A tower is 50 meters high. From a point on the ground, the angle of elevation to
Practice Questions
Q1
A tower is 50 meters high. From a point on the ground, the angle of elevation to the top of the tower is 30 degrees. What is the distance from the point to the base of the tower?
25√3 m
50 m
25 m
50√3 m
Questions & Step-by-Step Solutions
A tower is 50 meters high. From a point on the ground, the angle of elevation to the top of the tower is 30 degrees. What is the distance from the point to the base of the tower?
Step 1: Understand the problem. We have a tower that is 50 meters high and we want to find the distance from a point on the ground to the base of the tower.
Step 2: Identify the angle of elevation. The angle of elevation to the top of the tower is given as 30 degrees.
Step 3: Recall the relationship in a right triangle. The tangent of an angle in a right triangle is equal to the opposite side (height of the tower) divided by the adjacent side (distance from the point to the base of the tower).
Step 4: Write the formula for tangent. We can express this as tan(30°) = height / distance.
Step 5: Substitute the known values into the formula. We know the height is 50 meters, so we write tan(30°) = 50 / distance.
Step 6: Rearrange the formula to find distance. This gives us distance = height / tan(30°).
Step 7: Calculate tan(30°). The value of tan(30°) is √3 / 3.
Step 8: Substitute tan(30°) into the distance formula. Now we have distance = 50 / (√3 / 3).
Step 9: Simplify the equation. This is the same as distance = 50 * (3 / √3).
Step 10: Further simplify. This gives us distance = 150 / √3.
Step 11: Rationalize the denominator. Multiply the numerator and denominator by √3 to get distance = 150√3 / 3.
Step 12: Simplify the final answer. This results in distance = 50√3 meters.
Trigonometry – The problem involves using the tangent function to relate the height of the tower and the distance from the point to the base.
Angle of Elevation – Understanding the angle of elevation is crucial for setting up the right triangle in the problem.
Right Triangle Properties – The relationship between the sides of a right triangle and the angles is fundamental to solving the problem.
