Problem 73 - Prealgebra

Question

Kathy needs to create a rectangular garden plot covering 100 square meters \(\left(100 \mathrm{~m}^{2}\right)\). If the width of the plot is \(7.5\) meters, find the length of the plot correct to the nearest tenth of a meter.

Step-by-Step Solution

Answer

The length of the plot is approximately 13.3 meters.

Step 1: Understand the Formula for the Area of a Rectangle

The area \(A\) of a rectangle is calculated using the formula \(A = \text{length} \times \text{width}\). We are given the area \(A = 100\) square meters and the width \(w = 7.5\) meters. We need to find the length \(l\).

Step 2: Introduce the Equation

Using the given values, substitute them into the area formula: \(100 = l \times 7.5\). Our goal is to solve for \(l\).

Step 3: Solve for Length \(l\)

Rearrange the equation to find \(l\): \(l = \frac{100}{7.5}\).

Step 4: Calculate the Length

Perform the division to find the length: \(l = \frac{100}{7.5} \approx 13.3333\).

Step 5: Round the Length

Round the length to the nearest tenth of a meter: \(l \approx 13.3\).

Key Concepts

geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It involves studying the dimensions and relationships between various geometrical shapes. In the case of Kathy's rectangular garden, we apply the geometrical concept of area. To determine the area of a rectangle, you must multiply its length by its width. This is a fundamental principle in geometry and forms the basis for many practical applications, such as designing spaces or plots like a garden.

• The area formula is: \[ A = \text{length} \times \text{width} \]
• It helps us find how much space is covered by a shape on a flat surface.
• The units of area depend on the units used for length and width. If both are in meters, area is in square meters (\(m^2\)).

rectangular garden

A rectangular garden is simply a garden that is shaped like a rectangle. This particular shape is quite common because of its simplicity and the ease with which it can be measured and constructed. The essentials to understand about a rectangular garden include:

• Dimensions: To build or define a rectangular garden, you need to know its length and width. In Kathy's case, she had the width and needed to find the length using the area provided.
• Area: The area of the garden is the extent of space it covers, calculated in square units. For Kathy's garden, it was given as 100 square meters, and we used this measure to solve for the missing dimension (length).
• Shape Benefits: Rectangular gardens can be especially beneficial for evenly planting rows, placing pathways, or organizing different types of plants in specific sections.

mathematics concepts

In mathematics, understanding concepts such as area is crucial for figuring out real-world problems like those involving garden plots. This particular exercise required applying mathematical reasoning and basic algebra to deduce the unknown value of the garden's length.

• Equations: Using equations helps to find unknown variables. For example, Kathy used the equation \(100 = l \times 7.5\) to find the unknown length \(l\).
• Division: Once you have set up the equation, mathematical operations like division help solve it. Here, dividing both sides of the equation by 7.5 isolated \(l\).
• Rounding: Mathematics often requires you to round answers to a practical degree of precision—in this case, to a tenth of a meter. Rounding ensures results are easy to use and interpret.

These concepts not only help in calculations but also build foundational skills for more complex mathematical problems.