Problem 79 - College Algebra

Question

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280\). What are the lot's dimensions?

Step-by-Step Solution

Answer

Length of the lot is 120 feet and its width is 30 feet.

Step 1: Formulate the first equation

Taking the length of the lot to be \( L \) and width to be \( W \), we see that \( L + 2W = 180 \). This equation comes from the fact that the total length of the fencing (which is half the perimeter since only three sides are fenced) is equal to the length of the lot plus twice the width of the lot.

Step 2: Formulate the second equation

From the cost of the fence, we see that \( 20L + 8W*2 = 3280 \). This is because the expensive fencing (costing \( \$20 \) per foot) is used for length and inexpensive fencing (costing \( \$8 \) per foot) is used for the two widths.

Step 3: Solve the equations

Substitute eqn 1 into eqn 2 to get the value of \( L \) or \( W \) and then substitute this value back into eqn 1 to get the other value.

Key Concepts

Perimeter Problems

When faced with perimeter problems, especially in the context of systems of linear equations, the goal is to determine unknown dimensions of shapes based on given parameters. Here, we have a rectangular lot with only three of its sides fenced, and this plays a crucial role in setting up our equations. For a rectangle, the perimeter is calculated based on the sum of all its sides. If the entire perimeter was required, we'd find it using the formula:

• Perimeter = 2L + 2W

However, in this problem where only three sides are fenced (i.e., two widths and one length), the relevant equation becomes: \( L + 2W = 180 \)This reflects half of the given total perimeter because the full perimeter is 360 feet, and only three sides are considered. Understanding how to form this equation is key to solving problems involving perimeter.

Cost Problems

Cost problems often involve determining the expenses associated with different components of a project. In this task, we are calculating the cost of fencing along three sides of the rectangular lot. It involves two types of fencing, each with different costs per foot.

• Expensive fencing costs \( \\( 20 \) per foot along the lot's length.
• Inexpensive fencing costs \( \\) 8 \) per foot along the two widths.

The provided information gives us a second equation crucial for solving the problem: \( 20L + 16W = 3280 \)This equation is derived by multiplying the cost per foot by the respective lengths that are fenced. This constraint, coupled with the perimeter equation, allows us to understand and find the dimensions of the lot according to the budgetary constraint placed by the fencing costs.

Solving Linear Equations

Solving linear equations is a method used to find unknown values. Here, we use a system of linear equations formed from understanding perimeter and cost problems. Our task is to solve these equations for the lot's length and width.The first step involves using substitution or elimination methods. We already have our equations:

• \( L + 2W = 180 \)
• \( 20L + 16W = 3280 \)

Substitute the expression for one variable from one equation into the other. For example:- If \( L = 180 - 2W \) from the first equation, we plug this back into the second to find: \( 20(180 - 2W) + 16W = 3280 \) Once we solve for one variable, we can easily find the other. It’s crucial to set up and manipulate these equations correctly to solve for the unknown dimensions.

Rectangular Lot Dimensions

Rectangular lot dimensions refer to identifying the length and width of a rectangular area. For this problem, understanding the geometric and algebraic interpretation of the lot is crucial. From our calculations:

• Length, \( L \), is found by separating the cost associated with the expensive fencing from the total budget.
• Width, \( W \), utilizes the cheaper fencing rates along with the total cost and perimeter constraints to solve for its value.

These dimensions are not just numbers; they represent the actual physical space which is being measured and financially managed through careful application of linear equations. Determining these values ensures that the fencing budget is adhered to while accurately enclosing the lot as described.