When it hatches from its egg, the shell of a certain crab is 1cm across. When fully grown the shell is approximately 10cm across. Each new shell is one-third bigger than the previous one.
Answers
Hatching dimension of the shell,a0 = 1cm
The first shall dimension , a1=1+1*(1/3) = 4/3 cm
The 2nd shell dimension a2=a1*+a1*(1/3) =(4/3)^2 cm
The 3rd shell dimension ,a3= a2+a2(1/3) = a2(4/3) =(4/3)^3 cm
Therefore, the nth shell dimension , an =(4/3)^n cm
If nth shell is full life size , then (4/3)^n = 10. Taking logarithms , we can solve for n.
n log(4/3) = log10
n= log10/log(4/3) = 8.00392278 .
Therefore, an , is the 8th shell whose dimension is nearly 10cm.
By actual calculation also we can see the dimensions of the shells at varoius stages.
a1=1.3333 cm
a2=1.7777 cm
a3=2.3704 cm
a4=3.1605 cm
a5=4.2140 cm
a6=5.6187 cm
a7=7.4915 cm
a8=9.9887 cmis the 8th shell.
a9=13.3183 >10 . So not possible
So, a8=9.887cm is the dimension.
a9==13.3183cm is not possible as it is more than 10cm.