If for the pipe carrying water in a building, h = 220 meters, v_1 = 3.8 m/s, and the cross-sectional area at 1 is twice that at 2.

What must P_1 be in order that P_2 = 135000 Pa?
P_1 =

Wondering where P1 is?

at the bottom

To find the value of P₁, we can use Bernoulli's equation, which states that the total pressure at any point in a flowing fluid system is constant. In this case, we can apply Bernoulli's equation between points 1 and 2.

The equation is given by:

P₁ + (½)ρ(v₁)² + ρgh₁ = P₂ + (½)ρ(v₂)² + ρgh₂

Where:
P₁ and P₂ are the pressures at points 1 and 2 respectively,
ρ is the density of the fluid (water in this case),
v₁ and v₂ are the velocities at points 1 and 2 respectively,
g is the acceleration due to gravity and
h₁ and h₂ are the heights at points 1 and 2 respectively.

Based on the information given in the question, we can substitute the values into the equation:

P₁ + (½)ρ(v₁)² + ρgh = P₂ + (½)ρ(v₂)² + ρgh
P₁ + (½)ρ(v₁)² + ρgh₁ = P₂ + (½)ρ(v₂)² + ρgh₂

Since the cross-sectional area at point 1 is twice that at point 2, and we know that the velocity is given by v = Q/A (where Q is the flow rate and A is the cross-sectional area), we can write v₁ = (Q/A₁) and v₂ = (Q/A₂). Therefore, since Area at 1 is twice that at 2, A₂ = 2A₁, and v₂ = v₁/2.

Substituting these values into the equation, we get:

P₁ + (½)ρ(v₁)² + ρgh₁ = P₂ + (½)ρ(v₂)² + ρgh₂
P₁ + (½)ρ(v₁)² + ρgh₁ = P₂ + (½)ρ(v₁/2)² + ρgh₂

Now we can substitute the known values into the equation:
P₁ + (½)(density of water)(3.8 m/s)² + (density of water)(9.8 m/s²)(220 m) = 135,000 Pa + (½)(density of water)((3.8 m/s)/2)² + (density of water)(9.8 m/s²)(0 m)

Simplifying the equation gives us:
P₁ + ½(1000 kg/m³)(14.44 m²/s²) + (1000 kg/m³)(9.8 m/s²)(220 m) = 135,000 Pa + ½(1000 kg/m³)(0.9025 m²/s²)

Now we can solve for P₁ by rearranging the equation and isolating P₁:

P₁ = 135,000 Pa + ½(1000 kg/m³)(0.9025 m²/s²) - ½(1000 kg/m³)(14.44 m²/s²) - (1000 kg/m³)(9.8 m/s²)(220 m)

P₁ ≈ 13,900 Pa

Therefore, in order for P₂ to be 135,000 Pa, P₁ must be approximately 13,900 Pa.