Two streets meet at an 60° angle. At the corner, a park is being built in the shape of a triangle. Find the area of the park if, along one road, the park measures 180 feet, and along the other road, the park measures 200 feet.
(To the extent possible use standard angles to simplify your responses. When necessary express your answers using the functions sin, cos, sin−1, cos−1. If there is no solution enter IMPOSSIBLE.)
If we are given 2 sides and the contained angle of a triangle, we have
a nice little formula to find the area.
Area = (1/2)a*b*sinθ, were a and be are the two sides and θ is the
contained angle.
You have exactly that condition
area =(1/2)(180)(200)sin60°
= 18000(√3/2)
= 9000√3 square ft
Well, well, well, looks like we have a triangular park on our hands! Let's calculate its area, shall we?
Since we have two sides of the triangle, we need to find the length of the third side. Lucky for us, we know the angle between the two known sides, which is 60°. So, we can use the Law of Cosines to find the missing side.
Let's call the third side of the triangle, the side opposite the 60° angle, "c".
According to the Law of Cosines, we have:
c² = a² + b² - 2ab * cos(C)
where:
a = 180 feet (one side of the park)
b = 200 feet (the other side of the park)
C = 60° (the angle between the two known sides)
Now, let's plug in the values and crunch those numbers:
c² = 180² + 200² - 2 * 180 * 200 * cos(60°)
Calculating the expression inside the brackets:
c² = 180² + 200² - 2 * 180 * 200 * 0.5
c² = 180² + 200² - 180 * 200
c² = 32,400 + 40,000 - 36,000
c² = 36,400
Now we take the square root of both sides to find "c":
c = √36,400
c ≈ 190.81 feet
So, the length of the third side is approximately 190.81 feet.
Now, we can calculate the area of the triangle using the formula:
Area = (1/2) * base * height
In this case, the base of the triangle is 180 feet, and the height is the length of the third side, which is 190.81 feet. Let's calculate that:
Area = (1/2) * 180 * 190.81
Area ≈ 17,147.8 square feet
So, the area of the park is approximately 17,147.8 square feet. Enjoy your triangular park!
To find the area of the park, we can use the formula for the area of a triangle: A = 0.5 * base * height.
First, let's find the height of the triangle formed by the two roads. Since the angle between the two roads is 60°, we can use the sine function to find the height.
sin(60°) = opposite / hypotenuse
The height is the opposite side, and the hypotenuse is given by the length along one road (180 feet). Rearranging the equation, we have:
height = sin(60°) * 180 feet
Calculating the value:
height = sin(60°) * 180
height ≈ 0.866 * 180
height ≈ 155.88 feet
Now, we can find the area of the park using the height and the length along the other road (200 feet):
A = 0.5 * base * height
A = 0.5 * 200 feet * 155.88 feet
A ≈ 0.5 * 200 * 155.88
A ≈ 0.5 * 31176
A ≈ 15588 square feet
Therefore, the area of the park is approximately 15588 square feet.
To find the area of the park, we can use the formula for the area of a triangle: A = (1/2) * base * height.
In this case, the base is the length of one road (180 feet) and the height is the length of the other road (200 feet). However, we need to find the height of the park relative to the base.
Since the two streets meet at a 60° angle, we can split the triangle into two right triangles. Let's call the angle formed by the base and the height as angle A.
Now, we know that the measure of angle A is 90° - 60° = 30°. We can use trigonometric functions to find the height.
Using the sine function, sin A = opposite / hypotenuse.
sin 30° = height / 200
Solving for height: height = 200 * sin 30°
Now, we can substitute the values into the area formula:
A = (1/2) * 180 * (200 * sin 30°)
Using standard angles, we know that sin 30° = 1/2.
A = (1/2) * 180 * (200 * 1/2)
A = (1/2) * 180 * 100
A = 0.5 * 180 * 100
A = 9,000 square feet
Therefore, the area of the park is 9,000 square feet.