Kyle is pulled back on a swing so that the rope forms an angle of 30° with the vertical. The distance from the top of the swing set directly to the ground is 12 feet. Find Kyle’s height off the ground, , when he is in the pulled-back position. Round the answer to the nearest hundredth.
no way to know. I assume the swing is not dragging on the ground when just hanging.
However, assuming the swing chain is x feet long (x < 12), then the swing is hanging 12-x feet off the ground when at rest.
So, for our problem, the swing at 30° is
12-x + x(1-cos30°) feet off the ground.
To find Kyle's height off the ground when he is in the pulled-back position, we can use trigonometry. Let's break the problem down into steps:
Step 1: Draw a diagram
Draw a diagram representing the swing set, with Kyle on the swing in the pulled-back position. Label the top of the swing set as "T," Kyle's position as "K," and the vertical line from the top to the ground as "G." Also, label the angle between the rope and the vertical line as "30°".
Step 2: Identify the right triangle
In the diagram, the triangle formed by T, K, and G is a right triangle, with the right angle at point G. This is because a rope hanging freely forms a straight line with the vertical.
Step 3: Identify the known and unknown values
The known value is the distance from the top of the swing set directly to the ground, which is 12 feet. The unknown value is Kyle's height off the ground, which we'll denote as "h."
Step 4: Choose the appropriate trigonometric function
Since we know the angle and the opposite side (the height), we can use the sine function to solve for the unknown value. The sine function relates the ratio of the length of the opposite side of a right triangle to the length of the hypotenuse.
Step 5: Setup and solve the equation
Using the sine function, we have:
sin(30°) = h / 12
We want to solve for h, so we'll isolate it:
h = 12 * sin(30°)
Step 6: Calculate the answer
Using a scientific calculator or reference material, find the sine of 30°, which is approximately 0.5. Plug this value into the equation:
h = 12 * 0.5 = 6
Therefore, Kyle's height off the ground when he is in the pulled-back position is 6 feet.