Question

The normal amount of lead in a certain water supply is \(1.5 \times 10^{-5}\) milligrams per liter. Today, when the water was tested, the lead level found was exactly 100 times as great as the normal level, still well below the Environmental Protection Agency's action level. What concentration of lead, in milligrams per liter, was in the water tested today? B. \(1.5 \times 10^{-10}\) C. \(1.5 \times 10^{-7}\) D. \(1.5 \times 10^{-7}\) E. \(1.5 \times 10^{-\frac{5}{2}}\)

Short answer

Answer: The concentration of lead in the water tested today is \(1.5 \times 10^{-3}\) milligrams per liter.

Step-by-step solution

Identify the normal amount of lead in water supply

The normal amount of lead in a certain water supply is \(1.5 \times 10^{-5}\) milligrams per liter.

Identify the multiple by which the lead level has increased

Today, the lead level is exactly 100 times as great as the normal level. So, we need to multiply the normal amount of lead by 100.

Calculate the concentration of lead in water tested today

To find the concentration of lead in the water tested today, we multiply the normal amount of lead by 100: \((1.5 \times 10^{-5}) \times 100 = 1.5 \times (10^{-5} \times 10^2) = 1.5 \times 10^{-3}\) milligrams per liter.

Compare the result with the given answer choices

The concentration of lead in the water tested today, \(1.5 \times 10^{-3}\) milligrams per liter, is not among the given answer choices. We have made a mistake in the options. The correct answer option should be: F. \(1.5 \times 10^{-3}\)

Key concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form that is easier to read and work with. This notation uses powers of ten to represent the magnitude of a number. For instance, the number 0.000015 can be written in scientific notation as \(1.5 \times 10^{-5}\). Here's how to interpret it:

The coefficient (1.5 in this example) is always a number greater than or equal to 1 and less than 10. The exponent (the -5 in this case) indicates how many places the decimal must move to write the number in standard form. A negative exponent means the number is less than one. When calculating with scientific notation, you can manipulate the powers of ten using exponent rules.

• To multiply two numbers in scientific notation, you multiply their coefficients and add the exponents of their powers of ten.
• To divide, you divide the coefficients and subtract the exponents.

Using these rules makes working with very large or small numbers more manageable, especially in fields like environmental science, where measurements can vary greatly in scale.

Environmental Science

Environmental science involves the study of how various elements, including pollutants, impact the environment. In the context of our textbook problem, the concern is with the concentration of lead in a water supply. Knowing the safe levels of lead as determined by agencies like the Environmental Protection Agency (EPA) is crucial for protecting public health.

Measuring the concentration of contaminants in the environment typically involves small numbers that are best expressed in scientific notation. For example, contaminants like lead are often found at levels of milligrams per liter in water, and these levels can be critical when comparing against regulatory limits. In this case, the water was tested at a concentration of lead that was 100 times greater than the usual level; it's an exercise to ensure the student can apply both scientific notation and algebraic manipulation to solve for the new concentration realistically in an environmental science context.

Algebraic Manipulation

Algebraic manipulation is used to solve equations or modify expressions, utilizing various algebraic laws and properties. The process often includes operations such as multiplication, division, and exponentiation. In terms of our example problem from the textbook, after realizing the lead concentration in water increased by a factor of 100, the student must apply algebraic skills to adjust the original number accordingly.

• Starting with the normal level of lead \(1.5 \times 10^{-5}\) milligrams per liter, the increase by a factor of 100 involves algebraically multiplying the number by 100.
• The multiplication of \(1.5 \times 10^{-5}\) by 100 is an exercise in both simple arithmetic and exponents. You multiply the 1.5 (coefficient) by 100 (considered to be \(10^2\)) and adjust the exponent accordingly.

This is a straightforward example of using algebraic manipulation in a real-world scenario, teaching students how to apply mathematical concepts in practical situations.