Problem 68 - 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 15 nonsense syllables consecutively.

Step-by-Step Solution

Answer

The estimated IQ of the person is 144.

Step 1: Identify Given Values

We are given that the number of syllables the person can repeat, \(S\), is 15. We need to find the IQ score, \(I\), using the given equation, \(S = 2 \sqrt{I} - 9\).

Step 2: Substitute and Rearrange Equation

Substitute \(S = 15\) into the equation to get:\[15 = 2 \sqrt{I} - 9\]Rearrange the equation to solve for \(\sqrt{I}\):\[15 + 9 = 2 \sqrt{I}\]\[24 = 2 \sqrt{I}\]

Step 3: Isolate the Square Root

Divide both sides of the equation by 2 to isolate \(\sqrt{I}\):\[\sqrt{I} = \frac{24}{2}\]\[\sqrt{I} = 12\]

Step 4: Solve for IQ Score I

To find \(I\), square both sides of the equation:\[I = 12^2\]\[I = 144\]

Step 5: Verify the Solution

Substitute \(I = 144\) back into the original equation to ensure it satisfies the condition:\[S = 2 \sqrt{144} - 9\]\[S = 2 imes 12 - 9\]\[S = 24 - 9\]\[S = 15\]The solution is verified as correct.

Key Concepts

Square Roots

Square roots are fundamental components in mathematics. They represent one of two equal factors of a number. For instance, the square root of 36 is 6, because

• 6 times 6 equals 36

In solving algebraic equations, understanding square roots is crucial, especially when the relationship involves quadratic expressions.
In our given problem, the formula for the number of nonsense syllables is based on a square root, \[ S = 2 \sqrt{I} - 9 \].
Here, \( \sqrt{I} \) denotes the square root of the IQ score.
When dealing with equations involving a square root, the goal is often to solve for the variable inside the root, which requires isolating it first.
Once isolated, squaring both sides of the equation helps remove the square root.

Substitution Method

The substitution method in algebra involves replacing a variable with a given value or another expression.
This approach is highly useful when solving equations, as it helps simplify the problem or check potential solutions.
In the context of our problem, we are given that the person can repeat 15 syllables, effectively setting \( S = 15 \).
By substituting this known value into the equation

• \( S = 2\sqrt{I} - 9 \)

we get \[ 15 = 2\sqrt{I} - 9 \]
This substitution sets the stage for rearranging and solving the equation to find the IQ score. Through substitution, the unknown variable can be systematically uncovered.

Rearranging Equations

Rearranging equations is a central skill in solving algebraic problems. It involves changing the position or form of an equation's components to isolate the desired variable.
In our example, after substituting \( S = 15 \), we arrived at \[ 15 = 2 \sqrt{I} - 9 \].
To solve for \( \sqrt{I} \), we first add 9 to both sides, leading to \[ 15 + 9 = 2\sqrt{I} \]
which simplifies to \[ 24 = 2\sqrt{I} \]
Further rearranging by dividing both sides by 2 yields \[ \sqrt{I} = 12 \].
Rearranging ensures that we systematically tackle the problem until the variable of interest is isolated.

Verification of Solutions

After finding a solution, verifying it is the next crucial step. Verification ensures that the solution satisfies the original conditions of a problem.
For the given equation, once we determined that \( I = 144 \), it's essential to check our work.
Substituting back into the original formula

• \( S = 2 \sqrt{144} - 9 \)

leads to \[ S = 2 \times 12 - 9 \]
which simplifies to \[ S = 24 - 9 \]
and finally \[ S = 15 \].
Since this matches the given condition, our solution is verified as correct.
Verification prevents errors and confirms accuracy, adding confidence to problem-solving efforts.