Given;

#"Rate" rArr r = 4.5%#

#"Principal" rArr p = p#

#"Compund" rArr n = 1/4# *(quarterly #=3/12#)*

#"Compund Interest" rArr A = 2p# *(if the amount of d money doubles)*

#"No of years" rArr t = ?yrs#

Hence we solve;

#A = p(1+r/n)^(nt)#

Inputing the values above;

#2p = p(1 + 4.5 xx 4)^(1/4 xx t)#

#2p = p(1 + 18)^(t/4)#

#2p = p(19)^(t/4)#

Divide both sides by #p#

#(2p)/p = (p(19)^(t/4))/p#

#(2cancelp)/cancelp = (cancelp(19)^(t/4))/cancelp#

#2 = 19^(t/4)#

Multiply both sides by the power of #4/t#..

#2^(4/t) = 19^(t/4 xx 4/t)#

#2^(4/t) =19#

Take #log# of both sides..

#log 2^(4/t) = log 19#

#4/tlog2 = log19#

Divide both sides by #log2#

#(4/tlog2)/log2 = log19/log2#

#(4/tcancellog2)/cancellog2 = log19/log2#

#4/t = log19/log2#

#4/t = 4.248#

Cross multiplying..

#4/t = 4.248/1#

#4 xx 1 = 4.248 xx t#

#4 = 4.248t#

Divide both sides by #4.248#

#4/4.248 = (4.248t)/4.248#

#4/4.248 = (cancel4.248t)/cancel4.248#

#4/4.248 = t#

#t = 0.94yrs -> "to the nearest tenth as required"#

Hope this helps!