Problem 67 - Algebra A Combined Function

Question

In psychology, it has been suggested that the number S of nonsense syllables that a person can repeat consecutively depends on his or her IQ score I according to the equation \(S=2 \sqrt{I}-9\). Use this relationship to estimate the IQ of a person who can repeat 11 nonsense syllables consecutively.

Step-by-Step Solution

Answer

The estimated IQ is 100.

Step 1: Understanding the Problem

We are given the equation \(S = 2\sqrt{I} - 9\), which describes the relationship between IQ \(I\) and the number of nonsense syllables \(S\) a person can repeat. We need to find the IQ \(I\) when \(S = 11\).

Step 2: Setting the Equation

Substitute \(S = 11\) into the equation. This gives us: \(11 = 2\sqrt{I} - 9\).

Step 3: Isolating the Square Root Term

Rearrange the equation to isolate the square root term: \(11 + 9 = 2\sqrt{I}\). Simplify to obtain: \(20 = 2\sqrt{I}\).

Step 4: Solving for the Square Root

Divide both sides by 2 to solve for \(\sqrt{I}\): \(\sqrt{I} = \frac{20}{2}\). Simplify this to: \(\sqrt{I} = 10\).

Step 5: Solving for IQ

Square both sides of the equation to solve for \(I\): \(I = 10^2\). Therefore, \(I = 100\).

Step 6: Verifying the Solution

To verify, substitute \(I = 100\) back into the original equation: \(S = 2\sqrt{100} - 9 = 20 - 9 = 11\), which confirms our solution is correct.

Key Concepts

Problem Solving

Problem solving is a crucial skill that we rely on in various situations. In this context, we were presented with an algebraic equation that models a psychological phenomenon. The equation \(S = 2 \sqrt{I} - 9\) allows us to understand how the number of nonsense syllables \(S\) a person can repeat relates to their IQ score \(I\). When tackling such problems:

• Always begin by understanding the given data and what is required to find. For our exercise, we needed to determine an unknown IQ score \(I\) based on the known number of syllables \(S = 11\).
• Insert known values into the equation. This step helps in breaking down complex problems into manageable parts.
• Rearrange the equation to isolate the needed variable. This involves basic algebraic manipulations, like moving terms from one side of the equation to the other.
• Solve step-by-step. Make sure to simplify each step as much as possible, which was shown when we simplified \(20 = 2 \sqrt{I}\) into \( \sqrt{I} = 10\).
• Verify the result by substituting back into the original equation to ensure accuracy.

Approaching problem solving methodically ensures that even complex algebraic equations become clear puzzles to solve rather than overwhelming challenges.

Square Roots

Square roots are a fundamental part of algebraic procedures. In our exercise, the key operation was solving for the square root term \( \sqrt{I} \) due to its appearance in the equation. Here's how to effectively work with square roots:

• When an equation involves a square root, focus on isolating that part first to make the solution process smoother. In our problem, rearranging \(11 + 9 = 2 \sqrt{I}\) allowed us to conveniently solve for \( \sqrt{I} \).
• Recognize that numbers can be expressed in terms of their square roots, like \(10 = \sqrt{100}\), which helps in solving for the unknown.
• Squaring both sides of an equation can eliminate the square root, which is a crucial step when solving for the variable inside it. Doing this step correctly is essential, as any mistake can lead to an inaccurate solution.
• Once squared, always confirm the operations with the original equation to maintain accuracy, avoiding any oversight.

By understanding and manipulating square roots correctly, you can simplify and solve complex problems effortlessly.

Psychology Applications in Math

Math isn't just about numbers and abstract figures. It often intersects with fields like psychology to explain human behavior, as illustrated in our exercise. Here's how math finds applications in psychology:

• Mathematical models help psychologists quantify and predict behaviors, like the relation between IQ scores and memory tasks. This exercise establishes an estimation model for nonsense syllable recall.
• Equations can decipher relationships between different psychological variables, providing insights into cognitive processes that are difficult to measure through observation alone.
• Such models allow researchers to propose hypotheses and test them with empirical data, thereby advancing psychological theories.
• Understanding these mathematical applications empowers one to appreciate the intricate connections between cognition and quantifiable data.

Being familiar with math's applications in psychology enables a richer understanding of cognitive phenomena and highlights the synergy between these two fields.