two straight roads intersect to form an angle of 75 degree. find the shortest distance from one road to a gas station on the other road 1000 m from the junction.
the shortest distance from a point to a line is along a direction perpendicular to that line. So, you want a right triangle, and the desired distance x can be found by
x/1000 = sin 75°
Well, if you want the shortest distance to the gas station, you should probably take a helicopter! Just kidding! Let's solve this problem together.
Since the roads form an angle of 75 degrees, we can draw a right triangle where one of the legs represents the distance to the gas station, and the other leg is formed by an imaginary line connecting the gas station to the junction.
To find the shortest distance, we need to find the length of the leg connecting the junction to the gas station. We can do this using trigonometry.
Since the angle between the two roads is known (75 degrees) and we have the adjacent side to the angle (1000 m), we can use the cosine function.
The formula for calculating the adjacent side using the cosine function is:
adjacent side = hypotenuse * cosine(angle)
In this case, the adjacent side is the distance from the junction to the gas station, and the hypotenuse is the distance from the junction to the gas station plus the distance from the gas station to the junction.
So, let's calculate it:
adjacent side = 1000 * cosine(75 degrees)
Using a calculator, the value of the adjacent side is approximately 298.24 meters.
Therefore, the shortest distance from one road to the gas station on the other road, 1000 meters from the junction, is approximately 298.24 meters.
To find the shortest distance from one road to a gas station on the other road, we can use trigonometry.
First, let's label the two roads as Road A and Road B, and the intersection as Point O. The location of the gas station is Point G, located 1000 meters along Road B from the junction.
Now, let's draw a perpendicular line from Point G to Road A, and label the point where the line intersects Road A as Point X.
We need to find the length of GX, which represents the shortest distance from Road A to the gas station.
Given that angle AOG (where O is the vertex of the angle) is 75 degrees, we can find angle OAG to be 180 - 75 = 105 degrees.
Using trigonometry, we can find the length of GX using the tangent function:
tan(OAG) = GX / 1000
tan(105) = GX / 1000
Using a calculator, we can determine that tan(105) is approximately 7.10.
Therefore, GX = 7.10 * 1000 = 7100 meters.
So, the shortest distance from one road to the gas station is 7100 meters.
To find the shortest distance from one road to the gas station, we can use basic trigonometry.
Let's consider the diagram of the intersection of two straight roads forming an angle of 75 degrees:
```
. Gas Station (1000m)
/
/
/
A *-----------* B
| /
| /
| /
| /
| /
| /
| /
| /
| /
|/
Junction
```
In the diagram, we have two roads intersecting at point "Junction." The gas station is located 1000 meters along one of the roads, denoted as "A." We need to find the shortest distance from the other road, denoted as "B," to the gas station.
To solve this problem, we can create a right triangle by drawing a perpendicular line from the gas station to the road "B." The perpendicular bisects the angle 75 degrees, giving us two angles of 37.5 degrees each.
```
. Gas Station (1000m)
/
/
*---------.--------*
---------
37.5°
```
Now we can calculate the length of the perpendicular line using trigonometry. Since the angle is 37.5 degrees, we can use the trigonometric function tangent (tan) to find the length of the line.
The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the length we want to calculate, and the adjacent side is the length from the intersection to the gas station, which is 1000 meters.
So, we have:
tan(37.5°) = opposite side / adjacent side.
Let's solve for the opposite side (shortest distance to the gas station):
opposite side = tan(37.5°) * adjacent side
= tan(37.5°) * 1000 meters.
Using a scientific calculator or an online calculator, we can find the value of tan(37.5°) to be approximately 0.75355.
So, the shortest distance from road B to the gas station is:
opposite side = 0.75355 * 1000 meters
≈ 753.55 meters.
Therefore, the shortest distance from road B to the gas station is approximately 753.55 meters.