A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°

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Question: A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. How tall is the tree?

Options:

Correct Answer: 10 m

Solution:

Using tan(30°) = height/shadow, height = 10 * tan(30°) = 10 * (1/√3) = 10 m.

A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. How tall is the tree?

A tree casts a shadow of 10 meters when the angle of elevation of the sun is 30°. How tall is the tree?

Step 1: Understand that the tree, the shadow, and the sun create a right triangle.
Step 2: Identify the angle of elevation of the sun, which is 30°.
Step 3: Recognize that the shadow of the tree is 10 meters long.
Step 4: Use the tangent function, which relates the angle to the opposite side (height of the tree) and the adjacent side (length of the shadow).
Step 5: Write the formula: tan(30°) = height / shadow.
Step 6: Substitute the known values into the formula: tan(30°) = height / 10.
Step 7: Calculate tan(30°), which is equal to 1/√3.
Step 8: Rewrite the equation: 1/√3 = height / 10.
Step 9: Multiply both sides by 10 to solve for height: height = 10 * (1/√3).
Step 10: Calculate the height: height = 10 / √3, which is approximately 5.77 meters.
Step 11: Round the height to the nearest whole number if needed, which is about 6 meters.

Trigonometry – The problem involves using the tangent function to relate the height of the tree to the length of its shadow based on the angle of elevation of the sun.
Right Triangle Properties – The scenario can be visualized as a right triangle where the height of the tree is the opposite side, the shadow is the adjacent side, and the angle of elevation is given.