Question

need solution for this !!​

Answer 1

★ Concept :-

Here the concept of Power of the heart has been used. We see that we are given the volume of discharge of blood. Even we are given the output power of the heart, height of blood to be lifted and density of the blood. So firstly using the relationship of power in terms of differentiation, we can find the exact formula of power required to us and then find the exact power. Then we can compare it with given power and then find the answer.

Let's do it !!

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★ Formula Used :-

$\;\boxed{\sf{\pink{Power\;=\;\bf{\dfrac{dm}{dt}\:g\:h}}}}$

$\;\boxed{\sf{\pink{\dfrac{dm}{dt}\;=\;\bf{\rho \dfrac{dV}{dt}}}}}$

________________________________

★ Solution :-

Given,

» Amount of blood discharged by heart = V = 7500 L per day

» Height to which blood is lifted = h = 1.63 m

» Density of blood = ρ = 1.05 × 10³ Kg m⁻³

» Given power of heart = P' = 148/10ⁿ W = 148 × 10⁻ⁿ W

» Time taken = t = 1 day = 24 × 60 × 60 sec

» Acceleration due to gravity = g = 10 m/sec²

-----------------------------------------------------------

~ For the actual power of the heart, P ::

• Let the actual power of heart be P .

We know that,

$\;\sf{\rightarrow\;\;Power\;=\;\bf{\dfrac{dm}{dt}\:g\:h}}$

And we also know that,

$\;\sf{\rightarrow\;\;\dfrac{dm}{dt}\;=\;\bf{\rho \dfrac{dV}{dt}}}$

By combining these both equations we get,

$\;\sf{\rightarrow\;\;Power\;=\;\bf{\rho\:\dfrac{dV}{dt}\:g\:h}}$

By applying values in this equation, we get

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:10^{3}\:\times\:7500\:\times\:10^{-3}\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:10^{3}\:\times\:7500\:\times\:10^{-3}\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

Cancelling the unlike terms, we get

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:7500\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{128362.5}{86400}}}$

$\;\bf{\rightarrow\;\;Power,\;P\;=\;\bf{\red{1.48\:\:W}}}$

• Here W = Watt

This is the actual power of the heart.

-----------------------------------------------------------

~ For the value of n ::

We know that,

• Actual Power = Given Power

• P = P'

Then, it will be

→ 1.48 = 148 × 10⁻ⁿ

→ 10⁻ⁿ = 1.48 / 148

→ 10ⁿ = 148/1.48

→ 10ⁿ = 100

→ 10ⁿ = 10²

By laws of Exponents and comparing LHS and RHS, we get

→ n = 2

$\;\underline{\boxed{\tt{Hence,\;\:value\;\:of\;\:n\;=\;\bf{\purple{2}}}}}$

________________________________

★ More to know :-

$\;\tt{\leadsto\;\;Momentum\;=\;Mass\:\times\:Velocity}$

$\;\tt{\leadsto\;\;Force\;=\;Mass\:\times\:Acceleration}$

$\;\tt{\leadsto\;\;Work\;=\;Force\:\times\:Distance}$

$\;\tt{\leadsto\;\; Impulse\;of\;Force\;=\;Force\:\times\:Time}$

$\;\tt{\leadsto\;\;Power\;=\;\dfrac{Work}{Time}}$

Answer 2

Explanation:

★ Concept :-

Here the concept of Power of the heart has been used. We see that we are given the volume of discharge of blood. Even we are given the output power of the heart, height of blood to be lifted and density of the blood. So firstly using the relationship of power in terms of differentiation, we can find the exact formula of power required to us and then find the exact power. Then we can compare it with given power and then find the answer.

Let's do it !!

________________________________

★ Formula Used :-

$\;\boxed{\sf{\pink{Power\;=\;\bf{\dfrac{dm}{dt}\:g\:h}}}}$

$\;\boxed{\sf{\pink{\dfrac{dm}{dt}\;=\;\bf{\rho \dfrac{dV}{dt}}}}}$

________________________________

★ Solution :-

Given,

» Amount of blood discharged by heart = V = 7500 L per day

» Height to which blood is lifted = h = 1.63 m

» Density of blood = ρ = 1.05 × 10³ Kg m⁻³

» Given power of heart = P' = 148/10ⁿ W = 148 × 10⁻ⁿ W

» Time taken = t = 1 day = 24 × 60 × 60 sec

» Acceleration due to gravity = g = 10 m/sec²

-----------------------------------------------------------

~ For the actual power of the heart, P ::

Let the actual power of heart be P .

We know that,

$\;\sf{\rightarrow\;\;Power\;=\;\bf{\dfrac{dm}{dt}\:g\:h}}$

And we also know that,

$\;\sf{\rightarrow\;\;\dfrac{dm}{dt}\;=\;\bf{\rho \dfrac{dV}{dt}}}$

By combining these both equations we get,

$\;\sf{\rightarrow\;\;Power\;=\;\bf{\rho\:\dfrac{dV}{dt}\:g\:h}}$

By applying values in this equation, we get

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:10^{3}\:\times\:7500\:\times\:10^{-3}\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:10^{3}\:\times\:7500\:\times\:10^{-3}\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

Cancelling the unlike terms, we get

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{1.05\:\times\:7500\:\times\:10\:\times\:1.63}{24\:\times\:60\:\times\:60}}}$

$\;\sf{\rightarrow\;\;Power,\;P\;=\;\bf{\dfrac{128362.5}{86400}}}$

$\;\bf{\rightarrow\;\;Power,\;P\;=\;\bf{\red{1.48\:\:W}}}$

Here W = Watt

This is the actual power of the heart.

-----------------------------------------------------------

~ For the value of n ::

We know that,

• Actual Power = Given Power

• P = P'

Then, it will be

→ 1.48 = 148 × 10⁻ⁿ

→ 10⁻ⁿ = 1.48 / 148

→ 10ⁿ = 148/1.48

→ 10ⁿ = 100

→ 10ⁿ = 10²

By laws of Exponents and comparing LHS and RHS, we get[/tex]

→ n = 2

$\;\underline{\boxed{\tt{Hence,\;\:value\;\:of\;\:n\;=\;\bf{\purple{2}}}}}$

________________________________

★ More to know :-

$\;\tt{\leadsto\;\;Momentum\;=\;Mass\:\times\:Velocity}$

$\;\tt{\leadsto\;\;Force\;=\;Mass\:\times\:Acceleration}$

$\;\tt{\leadsto\;\;Work\;=\;Force\:\times\:Distance}$

$\;\tt{\leadsto\;\; Impulse\;of\;Force\;=\;Force\:\times\:Time}$

$\;\tt{\leadsto\;\;Power\;=\;\dfrac{Work}{Time}}$