Convert the following :
a. `5 L` of a solution into `m^(3)`
b. 4 days into seconds
c. `200 lb` into kilogram
d. `2.00 km` into miles
`(1 "mile" = 1760 yd, 1 yd = 3 ft 1 "in" = 2.54 cm)`
e. 0.800 carat intog grams and kilograms
e. `8.0 km` into inches `(1 m = 1.094 "yards" (yd), 1 yd = 36 "in")`
f. 40 Em (exa metre) (thickness of Milky way galaxy) into metre
g. `1.4 Gm` (gigmetre) diametre of sun) into metre
h. `41 Pm` (petametre) (distance of nearest star) into metre
i. `1 fg` (femotgram) (mass of human `DNA` molecule) into kilogram
j. `500 Mg` (megagram) (mass of a loaded jumbo jet) into kilogram

Explanation

a. `1 L = 100 cm^(3)` and `1 m = 100 cm` which gives
`(1 m)/(100 cm) = 1 = (100 cm)/(1 m)`
To get `m^(3)` from the above unit factors, the first unit factor is taken and it is cubed.
`((1)/(100 cm))^(3) = (1 m^(23))/(10^(6) cm^(3)) = (1)^(3) = 1`
Now, `5 L = 5 xx 1000 cm^(2)`
The above is multiplied by the unit factor.
`5 xx 1000 cancel(cm^(3)) xx (1 m^(3))/(10^(6) cancel(cm^(3))) = (5 m^(3))/(10^(6))`
`= 5 xx 10^(-3) m^(3)`
b. 1 day = 24 hours (h), `1 h = 60` min, and 1 min `= 60 s`
or `(1 day)/(24 h) = 1 = (24 h)/(1 h)`
or `(1 h)/(60 "min") = 1 = (60 "min")/(1 h)`
So, 4 days = ........ = ....... seconds
The unit factors can be multiplied in series in one step only as follows :
`4 cancel(days) xx (24 cancel(h))/(1 cancel(day)) xx (60 cancel("min"))/(1 cancel(h)) xx (60 s)/(1 cancel("min"))`
`= 4 xx 24 xx 60 xx 60 x = 345600 s`
c. `1 kg = 2.205 lb`
`:. (2.205 lb)/(1 kg) = 1 = (1 kg)/(2.205 lb)`
Hence, `200 lb = 200 cancel(lb) xx (1 kg)/(2.205 cancel(lb)) = 90.7 kg`
d. `1 km = 1000 m, 1 m = 100 cm, 1 "in" = 2.54 cm`
`(1 ft = 12 "in"), 1 yd = 3 ft, 1 "mile" = 1760 yd`
`:. 2.00 km = 2.00 cancel(km) xx (1000 cancel(m))/(1 cancel(km)) xx (100 cancel(cm))/(1 cancel(m))`
`xx (1 cancel("in"))/(2.54 cancel(cm)) xx (1 cancel(ft))/(12 cancel("in")) xx (1 cancel(yd))/(3 cancel(ft)) xx (1 "mile")/(1760 cancel(yd))`
`= (2.00 xx 1000 xx 100)/(2.54 xx 12 xx 3 xx 1760) = 1.2427`
`= 1.24 "miles"`
But the result is to be reported upto three significant figures since the least precise term (2.00) has three significant figures.
e. `(1 "carat")/(3.168 "grains") = 1 = (3.168 "grains")/(1 "carat")`
and `(1 g)/(15.4 "grains") = 1 = (15.4 "grains")/(1 g)`
`0.800 "carat" = 0.800 cancel("carat") xx (3.168 cancel("grains"))/(1 "carat")`
`xx (1 g)/(15.4 cancel("grains"))`
f. The unit conversion factors are:
`(1000 m)/(1 km) = 1 = (1 km)/(1000 m)`
and `(1.094 yd)/(1 m) = 1 = (1)/(1.094 yd)`
and `(36 "in")/(3 yd) = 1 = (1 yd)/(36 "in")`
Thus, `8.0 km = 8.0 cancel(km) xx (1000 cancel(m))/(1 cancel(km)) xx (1.094 cancel(yd))/(1 cancel(m))`
`xx (36 "in")/(1 cancel(yd))`
`= 315072 = 3.1 xx 10^(5) "in"`
The result is to be reported upto 2 significant figures since the least precise term (8.0) has 2 significant figure.
g. `1 Em` (extametre) `= 10^(8) m`
`:. 40 Em = 40 cancel(Em) xx (10^(18) m)/(1 cancel(Em)) = 4 xx 10^(19) m.`
h. `1 Gn` (gigametre) `= 10^(9) m`
`:. 1.4 Gm = 1.4 xx cancel(Gm) xx (10^(9) m)/(1 cancel(Gm)) = 1.4 xx 10^(9) m`.
i. 1 Pm (petametre) `= 10^(15) m`
`:. 41 Pm = 41 xx cancel(Pm) xx (10^(15) m)/(1 cancel(Pm)) = 41 xx 10^(15) m`.
j. `1 fg` (femtogram) `= 10^(-15) g` and `1 kg = 1000 g`
`:. 1 fg` (femtogram) `= 1 cancel(fg) xx (10^(-15) cancel(g))/(1 fg) xx (1 kg)/(1000 cancel(g))`
`= 10^(-18) kg`
k. `1 Mg` (Megagram) `= 10^(6) g` and `1 kg = 1000 g`
`:. 500 Mg = 5 cancel(00) cancel(Mg) xx (10^(6) cancel(g))/(1 cancel(Mg)) xx (1 kg)/(10 cancel(00) g)`
`5 xx 10^(5) kg`