Problem 54 - College Algebra

Question

A formula for the normal systolic blood pressure for a man age \(A\) , measured in mm Hg, is given as \(P=0.006 A^{2}-0.02 A+120 .\) Find the age to the nearest year of a man whose normal blood pressure measures 125 mm Hg.

Step-by-Step Solution

Answer

The man's age is approximately 31 years old.

Step 1: Understand the Equation

We start with the equation given in the problem, which is for normal systolic blood pressure: \(P = 0.006A^2 - 0.02A + 120\). The goal is to find the value of \(A\) when \(P = 125\).

Step 2: Substitute the Blood Pressure into the Equation

Substitute \(P = 125\) into the equation: \[ 125 = 0.006A^2 - 0.02A + 120 \]

Step 3: Rearrange the Equation

Move all terms to one side of the equation to set it to zero: Subtract 125 from both sides:\[ 0.006A^2 - 0.02A + 120 - 125 = 0 \]This simplifies to: \[ 0.006A^2 - 0.02A - 5 = 0 \]

Step 4: Solve the Quadratic Equation

We have a quadratic equation of the form \( ax^2 + bx + c = 0 \). Use the quadratic formula: \[ A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \( a = 0.006 \), \( b = -0.02 \), and \( c = -5 \).Calculate the discriminant:\[ b^2 - 4ac = (-0.02)^2 - 4(0.006)(-5) = 0.0004 + 0.12 = 0.1204 \]Calculate \( A \):\[ A = \frac{-(-0.02) \pm \sqrt{0.1204}}{2 \times 0.006} = \frac{0.02 \pm 0.347 \ldots}{0.012} \]

Step 5: Calculate the Possible Values for A

Solve for both values using the quadratic formula results:1. \( A_1 = \frac{0.02 + 0.347 \ldots}{0.012} \approx 30.59 \)2. \( A_2 = \frac{0.02 - 0.347 \ldots}{0.012} \) which is negative and non-integers are not valid for age as ages cannot be negative.

Step 6: Approximate the Correct Age

Round the positive solution \( A_1 \) to the nearest year:\( A_1 \approx 31 \) since 30.59 rounds to 31.

Key Concepts

Normal Systolic Blood Pressure

Normal systolic blood pressure refers to the pressure in the arteries when the heart is beating. It is an important measure in assessing the health of the cardiovascular system.
The measurement is presented in millimeters of mercury, abbreviated as mm Hg. For adult men, there are general estimates for normal values, though individual variations exist.
An equation like the one given, \(P = 0.006A^2 - 0.02A + 120\) can be used to approximate the normal systolic blood pressure based on age, helping to gauge expected values. It uses age \(A\) of an individual to estimate the pressure \(P\).
This formula allows for understanding how blood pressure might naturally change with age for males.

Solving Quadratic Equations

A quadratic equation is any equation that can be rearranged in standard form as \(ax^2 + bx + c = 0\), where \(x\) represents an unknown variable, and \(a, b,\) and \(c\) represent known coefficients.
These equations can have two solutions, and to find them, we can use different methods such as factoring, completing the square, or employing the quadratic formula.
In solving, our aim is to find the value of the variable, which is typically represented as \(x\) or sometimes as \(A\), depending on context. Solving involves transforming the equation and isolating \(x\) or \(A\).
There is often a necessity to ensure the equation is set to zero for many methods to work effectively.

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. It helps find the roots of the equation when it is in the form \(ax^2 + bx + c = 0\).
The formula is given as follows: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Where:

• \(a\), \(b\), and \(c\) are coefficients from the original equation.
• The symbol \(\pm\) indicates two potential solutions — one using the plus and one using the minus.
• The term under the square root, \(b^2 - 4ac\), is called the discriminant. It determines the number and type of solutions.

This method is especially useful when the quadratic equation cannot be easily factored. It ensures an accurate solution by mathematically deriving both potential roots.

Age Estimation

Age estimation using mathematical models involves using formulas to infer age based on other measurable parameters, in this case, blood pressure.
By manipulating the equation \(P = 0.006A^2 - 0.02A + 120\), where \(P\) is known, you can determine \(A\), the age.

• After substituting the known value of \(P\) into the equation, it transforms into a quadratic equation in the standard format.
• The solutions correspond to the potential ages that match the given blood pressure.
• It's crucial to check that the calculated age is logical, as negative or non-integer results may not be valid.

This technique is useful in medical and scientific fields where estimating an individual's age based on physiological data is necessary.