Question

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. How many shells does a fully grown crab have during its life

Answer 1

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.