Problem 109
Question
The algebraic expression \(0.07 d^{3 / 2}\) describes the duration of a storm, in hours, whose diameter is \(d\) miles. Evaluate the algebraic expression for \(d=9\) Describe what the answer means in practical terms.
Step-by-Step Solution
Verified Answer
A storm with a diameter of 9 miles is expected to last approximately 2.83 hours.
1Step 1: Substitution and Calculation
Substitute \(d=9\) into the expression \(0.07 d^{3 / 2}\). Calculate \(0.07 * 9^{3/2}\), yielding approximately \(2.83\) hours.
2Step 2: Interpretation
The number approximated from the calculation represents the duration of a storm in hours. In this case, a storm with a diameter of 9 miles is expected to last approximately 2.83 hours.
Key Concepts
Substitution in AlgebraExponents and RootsPractical Interpretation of Mathematical Results
Substitution in Algebra
When working with algebraic expressions, substitution is a technique used to evaluate the expression for given values. It involves replacing the variable in the expression with a specific number or another expression. Substitution is particularly useful because it can simplify complex problems and make them more manageable.
For example, the expression in the exercise, 0.07 d^{3 / 2}, represents the duration of a storm when we know its diameter d. To evaluate this expression for d=9, simply substitute 9 for every instance of d in the expression, resulting in 0.07*9^{3/2}. This allows us to compute the value using known arithmetic operations.
Remember, the key steps in substitution are identifying the variable, replacing it with the given value, and then performing the arithmetic operation to find the result. Substitution is practical and widely used, not just in algebra, but in various branches of mathematics.
For example, the expression in the exercise, 0.07 d^{3 / 2}, represents the duration of a storm when we know its diameter d. To evaluate this expression for d=9, simply substitute 9 for every instance of d in the expression, resulting in 0.07*9^{3/2}. This allows us to compute the value using known arithmetic operations.
Remember, the key steps in substitution are identifying the variable, replacing it with the given value, and then performing the arithmetic operation to find the result. Substitution is practical and widely used, not just in algebra, but in various branches of mathematics.
Exponents and Roots
Exponents and roots are fundamental concepts in mathematics, expressing how many times a number, known as the base, is multiplied by itself. An expression like d^{3/2} combines both an exponent and a root. The exponent 3/2 tells us the base number d is raised to the power of 3 and then the square root is taken because of the fraction.
In our exercise, we deal with 9^{3/2}, which means we need to cube 9 (multiply it by itself twice) and then take the square root. Doing this, we find the cube of 9 is 729, and the square root of 729 is 27. Therefore, 9^{3/2} simplifies to 27. Exponents and roots are not just abstract numbers; they can represent real-world quantities, such as the time forecasted for a storm based on its diameter, illustrating the power of algebra in interpreting natural phenomena.
In our exercise, we deal with 9^{3/2}, which means we need to cube 9 (multiply it by itself twice) and then take the square root. Doing this, we find the cube of 9 is 729, and the square root of 729 is 27. Therefore, 9^{3/2} simplifies to 27. Exponents and roots are not just abstract numbers; they can represent real-world quantities, such as the time forecasted for a storm based on its diameter, illustrating the power of algebra in interpreting natural phenomena.
Practical Interpretation of Mathematical Results
Interpreting mathematical results in practical terms connects numbers and formulas to real-world situations. After we calculate the value of the algebraic expression, it is crucial to understand what this value represents.
In the context of our exercise, the result of approximately 2.83 is not just a number; it predicts how long, in hours, a storm with a diameter of 9 miles will last. This information could be essential for emergency services, event planning, or informing the public about potential weather conditions. Being able to interpret mathematical results can lead to informed decision-making in various sectors such as meteorology, engineering, and finance.
Understanding the underlying meaning of the values calculated from algebraic expressions is just as important as the ability to compute them. This relevance of mathematics to daily life and professions highlights its significance beyond the classroom.
In the context of our exercise, the result of approximately 2.83 is not just a number; it predicts how long, in hours, a storm with a diameter of 9 miles will last. This information could be essential for emergency services, event planning, or informing the public about potential weather conditions. Being able to interpret mathematical results can lead to informed decision-making in various sectors such as meteorology, engineering, and finance.
Understanding the underlying meaning of the values calculated from algebraic expressions is just as important as the ability to compute them. This relevance of mathematics to daily life and professions highlights its significance beyond the classroom.
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