Question

By what factor must your blood pressure increase to compensate for a 5 % narrowing in your blood vessels? (The pressure difference across your blood vessels is essentially equal to your blood pressure.)
P =​ x P0

Answer 1

Let us assume -

First cross-sectional area = A1

Second cross-sectional area = A2

As given in the problem, A2 = A1*(100-5)/100 = 0.95*A1

Use continuity equation -  

A1* v1 = A2* v2

=> A1*v1 = 0.95*A1*v2

=> v2 = v1/(0.95)

Use Bernoulli's equation for dynamics fluid,

P1 + ρ g h + ½ ρ (v1)² = P2 + ρ g h + ½ ρ (v2)²

P1 + ½ ρ (v1)² = P2 + ½ ρ (v1/(0.95))²

P1 - P2 = 0.108 ρ (v1)^2

The pressure difference across your blood vessels is essentially equal to your blood pressure.

This implies -

Po = ½ ρ (v1)²

(P1 - P2)/Po = 0.108 ρ (v1)^2 / (½ ρ (v1)²)

or

ΔP = 0.108 x Po (Answer)

This can be also written as -

ΔP = 10.8% Po