Problem 10 - A Graphical Approach to College Algebra

Question

Solve each problem. Area of a Picture The mat around the picture shown measures \(x\) inches across. Which equation says that the area of the picture itself is 600 square inches? A. \(2(34-2 x)+2(21-2 x)=600\) $$\text { B. }(34-2 x)(21-2 x)=600$$ C. \((34-x)(21-x)=600\) D. \(x(34)(21)=600\) (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

B: \((34-2x)(21-2x)=600\)

Step 1: Understanding the Problem

The task is asking for an equation that correctly represents the area of a picture within a mat. The mat is described as being 'x inches across.' Our goal is to identify the equation that states the picture's area is 600 square inches.

Step 2: Identifying the Picture's Dimensions

First, we need to determine the dimensions of the picture itself. Since the mat is described as being 'x inches across,' the mat surrounds the entire picture. Therefore, for the picture, we subtract twice the width of the mat from each dimension: Width = 34 - 2x, Height = 21 - 2x.

Step 3: Calculating the Area of the Picture

The area of a rectangle (the picture) is calculated by multiplying its length by its width. Therefore, the area of the picture is given by the formula: \[ (34 - 2x)(21 - 2x) \]

Step 4: Matching the Area with the Given Choices

Given that the area of the picture is stated to be 600 square inches, we equate our area expression to 600: \[ (34 - 2x)(21 - 2x) = 600 \] This matches option B.

Key Concepts

Equation Formulation

When confronted with a word problem involving geometry, such as determining the area of a picture surrounded by a mat, translating the text into a mathematical equation is a crucial step. This process, known as equation formulation, involves identifying the relevant mathematical relationships described in the problem. In our exercise, we must determine an equation correct for the area of the picture.

The critical detail here is that the mat measures 'x inches across,' which means it extends outward from all sides of the picture. This requires adjustments to the picture's dimensions to account for the mat on all four sides. If the outer dimensions given are 34 inches by 21 inches, we need to subtract twice the width of the mat (2x) from each dimension to find the actual dimensions of the picture.

Hence, the picture's width becomes \( 34 - 2x \) and its height is \( 21 - 2x \). With these adjustments, the expression for the area is determined as \( (34 - 2x)(21 - 2x) = 600 \), which matches option B in the original problem.

Area of Rectangles

Understanding how to calculate the area of rectangles is essential in solving geometry-related algebra problems. The basic formula for the area of a rectangle is to multiply its width by its height. This formula is pivotal in analyzing and solving the given problem about the picture's area surrounded by a mat.

In this specific problem, once the correct dimensions of the picture are established, as discussed in the equation formulation section, you can apply the area formula of rectangles. The dimensions calculated are \( 34 - 2x \) for the width and \( 21 - 2x \) for the height, resulting in the area expression:

• Width: \( 34 - 2x \)
• Height: \( 21 - 2x \)

By multiplying these values together, the expression \( (34 - 2x)(21 - 2x) \) is generated. Equating this to 600 square inches confirms our problem's requirement that this product equals exactly 600.

Dimension Reduction

Dimension reduction is a crucial concept in understanding geometry problems involving perimeter constraints or additional features, like mats or frames. This concept involves altering standard lengths by subtracting or adding certain measurements to account for extra elements.

In the context of this geometry problem, dimension reduction helps understand how the mat alters the space of the picture. Without considering this adjustment, one might incorrectly calculate the picture's area. Here, because the mat's width is \( x \) inches across on all sides, we reduce the width and height of the entire outer rectangle by \( 2x \) inches each to get the dimensions of just the picture.

This step is vital to ensure accurate area calculations. The given problem outlines that by reducing the dimensions by twice the width of the mat, the equations reflect only the picture area, lending to a correct equation of \( (34 - 2x)(21 - 2x) = 600 \). It is a good practice to visualize and double-check each dimension manipulated, particularly in multi-step problems involving geometric constraints.