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Question: A human heart pumps \(70{\text{ }}CC\) of blood at each beat against a pressure of \(72{\text{ }}mm\...

A human heart pumps 70 CC70{\text{ }}CC of blood at each beat against a pressure of 72 mm72{\text{ }}mm of Hg. If the pulse frequency is 7272 per minute the power of the heart is
A. 1.2 W1.2{\text{ }}W
B. 1.4 W1.4{\text{ }}W
C. 1.6 W1.6{\text{ }}W
D. 0.8 W0.8{\text{ }}W

Explanation

Solution

We know that we have to find out Power so we use the formula of power as power is defined as the time required to complete a work done. There are various formulas to find Work Done but in this particular case as the pressure and change of volume given so we will make use of them to find work and then divide with time to get the result. Density of mercury is 13.6 g.cm−313.6{\text{ }}g.c{m^{ - 3}}.

Complete step by step answer:
We will use Work done (W)=PdV(W) = PdV. The above question is assigned with different values so we have to make use of them to find the respective solution. Firstly, we have to find out the pressure(P)(P) from the above values,
We know that,
P=hρgP = h\rho g
Where height (h)(h) == 72 mm72{\text{ }}mm of Hg=72×10−3 m = 72 \times {10^{ - 3}}{\text{ }}m of Hg.
Density(ρ)(\rho ) =13600 kg.m−3 = 13600{\text{ }}kg.{m^{ - 3}}
Acceleration due to gravity (g)=9.8 m.s−2(g) = 9.8{\text{ }}m.{s^{ - 2}}
P=72×10−3×13600×9.8 ⇒P=9596.16 Pascal−−−−−−(1) P = 72 \times {10^{ - 3}} \times 13600 \times 9.8 \\\ \Rightarrow P = 9596.16{\text{ }}Pascal - - - - - - (1) \\\
Now, as the frequency of heart beat is given then we must have to find out the total change in volume.In one beat the total amount of change in volume dVdV is 70 CC70{\text{ }}CC =70×10−6 m3 = 70 \times {10^{ - 6}}{\text{ }}{m^3}
Let dVTd{V_T} be the total change in volume.
Then dVTd{V_T} =70×10−6×72 = 70 \times {10^{ - 6}} \times 72
⇒dVT=5040×10−6 m3−−−−−−(2)\Rightarrow d{V_T} = 5040 \times {10^{ - 6}}\,{m^3} - - - - - - (2)
Now we have to convert time in seconds so, t=60 s−−−−−−(3)t = 60{\text{ }}s - - - - - - (3)

Therefore by putting the values of (1),(2),(3)\left( 1 \right),\left( 2 \right),\left( 3 \right) in the formula Power=PdVTt = \dfrac{{Pd{V_T}}}{t} we get,
Power=9596.16×5040×10−660 ∴Power=0.80 Watt\text{Power}= \dfrac{{9596.16 \times 5040 \times {{10}^{ - 6}}}}{{60}} \\\ \therefore \text{Power}= 0.80{\text{ }}Watt
Hence, the power generated when human heart pumps 70 CC70{\text{ }}CC of blood at each beat against a pressure of 72 mm72{\text{ }}mm of Hg with pulse frequency 7272 per minute is 0.80 Watt0.80{\text{ }}Watt.

Hence, the correct answer is option D.

Note: We must change the change dVdV(change in volume in per minute) to dVTd{V_T} (total change in volume in per minute). Where, dVTd{V_T} =n×dV = n \times dV where n is the number of frequencies. We must consider time in seconds every time.