A contractor examines the city plans for a light rail transit system. The plans include a tunnel to be constructed between University Station and Mall Station. From point A to University Station is 975 m, and from point A to Mall Station is 2310 m. The angle between the two lines is 105°, as shown in the diagram. Find the distance x between University Station and Mall Station.
To find the distance between University Station and Mall Station (x), we can use the Law of Cosines.
The Law of Cosines states that in any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:
c^2 = a^2 + b^2 - 2abcosC.
In our case, side a is 975 m, side b is 2310 m, and angle C is 105°. Let's substitute these values into the equation:
x^2 = 975^2 + 2310^2 - 2(975)(2310)cos105°.
Now, let's calculate this expression:
x^2 = 950625 + 5336100 - 2(975)(2310)(-0.258819).
Simplifying further:
x^2 = 6286725 + 4755635.369.
x^2 = 11072360.369.
Taking the square root of both sides:
x ≈ √(11072360.369).
x ≈ 3327.624 m.
Therefore, the distance x between University Station and Mall Station is approximately 3327.624 meters.
To find the distance between University Station and Mall Station, we can use the law of cosines. The law of cosines states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those two sides multiplied by the cosine of their included angle.
Let's call the length of x, the distance between University Station and Mall Station. According to the given information, the length of the line from point A to University Station is 975 m, and the length of the line from point A to Mall Station is 2310 m. The angle between these two lines is 105°.
Using the law of cosines, we can write the equation:
x^2 = 975^2 + 2310^2 - 2 * 975 * 2310 * cos(105°)
Simplifying the equation:
x^2 = 950625 + 5336100 - 4482750 * cos(105°)
Now we can calculate the value of cos(105°):
cos(105°) ≈ -0.258819
Substituting the values back into the equation:
x^2 = 950625 + 5336100 - 4482750 * (-0.258819)
x^2 = 950625 + 5336100 + 1161353.12705
x^2 ≈ 7445078.12705
Taking the square root of both sides to solve for x:
x ≈ √(7445078.12705)
x ≈ 2728.5 meters
Therefore, the distance between University Station and Mall Station is approximately 2728.5 meters.