The human ciculatory system can be thought of as a closed system of interconnecting pipes through which fluid is continuously circulated by two pumps the two pumps the right and left verticles of the heart, work as simple two-stroke force pumps. The muscles of the heart regulate the force by contracting and relaxing. the contraction (systole) lasts about 0.2s and a complete systole/diastole (contraction/relaxation) cycle lasts about 0.8s. For flood pressures and speeds in the normal range. the volume flow rate of blood through a blood vessel is directly proportional tot he pressure difference over a length of the vessel and to the fourth power of the radius of the vessel. The total mechanical energy per unit volume of blood just as it leaves the heart is `E//V=rhogh+P+rhov^(2)`
Q. Which of the following is a way to achieve approximately a 45% increase in the volume flow rate of blood through a blood vessel?

Options

A. increase the radius by 10%
B. increasse the cross-sectional area of the vessel by 10%
C. decrease the change in pressure by 10%
D. decrease the speed of flow by 10%

Answer

To solve the problem of achieving approximately a 45% increase in the volume flow rate of blood through a blood vessel, we can use the principles derived from Poiseuille's Law. This law states that the volume flow rate \( Q \) through a pipe is given by the equation:

\[ Q = \frac{\Delta P \cdot \pi r^4}{\eta \cdot L} \]

where:
- \( \Delta P \) is the pressure difference,
- \( r \) is the radius of the vessel,
- \( \eta \) is the viscosity of the fluid,
- \( L \) is the length of the vessel.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find a way to increase the volume flow rate \( Q \) by approximately 45%. This can be expressed mathematically as:

\[ \frac{\Delta Q}{Q} \times 100 \approx 45\% \]

2. **Analyzing the Options**: We will evaluate the impact of the provided options on the volume flow rate.

3. **Option 1: Increase in radius by 10%**:
- If the radius \( r \) increases by 10%, the new radius \( r' \) is \( r' = 1.1r \).
- The change in flow rate can be calculated as follows:

\[ Q' = \frac{\Delta P \cdot \pi (1.1r)^4}{\eta \cdot L} = \frac{\Delta P \cdot \pi (1.4641r^4)}{\eta \cdot L} \]

- The percentage increase in flow rate is:

\[ \frac{Q' - Q}{Q} \times 100 = \frac{(1.4641 - 1)Q}{Q} \times 100 \approx 46.41\% \]

- This option is valid as it gives a percentage increase close to 45%.

4. **Option 2: Increase in cross-sectional area by 10%**:
- The area \( A \) of a circular vessel is \( A = \pi r^2 \). A 10% increase in area means:

\[ A' = 1.1A \Rightarrow \pi (r')^2 = 1.1 \pi r^2 \Rightarrow (r')^2 = 1.1 r^2 \Rightarrow r' = \sqrt{1.1} r \approx 1.0488r \]

- The percentage increase in radius is approximately 4.88%, leading to:

\[ \frac{Q' - Q}{Q} \times 100 \approx 4 \times 4.88 \approx 19.52\% \]

- This option is incorrect as it does not achieve a 45% increase.

5. **Option 3: Decrease the pressure by 10%**:
- Decreasing the pressure would decrease the flow rate, hence this option is incorrect.

6. **Option 4: Decrease the speed of flow by 10%**:
- The flow rate does not depend directly on the speed of flow in this context, so this option is also incorrect.

### Conclusion:
The only viable option to achieve approximately a 45% increase in the volume flow rate of blood through a blood vessel is **Option 1: Increase in radius by 10%**.