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21) The focal length of a concave mirror is 30 cm. If a point object is placed at a distance of 50 cm from the concave mirror, find the distance of the image from the mirror.

(A) ______

Use the mirror equation: [tex]\(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\)[/tex]

Where:
- [tex]\(f\)[/tex] is the focal length,
- [tex]\(d_o\)[/tex] is the object distance,
- [tex]\(d_i\)[/tex] is the image distance.

Given:
[tex]\[ f = 30 \, \text{cm} \][/tex]
[tex]\[ d_o = 50 \, \text{cm} \][/tex]


Sagot :

To solve this problem, we need to use the mirror formula for a concave mirror. The mirror formula states:

[tex]\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \][/tex]

where:
- [tex]\( f \)[/tex] is the focal length of the mirror,
- [tex]\( v \)[/tex] is the image distance from the mirror,
- [tex]\( u \)[/tex] is the object distance from the mirror.

Given:
- The focal length [tex]\( f \)[/tex] is 30 cm,
- The object distance [tex]\( u \)[/tex] is 50 cm.

It's important to note that for concave mirrors, the object distance [tex]\( u \)[/tex] is taken as negative if the object is placed in front of the mirror, so [tex]\( u = -50 \)[/tex] cm.

Let's plug these values into the mirror formula and solve for [tex]\( v \)[/tex].

1. Write down the mirror formula:
[tex]\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \][/tex]

2. Substitute the given values [tex]\( f = 30 \)[/tex] cm and [tex]\( u = -50 \)[/tex] cm into the formula:
[tex]\[ \frac{1}{30} = \frac{1}{v} + \frac{1}{-50} \][/tex]

3. Rewrite the equation to make it easier to solve for [tex]\( \frac{1}{v} \)[/tex]:
[tex]\[ \frac{1}{30} = \frac{1}{v} - \frac{1}{50} \][/tex]

4. Combine the fractions on the right side:
[tex]\[ \frac{1}{30} + \frac{1}{50} = \frac{1}{v} \][/tex]

5. Find the common denominator and combine the fractions on the left side:
[tex]\[ \frac{1}{30} + \frac{1}{50} = \frac{50 + 30}{1500} = \frac{80}{1500} = \frac{8}{150} = \frac{4}{75} \][/tex]

6. Therefore:
[tex]\[ \frac{1}{v} = \frac{4}{75} \][/tex]

7. To find [tex]\( v \)[/tex], take the reciprocal of both sides:
[tex]\[ v = \frac{75}{4} \][/tex]

8. Simplify the fraction:
[tex]\[ v = 18.75 \text{ cm} \][/tex]

So, the distance of the image from the mirror is 18.75 cm.
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