Q91PP - University Physics with Modern Physics

Question

The human circulatory system is closed—that is, the blood pumped out of the left ventricle of the heart into the arteries is constrained to a series of continuous, branching vessels as it passes through the capillaries and then into the veins as it returns to the heart. The blood in each of the heart’s four chambers comes briefly to rest before it is ejected by contraction of the heart muscle

If the aorta (diameter d_a) branches into two equal-sized arteries with a combined area equal to that of the aorta, what is the diameter of one of the branches? (a)$\sqrt{\mathbf{d}}$; (b)$\frac{\mathbf{d}}{\sqrt{\mathbf{2}}}$; (c)$\mathbf{2}\mathbf{d}$; (d) $\frac{{\mathbf{d}}_{\mathbf{a}}}{\mathbf{2}}$

Step-by-Step Solution

Answer

The diameter of one of the branches is, $\frac{d_a}{\sqrt{2}}$.

Step 1: Identification of given data

The given data can be listed below as-

• The diameter of the aorta is,$d_a$.

Step 2: Concept of area

Two-dimensional regions, shapes and planar lamina may all be described by their area in the plane.

The area of circle is given by,

$A={\pi }^2$ … (i)

Step 3: Comparing initial area of arteries and final area of arteries

The area of the aorta is expressed as from the equation i),

 

$A=\pi {\left(\frac{d_a}{2}\right)}^2$............. (ii)

 

Let, the diameter of one of the arteries is,

 

The area of the arteries can be expressed from the equation i),

$A_1=\pi {\left(\frac{d}{2}\right)}^2$ 

 

According to the question,, then the equation will be,

 $$\begin{gathered} A=A_1+A_2 \\ A=2A_1 \end{gathered}$$

Substituting the values in the equation (ii),

 $$\begin{gathered} \pi {\left(\frac{d_a}{2}\right)}^2=2\pi {\left(\frac{d}{2}\right)}^2 \\ d=\frac{d_a}{\sqrt{2}} \end{gathered}$$

 Thus, the diameter of one of the branches is, $\frac{d_a}{\sqrt{2}}$.