Problem 43 - Algebra 2 and Trigonometry

Question

The dimensions of a rectangle are 12\(\sqrt{2}\) feet by \(\sqrt{50}\) feet. Express the area of the rectangle in simplest form.

Step-by-Step Solution

Answer

The area of the rectangle is 120 square feet.

Step 1: Write the Formula for the Area

The area of a rectangle is calculated by multiplying its length by its width. The formula is:\[A = ext{length} imes ext{width}\]In this case, the length is \(12\sqrt{2}\) feet, and the width is \(\sqrt{50}\) feet.

Step 2: Plug in the Measurements

Insert the given dimensions into the area formula:\[A = 12\sqrt{2} \times \sqrt{50}\]This expression will give the area of the rectangle.

Step 3: Simplify the Expression Under the Square Roots

Identify and simplify the square root expressions:\(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\). So, the width simplifies to \(5\sqrt{2}\).

Step 4: Multiply the Expressions

Now, multiply the simplified expressions:\[A = 12\sqrt{2} \times 5\sqrt{2}\]This involves multiplying the coefficients and the radicals separately.

Step 5: Perform the Multiplication

Multiply the numerical coefficients: \(12 \times 5 = 60\).Regarding the radicals:\(\sqrt{2} \times \sqrt{2} = \sqrt{4} = 2\).Hence, the area becomes:\[A = 60 \times 2 = 120\]So, the area is 120 square feet.

Key Concepts

Multiplying Square Roots

One of the foundational ideas when working with rectangles and their areas, particularly with dimensions involving square roots, is understanding how to multiply square roots together. Square roots, often denoted using the radical sign (√), represent a value that, when multiplied by itself, results in the original number inside the square root. For example,

• the square root of 4, \(\sqrt{4}\), is 2 because 2 multiplied by itself equals 4.
• Similarly, \(\sqrt{9}\) is 3 because 3 times 3 equals 9.

When multiplying square roots together, the rule is that you can multiply the numbers inside the radicals. This means that for any two square roots, \(\sqrt{a}\) and \(\sqrt{b}\), the multiplication can be expressed as \(\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}\). In our exercise, \(\sqrt{50}\) can be simplified and multiplied with \(12\sqrt{2}\) to find the area of the rectangle. Remember, when multiplying square roots that have the same radicand (the number inside the radical), you simply remove the radical and proceed with arithmetic as usual.

Simplifying Radicals

Simplifying radicals involves rewriting a square root in its simplest form. A radical may initially look complicated, but it can often be simplified by finding perfect squares within the number under the square root. For instance, in our rectangle exercise, we wanted to simplify \(\sqrt{50}\). To do this:

• We look for the largest perfect square that's a factor of 50. In this case, 25 (since \(25 \times 2 = 50\)).
• Then, rewrite \(\sqrt{50}\) as \(\sqrt{25 \times 2}\).
• By using the property of square roots \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), it becomes \(\sqrt{25} \times \sqrt{2}\).
• Since \(\sqrt{25}\) equals 5, we simplify \(\sqrt{50}\) to \(5\sqrt{2}\).

Simplifying radicals helps in making the arithmetic operations easier to perform and results more interpretable.

Geometry Formulas

Geometry involves many formulas that help us calculate different properties of shapes. One of the most fundamental aspects of geometry is knowing how to work with the areas of different shapes. The area of a rectangle, for example, is determined by the formula \(A = \text{length} \times \text{width}\). In this formula:

• The 'length' and 'width' are those of the rectangle's sides.
• The formula is a simple multiplication between the two measurements.

Understanding and applying the formula correctly is crucial, especially when the dimensions might involve certain mathematical expressions like square roots. The focus should always be on correctly substituting the length and width values into the formula and then simplifying any complex expressions, like radicals in this exercise—a key step in calculating the area precisely. With these basics clear, you can confidently tackle problems involving the geometry formulas.