 35/SinAngleabove
To solve this problem step by step, we can use the Pythagorean theorem and trigonometry:
Step 1: Draw a diagram of the two towns. Label one town as A and the other as B.
Step 2: Determine the distances between the towns. Given that one town is 35.0 km south and 72.0 km west of the other, we can label the horizontal distance as x (west) and the vertical distance as y (south).
Step 3: Use the Pythagorean theorem to find the shortest length of the highway. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). In this case, the length of the highway (c) is the hypotenuse.
So, using the Pythagorean theorem, we have:
c^2 = x^2 + y^2
Step 4: Calculate the values of x and y. Since the town B lies 72.0 km west of town A, we have x = 72.0 km. Similarly, since the town B lies 35.0 km south of town A, we have y = 35.0 km.
Step 5: Substitute the values of x and y into the equation from step 3 to find c^2.
c^2 = (72.0 km)^2 + (35.0 km)^2
Step 6: Calculate c by taking the square root of both sides of the equation.
c = √[(72.0 km)^2 + (35.0 km)^2]
Step 7: Use trigonometry to find the angle θ that the highway makes with the west direction. We can use the tangent function to find the angle.
The tangent of an angle θ is defined as the ratio of the opposite side (y in this case) to the adjacent side (x in this case).
tan(θ) = y / x
Step 8: Substitute the values of x and y.
tan(θ) = 35.0 km / 72.0 km
Step 9: Find the angle θ by taking the inverse tangent (arctan) of both sides of the equation.
θ = arctan(35.0 km / 72.0 km)
Step 10: Use a calculator or table to find the value of θ. The angle θ is approximately 26.6 degrees.
So, the shortest length of the highway is approximately √[(72.0 km)^2 + (35.0 km)^2] km, and the angle of the highway with respect to west is approximately 26.6 degrees.
To find the shortest length of highway between the two towns and the angle with respect to west, we can use the concept of vector addition.
Step 1: Draw a diagram
Draw a diagram representing the two towns and the distances given. Let's label the first town as A and the second town as B.
A -------------------------------- B
| |
| |
35 km 72 km
Step 2: Calculate the displacement vectors
In order to find the shortest length of the highway, we need to find the displacement vector between the two towns.
First, let's calculate the displacement in the north-south direction (y-direction) from A to B.
Displacement in y-direction = 35.0 km
Next, let's calculate the displacement in the east-west direction (x-direction) from A to B.
Displacement in x-direction = -72.0 km (negative sign indicating westward direction)
Step 3: Use vector addition to find the resultant displacement
We can find the resultant displacement (R) by using the Pythagorean theorem and trigonometry.
R = √((Displacement in x-direction)^2 + (Displacement in y-direction)^2)
= √((-72.0 km)^2 + (35.0 km)^2)
Calculating it:
R = √(5184 km^2 + 1225 km^2)
R = √(6409 km^2)
R ≈ 80.065 km
Step 4: Find the angle
To find the angle with respect to west, we can use trigonometry.
Angle (θ) = tan^(-1)((Displacement in y-direction) / (Displacement in x-direction))
= tan^(-1)((35.0 km) / (-72.0 km))
Calculating it:
θ = tan^(-1)(-0.4861)
θ ≈ -25.527 degrees
Therefore, the shortest length of the highway is approximately 80.065 km, and the angle with respect to due west is approximately -25.527 degrees. The negative sign indicates a direction west of due north.