A person whose eyes are 2.87 m above the floor stands 2.11 m in front of a vertical plane
mirror whose bottom edge is 41 cm above the floor. What is the horizontal distance x to the
base of the wall supporting the mirror of the nearest point on the floor that can be seen
reflected in the mirror?
Draw a good sketch and make the incident and reflected rays at the same angle to the perpendicular to the mirror at the bottom.
To find the horizontal distance x to the base of the wall supporting the mirror, we can use the concept of similar triangles. Here's how you can solve it step by step:
Step 1: Draw a diagram to visualize the situation. Draw a vertical line representing the wall, a horizontal line representing the floor, and another line representing the mirror. Label the height of the person's eyes above the floor as 2.87 m (h1), the distance between the person and the mirror as 2.11 m (d), and the height of the mirror above the floor as 41 cm (h2).
Step 2: Set up the proportion using similar triangles. The triangle formed by the person's eyes, the mirror, and the floor is similar to the triangle formed by the mirror, the wall, and the base point of the wall.
Step 3: Write the proportion. We can write:
(d / x) = (h1 / h2)
Step 4: Convert all the measurements to the same unit. In this case, we need to convert the height of the mirror from centimeters to meters. Since 1 meter is equal to 100 centimeters, h2 is equal to 41 cm / 100 = 0.41 m.
Step 5: Plug in the values into the proportion. We have:
(2.11 m / x) = (2.87 m / 0.41 m)
Step 6: Cross-multiply and solve for x. Multiply both sides of the equation by x to get:
2.11 m = (2.87 m / 0.41 m) * x
Simplify the right side of the equation:
2.11 m = 7 m * x
Divide both sides of the equation by 7 m:
(2.11 m) / (7 m) = x
Step 7: Perform the calculation. Divide 2.11 m by 7 m:
x = 0.3014 m
So, the horizontal distance x to the base of the wall supporting the mirror is approximately 0.3014 m.