Problem 46 - College Algebra and Calculus: An Applied Approach

Question

Geometry A picture frame has a total perimeter of 3 feet (see figure). The width of the frame is \(0.62\) times its length. Find the dimensions of the frame.

Step-by-Step Solution

Answer

The dimensions of the frame after solving the equations are: length = 0.93 feet and width = 0.57 feet.

Step 1: Understand the problem

We are asked to find the dimensions of a frame, where we know the total perimeter and the relationship between the length and the width of the frame. The picture frame is rectangular. The problem states that the width is \(0.62\) times the length and the total perimeter is 3 feet.

Step 2: Formulate the equations

Knowing that the perimeter of a rectangle is \(2(length + width)\), we can write the following equation: \(2(length + width) = 3\). Given that the width is \(0.62\) times the length we obtain another equation: \(width = 0.62 \times length\). So now we have a system of two equations.

Step 3: Solve the equation for one of the variables

Since we know that the width is \(0.62 \times length\), we substitute width in the perimeter equation and we get: \(2(length + 0.62 \times length) = 3\), that simplifies to \(2 \times 1.62 \times length = 3\), then solve for the length by dividing both sides of the equation by \(3.24\) to get \(length = \frac{3}{3.24}\).

Step 4: Solve the equation for the other variable

Substitute the found length in the width's equation to find the value of width, i.e., \(width = 0.62 \times length\).

Key Concepts

Rectangular Perimeter

Understanding the concept of rectangular perimeter is a key part of solving geometric problems, especially when dealing with shapes like rectangles and picture frames. The perimeter of a rectangle is the total distance around the outside of the rectangle. To calculate it, you simply add up the lengths of all the sides. Since a rectangle has two pairs of equal sides, the formula for finding the perimeter is:

• Perimeter = 2 × (length + width)

This formula derives from the fact that you have two lengths and two widths in a rectangle. Knowing the perimeter can help you find unknown dimensions if you have additional information.
For example, if you know the perimeter and the relationship between the length and width, like in the picture frame problem, you can use these to find both dimensions.

Algebraic Equations

Algebraic equations are essential tools for solving a variety of mathematical problems. In the context of geometry, they allow us to express relationships between different quantities and dimensions. There are a few key steps involved in working with algebraic equations:

• Formulate equations based on problem statements.
• Substitute known values or relationships into the equations.
• Solve for unknown variables using algebraic manipulation.

In the picture frame exercise, we established two algebraic equations: one for the perimeter and another for the relationship between width and length. By substituting the second equation into the first, we reduced the number of variables and made it possible to solve the problem systematically. This illustrates how algebra helps us systematically approach math problems.

Problem-solving in Mathematics

Problem-solving is at the heart of mathematics education and involves a set of approaches and strategies. When confronting a problem like finding dimensions of a picture frame, effective problem-solving involves:

• Understanding the problem thoroughly.
• Identifying the relevant mathematical concepts and relationships.
• Setting up equations or calculations that reflect these relationships.

Throughout this process, logical thinking and strategic planning are vital. In particular, breaking down the problem into manageable steps and checking each step avoids errors and ensures the solution's accuracy. By first understanding the condition of the width, formulating equations, and then solving them, students engage in a complete problem-solving cycle. Once the solution is reached, it is important to verify if the results make sense in the context of the problem, for thorough understanding and learning.