A child’s kite is on top of a straight pine tree. A ladder is placed against the tree and touches the kite. The ladder forms a 45-degree angle with the flat ground. The base of the tree is 9 feet away from the base of the ladder. With only this information, how can you determine the height of the tree?
A diagram of this problem is a right isosles triangle. The right angle is where the tree touches the ground and the two 45 degree angles are where the kite hits the top of the tree and where the ladder hits the ground. The legs of an isosles triangle are equal. The tree is one of the legs and is 9 feet tall.
Can you go from there to get answer?
To determine the height of the tree, we can use trigonometry, specifically the sine function. Here's how:
Step 1: Draw a diagram to visualize the problem. Draw a triangle representing the ladder leaning against the tree. Label the ladder as "X," the height of the tree as "H," and the base of the tree as "9 feet." Also, mark the angle formed between the ladder and the ground as 45 degrees.
Step 2: Recall that the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, the side opposite the 45-degree angle is "H," and the hypotenuse is "X."
Step 3: Write down the trigonometric relationship. We have sin(45 degrees) = H / X.
Step 4: Remember that the sine of 45 degrees is equal to the square root of 2 divided by 2 (√2/2).
Step 5: Substitute the known values into the equation. We have √2/2 = H / X.
Step 6: Rearrange the equation to solve for H. Multiply both sides of the equation by X and then divide by √2/2. This simplifies to H = X * (√2/2).
Step 7: Substitute the given value for X into the equation, which is the base of the ladder, 9 feet: H = 9 * (√2/2).
Step 8: Simplify the equation: H = 9 * (√2/2) ≈ 6.36 feet.
Therefore, the height of the tree is approximately 6.36 feet.