A convex security mirror has a focal length of -0.25m. A person with a height of 1.5m is 4m from the mirror. calculate the image distance and image height.

AND

An object 0.4m tall is placed 2.5m in front of a convex mirror that has a focal length of -90cm. Calculate image distance and image height.

To calculate the image distance and image height in each scenario, we can use the mirror equation and the magnification equation.

For the first scenario, where the focal length (f) is -0.25m and the object distance (d_o) is 4m:

Step 1: Calculate the image distance (d_i) using the mirror equation:

1/f = 1/d_i - 1/d_o

Substituting the given values:

1/-0.25 = 1/d_i - 1/4

Simplifying the equation:

-4 = d_i - 1/4

-4 + 1/4 = d_i

d_i = -4 + 0.25

d_i = -3.75m

Therefore, the image distance is -3.75m.

Step 2: Calculate the image height (h_i) using the magnification equation:

h_i / h_o = -d_i / d_o

Substituting the given values:

h_i / 1.5 = -(-3.75) / 4

h_i / 1.5 = 3.75 / 4

h_i = (3.75 / 4) * 1.5

h_i = 1.40625m

Therefore, the image height is approximately 1.40625m.

For the second scenario, where the focal length (f) is -90cm and the object distance (d_o) is 2.5m:

Step 1: Calculate the image distance (d_i) using the mirror equation:

1/f = 1/d_i - 1/d_o

Substituting the given values:

1/-0.9 = 1/d_i - 1/2.5

Simplifying the equation:

-1.1111 = d_i - 0.4

-1.1111 + 0.4 = d_i

d_i ≈ -0.7111m

Therefore, the image distance is approximately -0.7111m.

Step 2: Calculate the image height (h_i) using the magnification equation:

h_i / h_o = -d_i / d_o

Substituting the given values:

h_i / 0.4 = -(-0.7111) / 2.5

h_i / 0.4 ≈ 0.7111 / 2.5

h_i ≈ (0.7111 / 2.5) * 0.4

h_i ≈ 0.1138m

Therefore, the image height is approximately 0.1138m.

To solve these problems, we can use the mirror equation for convex mirrors:

1/f = 1/p + 1/q,
where:
f = focal length of the mirror,
p = object distance from the mirror (positive for real objects),
q = image distance from the mirror (positive for virtual images).

Let's solve the first problem step by step:

Given data:
focal length (f) = -0.25m,
object distance (p) = 4m.

1. Calculate the image distance (q):

Substituting the given values in the mirror equation:
1/(-0.25) = 1/4 + 1/q.

Simplifying the equation:
-4 = 1/4 + 1/q.

Combining the terms:
-4 = (1 + 4)/4q.

Solving for q:
-4 = 5/4q.

Cross-multiplying:
-4q = 5/4.

Dividing by -4:
q = (5/4) / -4 = -5/16.

Therefore, the image distance (q) is -5/16m.

2. Calculate the image height (h'):

Given data:
height of the person (h) = 1.5m.

The magnification equation for convex mirrors is:
h'/h = -q/p.

Substituting the given values in the magnification equation:
h'/1.5 = -(-5/16)/4.

Simplifying the equation:
h'/1.5 = 5/16 * 1/4.

Multiplying both sides by 1.5:
h' = (5/16) * (1/4) * 1.5.

Simplifying the equation:
h' = 0.234375.

Therefore, the image height (h') is approximately 0.234375m.

Now, let's solve the second problem using the same steps:

Given data:
focal length (f) = -90cm = -0.9m,
object height (h) = 0.4m.

1. Calculate the image distance (q):

Substituting the given values in the mirror equation:
1/(-0.9) = 1/2.5 + 1/q.

Combining the terms:
-10/9 = (2.5 + q)/2.5q.

Cross-multiplying:
-10q = 9(2.5 + q).

Expanding the equation:
-10q = 22.5 + 9q.

Simplifying the equation:
-19q = 22.5.

Dividing by -19:
q = 22.5 / -19 = -1.18421m.

Therefore, the image distance (q) is approximately -1.18421m.

2. Calculate the image height (h'):

The magnification equation for convex mirrors is the same as before:
h'/h = -q/p.

Substituting the given values in the magnification equation:
h'/0.4 = -(-1.18421)/2.5.

Simplifying the equation:
h'/0.4 = 1.18421/2.5.

h' = (1.18421/2.5) * 0.4.

h' ≈ 0.18947m.

Therefore, the image height (h') is approximately 0.18947m.

In each case use the mirror formula

1/do + 1/di = 1/f

do = object distance
di = image doistance

Make sure you use negative f for the convex mirrors. The image height is di/do times the object heeight.

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