A 60-footpiece of wire is strung between the top of a tower and the ground, making a 30-60-90 triangle. How far from the center of the base of the tower is the wire attached to the ground? How high is the tower?
Make a point of memorizing the ratio of sides of the 30-60-90 degree triangle,
the sides are in the ratio of 1 : √3 : 2 in the relation of the angles respectively.
so all you have to do is set up simple ratios, assuming that the base angle in your
diagram is 60°
so base/1 = 60/2
base = 30
height/√3 = 60/2
height = 30√3
notice that 30 : 30√3 : 60 = 1 : √3 : 2
You could of course just take sin30° and cos30°
e.g. cos 30° = height/60
height = 60 cos30
= 60(√3/2) = 30√3 , just as above
To find the distance from the center of the base of the tower to where the wire is attached to the ground, we need to determine the length of the base and the height of the triangle.
In a 30-60-90 triangle, the sides are in the ratio 1:√3:2.
Since the wire is the hypotenuse of the triangle and its length is 60 feet, we can let the length of the shorter leg (opposite the 30-degree angle) be x feet.
Using the ratio, we know that the longer leg (opposite the 60-degree angle) will be √3x feet, and the hypotenuse (the wire) will be 2x feet.
We have the equation: 2x = 60
By solving for x, we find x = 30.
Therefore, the length of the base (distance from the center of the base of the tower to where the wire is attached to the ground) is 30 feet.
To find the height of the tower (opposite the 90-degree angle), we multiply the length of the shorter leg (x) by √3:
Height = √3 * x = √3 * 30 = 30√3 feet.
Therefore, the height of the tower is 30√3 feet.
To solve this problem, we need to understand the properties of a 30-60-90 triangle and apply them to find the answers.
In a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, and the side opposite the 60-degree angle is (√3 / 2) times the length of the hypotenuse.
Let's use this knowledge to find the distance from the center of the base of the tower to where the wire is attached to the ground.
Let x be the distance from the center of the base to the attachment point. Since the wire is 60 feet long, the hypotenuse of the triangle is 60 feet.
According to the properties of a 30-60-90 triangle, the side opposite the 30-degree angle is half the length of the hypotenuse, so it is 60/2 = 30 feet long.
Therefore, the attachment point is 30 feet from the center of the base of the tower.
Now let's find the height of the tower. In a 30-60-90 triangle, the side opposite the 60-degree angle is (√3 / 2) times the length of the hypotenuse.
So, the height of the tower is (√3 / 2) * 60 = 30√3 feet.
Hence, the distance from the center of the base of the tower to where the wire is attached to the ground is 30 feet, and the height of the tower is 30√3 feet.