Given two points A and B along a roadside at a distance of 500 yards, with a designated third point C along a perpendicular to the road at B. A direct path from C to A forms an angle of 38° with the road. Now, let's find the distance from the road to point C and the distance from A to C.
The direct path from A to C, and the roadside, form a right angle triangle where the angle at A is 38° and the hypotenuse (AC) and sides adjacent (AB - which is same as along the road) and opposite to the angle (BC - which is same as the perpendicular from B to C or the distance from the road to point C) are to be found.
Using the trigonometric concept of sine, which is defined as the ratio of the length of the side of a triangle opposite the angle to the length of the hypotenuse, the distance from the road to point C (BC) can be found by multiplying the length along the road (AB) with the sine of the angle θ (38°). This calculation gives approximately 307.831 yards.
On the other hand, the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. So, the distance from A to C (AC) can be calculated by multiplying the length along the road (AB) with the cosine of the angle θ (38°). This calculation gives approximately 394.005 yards.
After rounding to three decimal places, the solutions to the problem are:
- The distance from the road (B) to point C is 307.831 yd,
- The distance from A to C is 394.005 yd.
So, none of the given choices (A,B,C,D) match the calculated answer. The answer must have been mistyped or miscalculated in the choices.