Problem 46 - Algebra 2 and Trigonometry

Question

The radius of the surface of a circular pool is \(\left(2+\sqrt{x y^{5}}\right)\) meters. Express the area of the pool in simplest form.

Step-by-Step Solution

Answer

The area of the pool is \\(\pi (4 + 4\sqrt{xy^5} + xy^5)\\).

Step 1: Understanding the Problem

We need to find the area of a circle, and we know the formula for the area is given by \(A = \pi r^2\) where \(r\) is the radius of the circle. The radius in this problem is given as \(r = 2 + \sqrt{xy^5}\).

Step 2: Substituting the Radius

Substitute \(r = 2 + \sqrt{xy^5}\) into the area formula: \(A = \pi (2 + \sqrt{xy^5})^2\).

Step 3: Expanding the Expression

Expand \( (2 + \sqrt{xy^5})^2\) using the formula \( (a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = 2\) and \(b = \sqrt{xy^5}\).

Step 4: Calculating Each Term

Calculate each term of the expansion:- \(2^2 = 4\)- \(2 \cdot 2 \cdot \sqrt{xy^5} = 4\sqrt{xy^5}\)- \((\sqrt{xy^5})^2 = xy^5\).

Step 5: Combining the Terms

Combine the terms from the expansion:\[(2 + \sqrt{xy^5})^2 = 4 + 4\sqrt{xy^5} + xy^5.\]

Step 6: Expressing the Area

Substitute the combined expression back into the area formula:\[A = \pi (4 + 4\sqrt{xy^5} + xy^5).\]This is the area of the pool expressed in simplest form.

Key Concepts

Algebraic Expressions

When dealing with algebraic expressions, you're working with combinations of numbers, variables, and operators such as addition and multiplication. In the world of algebra, expressions are representations of values that can change depending on the numbers you select for the variables involved.
In this context, the expression \(2 + \sqrt{xy^5}\) represents the radius of a circular pool. An algebraic expression can include constants, variables, and arithmetic operations:

• **Constants**: Numbers that do not change (not influenced by variables, e.g., 2).
• **Variables**: Symbols representing unknown numbers (e.g., \(x\) and \(y\)).
• **Operations**: Actions like addition (\(+\)), subtraction (\(-\)), multiplication (\(\times\)), and division (\(\div\)).

The task is to manipulate and use these expressions to solve problems. Simplifying these expressions can help in understanding and finding solutions.

Expanding Binomials

Understanding binomials is crucial in algebra. A binomial is an expression containing two terms. To expand a binomial means to multiply it out into a longer expression. The method commonly used for expansion is known as the **Distributive Property** or more specifically in this context, the FOIL (First, Outer, Inner, Last) method when multiplying two-binomial expressions.
When given the expression \((a + b)^2\), you can apply the identity: \[(a+b)^2 = a^2 + 2ab + b^2\] Here, the binomial given is \((2 + \sqrt{xy^5})\). To expand it, substitute into the identity:

• \(a = 2\), so \(a^2 = 4\).
• \(b = \sqrt{xy^5}\), so \(b^2 = xy^5\).
• The middle term is \(2ab = 4\sqrt{xy^5}\).

Combine these results to provide a final expression of the expanded binomial: \[4 + 4\sqrt{xy^5} + xy^5\] By expanding binomials, you are effectively applying distribution across each combined term.

Exponents

Exponents or powers are used in mathematics to express repeated multiplication of a number by itself. The notation consists of a base and an exponent; for example, in \(y^5\), \(y\) is the base, and 5 is the exponent, meaning \(y\times y\times y\times y\times y\).
Exponents signify how many times the base is used as a factor. They can be integrated into more complex algebraic expressions but follow specific rules and properties such as:

• **Product of Powers**: \(x^m \times x^n = x^{m+n}\)
• **Quotient of Powers**: \(x^m \div x^n = x^{m-n}\)
• **Power of a Power**: \((x^m)^n = x^{m \times n}\)

In the expression \((\sqrt{xy^5})^2\), we're dealing with a situation where an exponent is applied to a radical expression. Here, the square of \(\sqrt{xy^5}\) returns us to the base: \[\sqrt{xy^5} \times \sqrt{xy^5} = xy^5\] This shows how exponent rules simplify complications arising from radicals or square roots, an essential step when working with complex expressions.