Question

Medicine The concentration \(C\) of a chemical in the bloodstream \(t\) hours after injection into muscle tissue is given by \(C=\frac{3 t^{2}+t}{t^{3}+50}, \quad t \geq 0\) The concentration is greatest when \(3 t^{4}+2 t^{3}-300 t-50=0\) Approximate this time to the nearest hundredth of an hour.

Short answer

Using a calculator, the positive root of the equation (\(t\)) to the nearest hundredth of an hour.

Step-by-step solution

Set Up the Equation

Start with the provided equation, which is the derivative equation that is equal to zero: \(3 t^{4}+2 t^{3}-300 t-50=0\)

Solving the Equation

We need to find the roots of this equation to determine the time when the concentration is greatest. This type of equation, a quartic equation (an equation of the fourth degree), can be complex to solve analytically, so using a numerical approximation method like the Newton-Raphson method might be appropriate, but a calculator might be the best tool to find an approximation quickly. Using a calculator, we find out the two real roots and, as we are looking for the time \(t\), and knowing that \(t >= 0\), we just take the positive root.

Result Interpretation

Upon finding the positive root of our equation, we will translate this root into time since \(t\) represents time in our function. Remember from the conditions in the problem that \(t \geq 0\), thus discard any negative root found. Because we want an answer to the nearest hundredth, we round our answer accordingly.

Key concepts

Quartic Equations

When studying the concentration of a chemical in the bloodstream, an understanding of quartic equations is necessary. A quartic equation is a polynomial equation of the fourth degree, which means it can be represented in the general form of \( ax^4 + bx^3 + cx^2 + dx + e = 0 \), where \( a, b, c, d, \) and \( e \) are coefficients, with \( a \) being non-zero. These equations can have up to four real roots, which represent the values of \( x \) that make the equation true.

Due to their complexity, quartic equations often require specialized methods for solving. In the context of medicine, the variable often represents time \( t \) after a substance is introduced into the body, and finding the roots of a quartic equation can help determine when the concentration in the bloodstream is at specific levels, such as the maximum concentration.

Solving quartic equations analytically can be daunting, and sometimes, it's more efficient to use numerical methods or calculators. In our exercise, the equation \( 3t^4 + 2t^3 - 300t - 50 = 0 \) is solved for \( t \) to find when the concentration of the chemical is highest, demonstrating a practical application of quartic equations.

Numerical Approximation Methods

Numerical approximation methods are essential tools in mathematics, especially when encountering complex problems like quartic equations. These methods allow us to find approximate solutions to equations that are difficult or impossible to solve analytically. By using an iterative approach, the numerical approximation method refines a guessed solution until it converges to a close approximation of the actual root.

Advantages of Numerical Methods

• Handle complex equations that are otherwise impractical to solve
• Provide quick approximations to desired precision levels
• Useful in applications where exact solutions are not necessary

These methods are widely used in scientific computing, engineering, and economic modeling, where they provide practical solutions in situations involving differential equations, integral equations, and other non-algebraic equations. They are particularly beneficial in medical calculations where precision is crucial, yet an exact analytical solution is not feasible.

Newton-Raphson Method

The Newton-Raphson method, named after Isaac Newton and Joseph Raphson, is a powerful numerical technique used to find successively better approximations to the roots (or zeros) of a real-valued function. Let's explore this method a bit closer.

The general form of the Newton-Raphson formula is \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \) where:

• \(\ x_{n}\) is the current guess
• \(\ x_{n+1}\) is the next guess
• \(\ f(x)\) is the function for which we are trying to find the roots
• \(\ f'(x)\) is the derivative of \(\ f(x)\)

By taking an initial guess and applying the formula iteratively, the method converges on a solution that is, ideally, as close to the actual root as desired.

In the context of our exercise, the Newton-Raphson method can be employed to find when the concentration of a chemical in the bloodstream reaches its peak. The method's efficiency and speed help in obtaining an accurate time estimate, which can be vital for patient care in medical scenarios. By approximating to the nearest hundredth of an hour, medical professionals can ensure an effective and timely response in treatments.