If a cuboid of dimensions 10 cm × 15 cm × 5 cm is cut to form cubes of sides 5 cm, then what is the difference between the sum of the surface areas of these cubes and the surface area of the original cuboid?

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Solution

$G i v e n, d i m e n s i o n s o f c u b o i d i s 10 c m \times 15 c m \times 5 c m S o, S u r f a c e a r e a o f c u b o i d = 2 (10 \times 15 + 15 \times 5 + 5 \times 10) = 2 (150 + 75 + 50) = 2 (275) = 550 c m^{2} N o w, c u b o i d i s c u t t o f o r m c u b e s o f s i d e 5 c m . T h u s, T o t a l n u m b e r o f c u b e s = \frac{V o l u m e o f c u b o i d}{V o l u m e o f e a c h c u b e} = \frac{10 \times 15 \times 5}{5 \times 5 \times 5} = 6 S u r f a c e a r e a o f e a c h c u b e = 6 a^{2} = 6 \times {(5)}^{2} = 150 c m^{2} S o, S u m o f a l l s u r f a c e a r e a o f c u b e s = 6 \times 150 = 900 c m^{2} N o w, D i f f e r e n c e b e t w e e n t h e s u m o f t h e s u r f a c e a r e a o f c u b e s a n d s u r f a c e a r e a o f c u b o i d = 900 c m^{2} - 550 c m^{2} = 350 c m^{2} H e n c e, D i f f e r e n c e b e t w e e n t h e s u m o f t h e s u r f a c e a r e a o f c u b e s a n d s u r f a c e a r e a o f c u b o i d i s 350 c m^{2} .$ $G i v e n, d i m e n s i o n s o f c u b o i d i s 10 c m \times 15 c m \times 5 c m S o, S u r f a c e a r e a o f c u b o i d = 2 (10 \times 15 + 15 \times 5 + 5 \times 10) = 2 (150 + 75 + 50) = 2 (275) = 550 c m^{2} N o w, c u b o i d i s c u t t o f o r m c u b e s o f s i d e 5 c m . T h u s, T o t a l n u m b e r o f c u b e s = \frac{V o l u m e o f c u b o i d}{V o l u m e o f e a c h c u b e} = \frac{10 \times 15 \times 5}{5 \times 5 \times 5} = 6 S u r f a c e a r e a o f e a c h c u b e = 6 a^{2} = 6 \times {(5)}^{2} = 150 c m^{2} S o, S u m o f a l l s u r f a c e a r e a o f c u b e s = 6 \times 150 = 900 c m^{2} N o w, D i f f e r e n c e b e t w e e n t h e s u m o f t h e s u r f a c e a r e a o f c u b e s a n d s u r f a c e a r e a o f c u b o i d = 900 c m^{2} - 550 c m^{2} = 350 c m^{2} H e n c e, D i f f e r e n c e b e t w e e n t h e s u m o f t h e s u r f a c e a r e a o f c u b e s a n d s u r f a c e a r e a o f c u b o i d i s 350 c m^{2} .$

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