Problem 56 - Algebra 2

Question

Geometry A rectangular shelf is \(\sqrt{440} \mathrm{cm}\) by \(\sqrt{20} \mathrm{cm} .\) Find its area.

Step-by-Step Solution

Answer

The area of the rectangular shelf is \(20*\sqrt{22} cm²\)

Step 1: Identify the values for length and width

The given lengths are \( \sqrt{440} \mathrm{cm} \) for length and \( \sqrt{20} \mathrm{cm} \) for width.

Step 2: Simplify the square roots

We simplify \( \sqrt{440} \) and \( \sqrt{20} \) using the property of square roots that states that : \( \sqrt{a*b}=\sqrt{a}*\sqrt{b} \). So, \( \sqrt{440} = \sqrt{4*110} = \sqrt{4}*\sqrt{110} = 2*\sqrt{110} \) and \( \sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2*\sqrt{5} \).

Step 3: Apply the formula for the area of a rectangle

Area of a rectangle is Length x Width. Substituting the provided values, we get 'Area = 2*\sqrt{110} cm * 2*\sqrt{5} cm', which simplifies to 'Area = 4*\sqrt{550} cm²'.

Step 4: Simplify the expression

We further simplify \( \sqrt{550} = \sqrt{50*11} = \sqrt{50}*\sqrt{11} = 5*\sqrt{22} \), so the final answer is 'Area = 4*5*\sqrt{22} cm² = 20*\sqrt{22} cm²'

Key Concepts

Square Roots

The concept of square roots can sometimes puzzle students, but it’s fundamental in Geometry Area Calculation. A square root simply refers to a number which when multiplied by itself results in the given number. For example, the square root of 9 is 3 because 3 multiplied by 3 equals 9. In our exercise, the lengths of the rectangle are given in terms of square roots: \( \sqrt{440} \) cm and \( \sqrt{20} \) cm. Understanding square roots helps in various mathematical operations, especially when calculating areas or performing other geometric tasks. When dealing with square roots, the first step is often to simplify them, which leads us to our next topic.

Simplifying Radicals

Radicals, such as square roots, can be simplified to make calculations much easier. Simplifying radicals involves breaking down the number into its prime factors to simplify the square root. In the exercise, we were given \( \sqrt{440} \) and \( \sqrt{20} \), both of which can be simplified using factorization. - For \( \sqrt{440} \), notice that 440 can be written as \( 4 \times 110 \). Therefore, \( \sqrt{440} = \sqrt{4 \times 110} = \sqrt{4} \times \sqrt{110} \).- Since the square root of 4 is 2, we end up with \( 2 \times \sqrt{110} \).Similarly, for \( \sqrt{20} \):- 20 equals \( 4 \times 5 \), so \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} \).- This simplifies to \( 2 \times \sqrt{5} \).By simplifying radicals, we make the numbers easier to work with in further calculations, such as finding the area of a rectangle.

Area of Rectangle

Finding the area of a rectangle is a straightforward process; it’s simply the product of its length and width. This is often written as: \[ \text{Area} = \text{Length} \times \text{Width} \]In the case of our rectangle, we took our simplified dimensions \( 2 \times \sqrt{110} \) cm for the length and \( 2 \times \sqrt{5} \) cm for the width. After plugging these numbers into the formula, the calculation becomes:\[ \text{Area} = (2 \times \sqrt{110}) \times (2 \times \sqrt{5}) \]This simplifies to \( 4 \times \sqrt{550} \) cm². Upon further simplification of \( \sqrt{550} = 5 \times \sqrt{22} \), the final area we get is:\[ \text{Area} = 20 \times \sqrt{22} \text{ cm}^2 \]Understanding each component—how to manage square roots, simplify them, and apply them in area formulas—is crucial. This ensures that students grasp these important geometric and arithmetic concepts.