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.

Question

GPT 3.5 · Answer

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.