Question

A slide projector gives magnification of 10. If it projects a slide of \[3\,{\text{cm}} \times {\text{2}}\,{\text{cm}}\] on a screen, the area of image on the screen is:
A. \[6000\,{\text{c}}{{\text{m}}^2}\]
B. \[600\,{\text{c}}{{\text{m}}^2}\]
C. \[3600\,{\text{c}}{{\text{m}}^2}\]
D. \[1200\,{\text{c}}{{\text{m}}^2}\]

Answer 1

Hint: Use the formula for magnification of an optical instrument in terms of area. This formula gives the relation between the magnification of the optical instrument, area of the object and area of the image. Determine the area of the slide using the formula for the area of a rectangular object and substitute it in the formula for magnification of the optical instrument.

Formulae used:
The magnification of an optical instrument is given by
\[\mu = \sqrt {\dfrac{{{A_i}}}{{{A_o}}}} \] …… (1)
Here, \[\mu \] is the magnification of the optical instrument, \[{A_i}\] is the area of the image and \[{A_o}\] is the area of the object.
The area of a rectangular slide is given by
\[A = l \times b\] …… (2)
Here, \[A\] is the area of the slide, \[l\] is the length of the slide and \[b\] is the breadth of the slide.

Complete step by step answer:
We have given that the magnification of the projector is 10.
\[\mu = 10\]
The size of the slide is given as \[3\,{\text{cm}} \times {\text{2}}\,{\text{cm}}\].
Hence, the length of the slide is \[3\,{\text{cm}}\] and the breadth of the slide is \[{\text{2}}\,{\text{cm}}\].
\[l = 3\,{\text{cm}}\]
\[b = {\text{2}}\,{\text{cm}}\]
We can determine the area of the slide (object) using equation (2).
Rewrite equation for the area \[{A_o}\] of the slide (object).
\[{A_o} = l \times b\]
Substitute \[3\,{\text{cm}}\] for \[l\] and \[{\text{2}}\,{\text{cm}}\] for \[b\] in the above equation.
\[{A_o} = \left( {3\,{\text{cm}}} \right) \times \left( {{\text{2}}\,{\text{cm}}} \right)\]
\[ \Rightarrow {A_o} = 6\,{\text{c}}{{\text{m}}^2}\]
Hence, the area of the slide (object) is \[6\,{\text{c}}{{\text{m}}^2}\].
We can determine the area of the image of the slide on the screen using equation (2).
Substitute \[10\] for \[\mu \] and \[6\,{\text{c}}{{\text{m}}^2}\] for \[{A_o}\] in the above equation.
\[10 = \sqrt {\dfrac{{{A_i}}}{{6\,{\text{c}}{{\text{m}}^2}}}} \]
Take square on both sides of the above equation.
\[100 = \dfrac{{{A_i}}}{{6\,{\text{c}}{{\text{m}}^2}}}\]
Rearrange the above equation for the area \[{A_i}\] of the image of the slide on the screen.
\[ \Rightarrow {A_i} = 100 \times 6\,{\text{c}}{{\text{m}}^2}\]
\[ \therefore {A_i} = 600\,{\text{c}}{{\text{m}}^2}\]

Therefore, the area of the image of the slide on the screen is \[600\,{cm^2}\].Hence, the correct option is B.


Note:The students should keep in mind that there is no need of conversion of units of dimensions of the slide in the SI system of units. One can also solve the same question in another way. We can determine the dimensions of the image of the slide formed on the screen by multiplying the dimensions of the slide by the magnification of the projector and then determine the area of the image of the slide.