Application of First Order ODE

Question

A certain type of glass is such that a slab 1" thick absorbs one-quarter of the light which starts to pass through it. How thin must a pane be made of to absorb only 1% of the light.

Answer 1

dI/dx = kI

dI/I = kdx;∫

∫dI/I = ∫kdx; lnI = kx + lnC;

I = Ce^kx; If x = 0 then I = I_0, so C = I₀e^kx. I x = 1" then I = 3/4I₀, 3/4I₀ = I₀e^k·1"; k = ln(3/4);

I = I₀e^xln(3/4)= I₀(3/4)^x.

Now let absorbes 1% of light, so I = 0.99I₀; 0.99I₀ = I₀(3/4)^x; x = ln(0.99)/ln(3/4) = 0.035"