Problem 43 - Algebra and Trigonometry

Question

Mary’s backyard vegetable garden measures 20 \(\mathrm{ft}\) by 30 \(\mathrm{ft}\) , so its area is \(20 \times 30=600 \mathrm{ft}^{2}\). She decides to make it longer, as shown in the figure, so that the area increases to \(A=20(30+x) .\) Which property of real numbers tells us that the new area can also be written \(A=600+20 x ?\)

Step-by-Step Solution

Answer

The distributive property of real numbers allows the expression to be written as \( A = 600 + 20x \).

Step 1: Understand the Expression

Initially, Mary's garden is represented by the expression for area as \( A = 20 \times 30 \). Mary plans to extend the garden's length by \( x \) feet, which changes the area to \( A = 20(30 + x) \).

Step 2: Distributive Property

By applying the distributive property of multiplication over addition, \( 20(30 + x) \) can be expanded to \( 20 \times 30 + 20 \times x \). This breaks down the expression to separate components.

Step 3: Calculate Each Part

When you calculate \( 20 \times 30 \), you get 600, and when you calculate \( 20 \times x \), it remains as \( 20x \). So, the expression \( 20(30 + x) \) simplifies to \( 600 + 20x \).

Step 4: Identify the Property

The transformation from \( 20(30 + x) \) to \( 600 + 20x \) is an application of the distributive property of real numbers, where \( a(b + c) = ab + ac \).

Key Concepts

Area Calculation

Calculating the area of a space is foundational to understanding how its size changes with adjustments. In Mary's backyard garden exercise, she initially measures her garden's area using the formula for rectangles, which is

• Length × Width

Thus, the initial area of Mary's garden is computed as \[A = 20 \times 30 = 600 \text{ ft}^2.\]

To visualize how the area is affected by her decision to extend her garden, it's important to remember that length and width dictate the overall space. When she adds the extension of length, the new area is represented by the expression \[A = 20(30 + x),\] where she's essentially recalculating the total space, taking into account the original 30 ft and the additional \(x\) ft.
This recalculation helps her understand how much extra area her garden will gain with the extension.

Extension of Length

In Mary's scenario, the extension of the garden's length is crucial. By specifying an additional \(x\) feet to the original 30 ft length, we focus on how extensions modify dimensions and, consequently, areas.
The new expression, \[A = 20(30 + x),\] captures this change succinctly, considering that each additional foot added increases both the perimeter and the area.
By recognizing this, we understand that even a small extension can have a significant impact, increasing the total area incrementally with each additional foot added, which is represented as the term \(20x\).
This simple addition reflects real-world decisions gardeners face when optimizing or altering space use for their gardening needs.

Real Numbers Property

The real numbers property, specifically the distributive property, plays a crucial role in reshaping expressions involving additions and multiplications. This property allows us to distribute multiplication over addition, represented as \[a(b + c) = ab + ac.\]

In Mary's garden, applying this property transforms the expression \[20(30 + x)\] into the expanded form \[600 + 20x.\]

This not only simplifies understanding how each piece of the expression affects the outcome but also demonstrates the power of algebra in real-world applications. By using the distributive property, complex alterations in size are broken down into manageable parts, making it easier to calculate and conceptualize changes in dimensions systematically.