Problem 11 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

MUTATION In a study of mutation in fruit flies, researchers radiate flies with X-rays and determine that the mutation percentage \(M\) increases linearly with the X-ray dosage \(D\), measured in kilo- Roentgens (kR). When a dose of \(D=3 \mathrm{kR}\) is used, the percentage of mutations is \(7.7 \%\), while a dose of \(5 \mathrm{kR}\) results in a \(12.7 \%\) mutation percentage. Express \(M\) as a function of \(D\). What percentage of the flies will mutate even if no radiation is used?

Step-by-Step Solution

Answer

The function is \[ M(D) = 2.5D + 0.2 \]. Without radiation, 0.2% of the flies will mutate.

Step 1 - Understand given data

Identify the values from the problem: When D = 3 kR, M = 7.7%, and when D = 5 kR, M = 12.7%.

Step 2 - Determine the slope of the linear relationship

Use the formula for the slope of a line: \[ m = \frac{M_2 - M_1}{D_2 - D_1} \] where (D1, M1) = (3, 7.7) and (D2, M2) = (5, 12.7). Substitute the values: \[ m = \frac{12.7 - 7.7}{5 - 3} = \frac{5}{2} = 2.5 \]

Step 3 - Write the equation in slope-intercept form

To write the equation in slope-intercept form, use the general form \[ M = mD + b \] We need to find the y-intercept (b).

Step 4 - Solve for the y-intercept (b)

Choose a point (3, 7.7) and substitute it with the slope in the equation to solve for b: \[ 7.7 = 2.5(3) + b \] \[ 7.7 = 7.5 + b \] \[ b = 0.2 \]

Step 5 - Write the final function

Use the slope (m = 2.5) and y-intercept (b = 0.2) to write the final linear function: \[ M(D) = 2.5D + 0.2 \]

Step 6 - Find the mutation percentage at D=0

Substitute D = 0 into the function: \[ M(0) = 2.5(0) + 0.2 = 0.2 \] So, the percentage of the flies that will mutate with no radiation is 0.2%.

Key Concepts

Linear function

A linear function is a type of function where the output value changes in direct proportion to the input value. In other words, if you graph this function, you get a straight line. This relationship can be mathematically expressed as: \[ y = mx + b \] where

• \(y\) is the dependent variable
• \(x\) is the independent variable
• \(m\) is the slope
• \(b\) is the y-intercept

The slope \(m\) determines how steep the line is. Here, each unit increase in \(x\) results in a consistent increase defined by \(m\). For example, in the context of this problem, the mutation percentage changes linearly with the X-ray dosage.

Slope-intercept form

The slope-intercept form is a way to write the equation of a line. It is given by: \( y = mx + b \)

• \(m\) is the slope, representing the rate of change
• \(b\) is the y-intercept, representing the point where the line crosses the y-axis

In practical terms, the slope \(m\) shows how much the dependent variable (like mutation percentage) changes for a unit change in the independent variable (like X-ray dosage). The y-intercept \(b\) shows the value of the dependent variable when the independent variable is zero. For example, in our given problem:

• The slope \(m = 2.5\) means that for every kR increase in X-ray dosage, the mutation percentage increases by 2.5%.
• The y-intercept \(b = 0.2\) means that even with no radiation, the mutation percentage is 0.2%.

This form makes it easy to quickly understand how changes in \(x\) will affect \(y\).

Mutation percentage

Mutation percentage refers to the proportion of a population that shows mutations as a result of an influencing factor, such as X-ray dosage. In this particular study of fruit flies:

• A higher X-ray dosage leads to a higher mutation percentage.
• The researchers found that the relationship between X-ray dosage and mutation percentage is linear.
• Mathematically, it's represented by the function \( M(D) = 2.5D + 0.2 \).

To interpret this, suppose we increase the dosage by 1 kiloRoentgen (kR). According to the equation:

• The mutation percentage would increase by 2.5%.

This demonstrates the effect of radiation on mutation rates in a clear, linear fashion.

Roentgens

Roentgens (R) are units used to measure radiation exposure. One kiloRoentgen (kR) is equal to 1,000 Roentgens. In this context:

• The X-ray dosage is measured in kiloRoentgens (kR).
• Researchers use X-ray dosages to study their effects on mutation percentages in fruit flies.

Radiation can alter genetic material, leading to mutations. By measuring X-ray exposure and observing subsequent mutations, scientists can establish a relationship and predict outcomes mathematically. This helps us understand the risks and impacts of radiation on living organisms. For instance:

• With \(3 kR\), the mutation percentage is 7.7%.
• With \(5 kR\), the mutation percentage is 12.7%.

Knowing this relationship helps in many fields, including biology, medicine, and environmental science.