Question

ANSWER FAST!!!!!!!

A city wanted to enclose a triangular portion of the city park to make a dog park. The triangular portion of the park that they are enclosing has sides of 280 ft., 190 ft., and 330 ft. In order to construct the fences correctly, the city wants to determine the angle between the 190 ft. side and the 330 ft. side.

Answer 1

The angle between the 190 ft. side and the 330 ft. side is
`57.9^o`

Explanation:

The Law of Cosines

When we know the value of all sides of a triangle, we can compute all of its interior angles by using the Law of Cosines, which is a generalization of the Pythagoras's theorem. If a,b, and c are the known sides of a triangle and
`\alpha` is the angle formed by sides a and b (opposite to c), then

`\displaystyle c^2=a^2+b^2-2ab\ cos\alpha`

We'll use the values a=190, b=330, c=280 because we want to compute the angle opposite to c

`\displaystyle cos\alpha =(a^2+b^2-c^2)/(2ab)`

`\displaystyle cos\alpha =(190^2+330^2-280^2)/(2(190)(330))`

`\displaystyle cos\alpha =(36100+108900-78400)/(125400)`

`\displaystyle cos\alpha =(66600)/(125400)=0,5311`

`\displaystyle \alpha= arccos\ 0.5311`

`\boxed{\displaystyle \alpha= 57.9^o}`