Problem 29 - Elementary Algebra

Question

Use numerical evaluation to evaluate the equations for the following problems. $$ E=m c^{2} \text { . Find } E \text { if } m=120 \text { and } c=186,000 \text { . } $$

Step-by-Step Solution

Answer

Answer: The value of E is 4.15152 × 10^12 energy units.

Step 1: Write down the given equation and values

We are given the equation: $$ E = mc^{2} $$ And the values: $$ m = 120 \\ c = 186,000 $$

Step 2: Substitute the given values into the equation

Replace the variables m and c in the equation with their respective values: $$ E = (120)(186,000)^{2} $$

Step 3: Evaluate the equation

Perform the calculations: $$ E = 120 * (186,000)^{2} \\ E = 120 * (3.4596 * 10^{10}) \\ E = 4.15152 * 10^{12} $$

Step 4: Write the final answer

The value of E is: $$ E = 4.15152 \times 10^{12} \text{ (energy units)} $$

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a concise format. This method is useful in fields like physics and chemistry, where such numbers frequently occur. The format uses terms of a base number and an exponent base 10. For instance, the number 3,000,000 can be expressed as \(3 \times 10^6\).

• The base number is typically a value between 1 and 10. In our example, that's 3.
• The exponent indicates how many times the base number must be multiplied by 10. For 3,000,000, it's 6.

This approach simplifies calculations, especially when dealing with powers and scientific equations. It allows us to perform multiplications and divisions more easily.

Substitution Method

The substitution method involves replacing variables in an equation with specific values, allowing you to simplify and solve it. This technique is vital when solving practical problems in mathematics and science.
By substituting the known values into the equation, you can transform a general expression into a specific one:

• Identify the variables that need replacement. In our case, \(m = 120\) and \(c = 186,000\).
• Replace those variables with the given numbers in the equation \(E = m c^2\), leading to \(E = (120)\times(186,000)^2\).

Once the variables are substituted, you can proceed with the calculation to find the solution.

Physics Equation

Physics equations describe the relationships between different physical quantities. They often involve constants like the speed of light \(c\), used to relate mass and energy in Einstein’s famous equation \(E = mc^2\).
These equations help us explore fundamental concepts of physics and allow predictions about how systems behave. The energy equation \(E = mc^2\):

• Expresses the concept that mass can be converted into energy.
• Shows the direct proportionality between mass \(m\) and the square of the speed of light \(c^2\).

Understanding how to work with such equations involves recognizing how to translate physical concepts into mathematical formulations.

Mathematics Problem Solving

Mathematical problem solving involves a series of logical steps to reach a solution. This process can include:

• Identifying the problem's essential parts.
• Deciding on the most effective techniques to apply, such as substitution or transformation of equations.
• Carefully executing calculations, often using scientific notation for accuracy and simplicity.

For our problem, we substituted values into \(E = mc^2\), calculated \((186,000)^2\), and finally multiplied by 120. The systematic evaluation using substitution and power calculations led us to the numerical result \(E = 4.15152 \times 10^{12}\). Using these steps ensures clear reasoning and precise results.