Problem 15 - Introductory Algebra for College Students

Question

A rectangular lot whose perimeter is 320 feet is fenced along three sides. An expensive fencing along the lot's length costs 16 dollar per foot. An inexpensive fencing along the two side widths costs only 5 dollar per foot. The total cost of the fencing along the three sides comes to 2140 dollar What are the lot's dimensions?

Step-by-Step Solution

Answer

The lot's width is 60 feet and the length is 130 feet.

Step 1: Define Length and Width

Let’s denote the length of the rectangular lot as \(L\) (in feet) and the width as \(W\) (in feet).

Step 2: Develop Equations

Use the information given to set up two equations. Central to our problem is the formulation of these two equations from given information: 1) Perimeter information gives \(2L + W = 320\) feet, and 2) The total cost information gives \(16L + 5(2W) = 2140\) dollars.

Step 3: Solve Simultaneously

From the perimeter's equation, we express \(L\) in terms of \(W\) : \(L = (320 - W)/2\) . Substituting this into the cost's equation, solve for \(W\).

Step 4: Find the Width and Length

After that, substitute the obtained value of \(W\) into perimeter equation to find the length \(L\).

Step 5: Conclude

The values of \(W\) and \(L\) are the dimensions of the lot.

Key Concepts

Rectangular Lot

A rectangular lot is a common shape used in problems dealing with land or building space. These lots have two key dimensions: length and width. Both sides directly determine the lot's perimeter and area. The rectangular shape makes calculations simpler, which can be a reason for its frequent appearance in math problems.

• Length is the longer side of the rectangle, while width is the shorter.
• These measures defined are crucial as they play pivotal roles in determining perimeter and costs involved in fencing.

Grasping the basic layout of a rectangular lot helps in solving related word problems, where dimensions need to be calculated based on other given information. By understanding each provided detail and its role, problems become easier to manage.

Perimeter

Perimeter, in geometry, is the total distance around the edge of a figure. For a rectangle, this involves calculating the sum of all sides. For our rectangular lot, the formula for perimeter is given by:\[ 2L + 2W \] where \(L\) is the length and \(W\) is the width. In the case of our exercise, we see the perimeter stated as 320 feet. Since we are interested in fencing along three sides, only one length remains completely open.

• The information provided allows us to create equations using the known total perimeter.
• This step is vital for proceeding to determine individual dimensions.

Understanding perimeter not only helps in direct calculations but assists in setting up necessary equations to use effectively when solving related algebra word problems.

Cost Calculation

In many word problems involving rectangles, calculating cost becomes significant when additional factors are introduced, like varying materials or rates. For fencing our rectangular lot, two types of fencing costs are involved:

• The cost for fencing along the length is more expensive at 16 dollars per foot.
• The cost along the two widths is cheaper at 5 dollars per foot.

Our exercise sets the total expense to 2140 dollars. Using these rates helps in forming the second necessary equation to solve for the dimensions. Keep in mind that understanding the cost breakdown is essential when multiple materials or various pricing play a part, as it allows you to correctly allocate conditions in equations to solve the given problem.

Fencing Problem

Fencing problems often involve determining how much material is needed to enclose or partially enclose a space. This leads to considerations about cost, especially when different sides have different rates, as seen here. The key is to use known information, like perimeter or total cost, and solve for unknown dimensions.
To simplify:

• Identify which sides are to be fenced and their respective costs.
• Set up equations reflecting these constraints and conditions.
• Solve equations to uncover the dimensions or costs.

The approach in fencing problems is methodical, breaking down the whole process into manageable chunks. By acknowledging the constraints given, such as specific cost conditions or physical layout, you can solve these problems effectively using algebra.