Problem 42 - A Graphical Approach to College Algebra

Question

Solve each problem. Rate of Nerve Impulses The rate at which impulses are transmitted along a nerve fiber is directly proportional to the diameter of the fiber. The rate for a certain fiber is 40 meters per second when the diameter is 6 micrometers. Find the rate if the diameter is 8 micrometers.

Step-by-Step Solution

Answer

The rate is approximately 53.33 meters per second.

Step 1: Understanding Direct Proportionality

When a quantity is directly proportional to another, this means their ratio is constant. Mathematically, if rate (R) is directly proportional to diameter (D), then \( R = kD \), where \( k \) is a constant.

Step 2: Find the Constant

Substitute the known values into the equation to find \( k \). We know the rate \( R = 40 \) m/s when \( D = 6 \) micrometers. So, \( 40 = k \times 6 \). Solving for \( k \) gives \( k = \frac{40}{6} = \frac{20}{3} \).

Step 3: Apply the Constant to Find the New Rate

Now that we have \( k = \frac{20}{3} \), let's find the new rate when \( D = 8 \) micrometers. Substitute into the formula \( R = kD \), which becomes \( R = \frac{20}{3} \times 8 \).

Step 4: Calculate the New Rate

Perform the multiplication: \( R = \frac{20}{3} \times 8 = \frac{160}{3} \). Dividing 160 by 3 gives approximately \( R \approx 53.33 \) m/s.

Key Concepts

Rate of Nerve Impulses

The rate of nerve impulses refers to how quickly electrical signals are transmitted along a nerve fiber. This speed is influenced by several factors, including the diameter of the fiber itself. In the context of direct proportionality, if the diameter increases, the rate at which impulses are transmitted increases as well. This occurs because a larger diameter provides a pathway with less resistance, allowing the signals to travel faster.

Understanding this relationship helps in predicting changes. For example, if we know the rate for a fiber with a certain diameter, we can find the rate for a different diameter once we know the constant of proportionality. This principle is key in fields like biology and neurology, where knowing the rate of nerve impulses can help understand nerve function and health.

Constant of Proportionality

The constant of proportionality, often denoted by the symbol \( k \), is a crucial number in problems involving direct proportionality. This constant represents the relationship between the rate of nerve impulses and the diameter of the nerve fiber. By establishing this constant, we can see how exactly one variable changes in response to changes in another.

For calculating \( k \) in our exercise, we used the given rate of 40 meters per second when the diameter of the nerve fiber was 6 micrometers. By substituting these known values into the formula \( R = kD \), we derived that \( k = \frac{20}{3} \). This constant remains the same as long as the ratio between the diameter and the rate remains consistent. Knowing \( k \) allows us to predict outcomes for different fiber diameters efficiently.

Some key points about \( k \):

• It remains constant for directly proportional relationships.
• It provides a conversion factor between the rate and diameter.
• Once known, it can simplify calculating new rates with different diameters.

Calculating Rates

Calculating rates when two quantities are directly proportional involves a straightforward application of the relationship \( R = kD \). Once we have determined the constant \( k \), calculations for any new rate become simple.

In our example, with a constant \( k = \frac{20}{3} \), we computed the rate for a nerve fiber with a diameter of 8 micrometers by inserting it back into the formula as \( R = \frac{20}{3} \times 8 \). Performing this calculation gave us \( R = \frac{160}{3} \), approximately 53.33 meters per second.

While performing such calculations, always remember:

• Identify the constant of proportionality first.
• Ensure units are consistent when substituting into the formula.
• Check calculations for accuracy to ensure the derived rate makes sense.

By applying these steps, students can successfully calculate direct proportionality rates in various scenarios, making this concept a handy tool in both academic and real-world settings.