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BIO Power Output of the Heart The power output of the heart is given by the product of the average blood pressure, \(1.33 \mathrm{N} / \mathrm{cm}^{2}\), and the flow rate, \(105 \mathrm{cm}^{3} / \mathrm{s} .\) (a) Find the power of the heart. Give your answer in watts. (b) How much energy does the heart expend in a day? (c) Suppose the energy found in part (b) is used to lift a \(72-\mathrm{kg}\) person vertically to a height \(h\) Find \(h,\) in meters.

Short Answer

Expert verified
(a) 1.39815 W; (b) 120679.56 J; (c) 171.9 m

Step by step solution

01

Power Output Calculation

To calculate the power output of the heart, use the formula for power, which is given as the product of the pressure and the flow rate. Given that average blood pressure is \(1.33\ N/cm^2\) and flow rate is \(105\ cm^3/s\), the formula is:\[P = 1.33\ N/cm^2 \times 105\ cm^3/s\]Convert units to watts by realizing that \(1\ N/cm^2 = 10^4\ N/m^2\) and \(1\ cm^3/s = 10^{-6}\ m^3/s\):\[P = 1.33 \times 10^4\ N/m^2 \times 105 \times 10^{-6}\ m^3/s\]Calculating this gives:\[P = 1.39815\ W\]
02

Daily Energy Expenditure

To determine how much energy the heart expends in a day, use the formula \(E = P \times t\), where \(t\) is the time in seconds for one day (\(86400\ seconds\)). Compute this as follows:\[E = 1.39815\ W \times 86400\ s\]This results in:\[E = 120679.56\ J\]
03

Calculate Height Lifted

If the energy is used to lift a 72-kg person, use the potential energy formula \(E = m \cdot g \cdot h\), where \(m = 72\ kg\) is the mass of the person, \(g = 9.81\ m/s^2\) is the acceleration due to gravity, and \(h\) is the height.Rearrange for \(h\):\[h = \frac{E}{m \cdot g} = \frac{120679.56}{72 \cdot 9.81}\]Calculate \(h\):\[h \approx 171.9\ m\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Pressure
Blood pressure is the force that your blood exerts on the walls of your arteries as it circulates through your body. Think of it as the push that keeps blood moving from one part of your body to another.
It plays a crucial role in how your heart operates, especially when calculating the power output of the heart.
The average blood pressure given in the problem is measured as \(1.33\, ext{N/cm}^2\). To convert this for use in standard calculations, you often change it to \(N/m^2\) using conversion factors:
  • 1 N/cm² is equivalent to 10,000 N/m²
Understanding blood pressure helps in determining how hard the heart needs to work to pump blood through the vessels, which ties directly into the concept of power output.
Flow Rate
Flow rate represents the volume of blood that passes a point in the circulatory system in a given amount of time. This rate is typically measured in cubic centimeters per second (cm³/s) or liters per minute (L/min). It is crucial for calculating the power output of the heart. In the exercise, the flow rate is given as \(105\, ext{cm}^3/ ext{s}\).
This number tells us how much volume of blood the heart is able to pump out at a certain pressure.
Understanding how to convert this number to \(m^3/s\) for consistency in calculations is important. Here's how you can do it:
  • Use the conversion 1 cm³ = \(10^{-6}\) m³
  • Therefore, \(105\, ext{cm}^3/ ext{s} = 105 imes 10^{-6} ext{ m}^3/ ext{s}\)
When the heart functions efficiently, it maintains an ideal flow rate that ensures all body systems receive adequate blood supply, which is vital for maintaining life.
Energy Expenditure
Energy expenditure refers to the amount of energy the heart uses to maintain circulation of blood throughout the body. This is continuously happening, not just for short bursts, and amounts to a large daily total.
In our everyday activities, the heart is working around the clock, converting chemical energy into mechanical energy to pump blood.
The exercise calculates how much energy the heart expends over a day— a whopping \(120,679.56\, ext{J}\) (joules). This is an excellent real-world example of persistent energy use over time.When calculating energy expenditure, we're often interested in comparisons. For example:
  • In terms of power, measured in watts, with 1 watt equaling 1 joule per second.
  • Understanding that this continuous output is crucial for sustaining life functions.
  • This energy can be related to lifting a mass (like a 72-kg person, as in the problem) to a height. Using potential energy formulas offers relatable context for how much work the heart performs through the day.
This highlights the incredible efficiency and power of the heart as an organ.

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Most popular questions from this chapter

BIO Vasodilation When the body requires an increased blood flow rate in a particular organ or muscle, it can accomplish this by increasing the diameter of arterioles in that area. This is referred to as vasodilation. What percentage increase in the diameter of an arteriole is required to double the volume flow rate of blood, all other factors remaining the same?

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