Question

In a study, doctors found that in young people the arterial blood pressure, \(P\) in millimetres of mercury (mmHg), is related to the vessel volume, \(V\) in microlitres ( \(\mu \mathrm{L}\) ), of the radial artery by the logarithmic function \(V=0.23+0.35 \log (P-56.1), P>56.1\) a) To the nearest tenth of a microlitre, predict the vessel volume when the arterial blood pressure is \(110 \mathrm{mmHg}\). b) To the nearest millimetre of mercury, predict the arterial blood pressure when the vessel volume is 0.7 \(\mu \mathrm{L}\)

Short answer

a) To the nearest tenth, the vessel volume is 0.8 \mathrm{\mu L}. b) To the nearest millimetre of mercury, the arterial blood pressure is 78 \mathrm{mmHg}.

Step-by-step solution

Identify Given Values for Part (a)

The arterial blood pressure is given as \( P = 110 \, \mathrm{mmHg} \). Substitute this value into the given formula \( V = 0.23 + 0.35 \log(P - 56.1) \)

Substitute and Simplify for Part (a)

Substitute \( P = 110 \) into the equation: \[ V = 0.23 + 0.35 \log(110 - 56.1) \]Calculate the value inside the logarithm: \[ \log(110 - 56.1) = \log(53.9) \]Continue simplifying: \[ V = 0.23 + 0.35 \log(53.9) \]

Calculate Logarithm for Part (a)

Use a calculator to find \( \log(53.9) \). Approximate value: \[ \log(53.9) \approx 1.7324 \]Now substitute: \[ V = 0.23 + 0.35 \times 1.7324 \]

Final Calculation for Part (a)

Multiply and add: \[ V = 0.23 + 0.6063 \]\[ V \approx 0.8363 \ \text{To the nearest tenth: } V \approx 0.8 \ \mathrm{\mu L}\]

Identify Given Values for Part (b)

The vessel volume is given as \( V = 0.7 \ \mathrm{\mu L}\ \). Substitute this value into the formula to solve for \( P \): \( 0.7 = 0.23 + 0.35 \log(P - 56.1) \)

Isolate Logarithmic Term for Part (b)

Subtract 0.23 from both sides of the equation: \[ 0.7 - 0.23 = 0.35 \log(P - 56.1) \]\[ 0.47 = 0.35 \log(P - 56.1) \]

Solve for Logarithm for Part (b)

Divide both sides by 0.35: \[ \frac{0.47}{0.35} = \log(P - 56.1) \]\[ 1.3429 = \log(P - 56.1) \]

Solve for Pressure P for Part (b)

Exponentiate both sides to clear the logarithm: \[ 10^{1.3429} = P - 56.1 \]\[ 22.08 \approx P - 56.1 \]\[ P \approx 22.08 + 56.1 \]\[ P \approx 78.18 \text{To the nearest millimetre of mercury: } P \approx 78 \ \mathrm{mmHg}\]

Key concepts

arterial blood pressure

Arterial blood pressure is a critical measurement in understanding overall cardiovascular health.
It measures the force exerted by circulating blood on the walls of blood vessels.
This pressure is usually measured in millimetres of mercury (mmHg).

Normal arterial blood pressure ranges around 120 mmHg systolic (when the heart beats) and 80 mmHg diastolic (when the heart rests between beats).
Vital organs depend on consistent blood pressure to function correctly.
High blood pressure can lead to severe health issues, including heart disease and stroke.
Conversely, low blood pressure can cause dizziness and fainting.
In mathematical modeling, we often use arterial blood pressure in functions to predict other biological variables such as vessel volume.
This relationship can be expressed through different mathematical equations, including logarithmic functions.

vessel volume

The vessel volume in the radial artery refers to the amount of blood that can be held within that blood vessel.
It is typically measured in microlitres (\textmu L).
Knowing the vessel volume helps medical professionals assess the condition of the blood vessels and ensure proper blood circulation.

In the given problem, vessel volume (\textmu L) is related to arterial blood pressure (mmHg) by the logarithmic equation:
\[ V = 0.23 + 0.35 \log(P - 56.1), \space P > 56.1 \]
Where.
\ P = \text{Arterial blood pressure in mmHg} \ V = \text{Vessel volume in microlitres} \ This formula allows us to predict vessel volume based on known arterial blood pressure and vice versa.
In the medical field, understanding these relationships is crucial for diagnosing and prescribing appropriate treatments.

logarithm calculations

Logarithm calculations are a powerful tool in understanding relationships in various scientific fields, including medicine.
The logarithm of a number is the power to which a base (usually 10 in common logarithms) must be raised to produce that number.
For example,\[ \log(100) = 2 \]
because 10 raised to the power of 2 equals 100.

In our exercise, we use the logarithmic function to relate arterial blood pressure to vessel volume.
To solve problems involving logarithms, follow these steps:

• Identify the logarithmic function and substitute the given values.
• Simplify the logarithmic expression inside the function.
• Use a calculator to find the logarithmic value.
• Substitute the logarithmic value back into the equation to find the desired variable.

Consider the formula:
\[ V = 0.23 + 0.35 \log(P - 56.1) \]
When predicting vessel volume for a given pressure or vice versa, accurately computing the logarithm is essential for obtaining correct results.
This practical use of logarithm showcases how mathematical tools are used in real-life applications, bridging the gap between theory and practice.