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Chapter 9: Problem 81

The pressure in a water pipe in the basement of an apartment house is $4.10 \times 10^{5} \mathrm{Pa},\( but on the seventh floor it is only \)1.85 \times 10^{5} \mathrm{Pa} .$ What is the height between the basement and the seventh floor? Assume the water is not flowing; no faucets are opened.

Short Answer

Expert verified
Answer: The height difference between the basement and the seventh floor of the apartment house is approximately \(22.93\,\mathrm{m}\).

Step by step solution

01

Identify the formula for hydrostatic pressure

The hydrostatic pressure (P) in a fluid is given by the formula: \(P = \rho g h,\) where \(\rho\) is the density of the fluid, \(g\) is the acceleration due to gravity, and \(h\) is the height difference between the two points in the fluid.
02

Identify the pressure difference

The pressure in the basement is \(4.10 \times 10^{5} \mathrm{Pa}\) and on the seventh floor, it is \(1.85 \times 10^{5} \mathrm{Pa}\). The difference in pressure is: \(\Delta P = P_\text{basement} - P_\text{7th floor} = (4.10 \times 10^{5}) - (1.85 \times 10^{5}) \mathrm{Pa}\).
03

Calculate the pressure difference

Now we will calculate the pressure difference: \(\Delta P = (4.10-1.85) \times 10^5 = 2.25 \times 10^5 \mathrm{Pa}\).
04

Determine the water density and gravitational acceleration

The density of water \(\rho\) is approximately \(1000 \,\mathrm{kg/m^{3}}\). The acceleration due to gravity \(g\) is \(9.81 \,\mathrm{m/s^{2}}\).
05

Solve for height difference (h)

Rearrange the hydrostatic pressure formula to solve for height difference: \(h = \frac{\Delta P}{\rho g}\). Now, plug in the values: \(h = \frac{2.25 \times 10^{5}\,\mathrm{Pa}}{1000\, \mathrm{kg/m^{3}} \cdot 9.81\, \mathrm{m/s^2}}\).
06

Calculate the height difference

Calculate the height difference: \(h = \frac{2.25 \times 10^{5}}{1000 \cdot 9.81} = \frac{2.25 \times 10^{5}}{9810} = 22.93\,\mathrm{m}\).
07

Interpret the result

The height difference between the basement and the seventh floor of the apartment house is approximately \(22.93\,\mathrm{m}\).

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Most popular questions from this chapter

A bug from South America known as Rhodnius prolixus extracts the blood of animals. Suppose Rhodnius prolixus extracts \(0.30 \mathrm{cm}^{3}\) of blood in 25 min from a human arm through its feeding tube of length \(0.20 \mathrm{mm}\) and radius \(5.0 \mu \mathrm{m} .\) What is the absolute pressure at the bug's end of the feeding tube if the absolute pressure at the other end (in the human arm) is 105 kPa? Assume the viscosity of blood is 0.0013 Pa-s. [Note: Negative absolute pressures are possible in liquids in very slender tubes.]
You are hiking through a lush forest with some of your friends when you come to a large river that seems impossible to cross. However, one of your friends notices an old metal barrel sitting on the shore. The barrel is shaped like a cylinder and is \(1.20 \mathrm{m}\) high and \(0.76 \mathrm{m}\) in diameter. One of the circular ends of the barrel is open and the barrel is empty. When you put the barrel in the water with the open end facing up, you find that the barrel floats with \(33 \%\) of it under water. You decide that you can use the barrel as a boat to cross the river, as long as you leave about $30 \mathrm{cm}$ sticking above the water. How much extra mass can you put in this barrel to use it as a boat?
A manometer using oil (density \(0.90 \mathrm{g} / \mathrm{cm}^{3}\) ) as a fluid is connected to an air tank. Suddenly the pressure in the tank increases by \(0.74 \mathrm{cm}\) Hg. (a) By how much does the fluid level rise in the side of the manometer that is open to the atmosphere? (b) What would your answer be if the manometer used mercury instead?
A hypodermic syringe is attached to a needle that has an internal radius of \(0.300 \mathrm{mm}\) and a length of \(3.00 \mathrm{cm} .\) The needle is filled with a solution of viscosity $2.00 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s} ;\( it is injected into a vein at a gauge pressure of \)16.0 \mathrm{mm}$ Hg. Ignore the extra pressure required to accelerate the fluid from the syringe into the entrance of the needle. (a) What must the pressure of the fluid in the syringe be in order to inject the solution at a rate of $0.250 \mathrm{mL} / \mathrm{s} ?$ (b) What force must be applied to the plunger, which has an area of \(1.00 \mathrm{cm}^{2} ?\)
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