Problem 47 - Precalculus Mathematics for Calculus

Question

Ali paints with watercolors on a sheet of paper 20 in. wide by 15 in. high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture? (Picture cant copy)

Step-by-Step Solution

Answer

The strip is 4 inches wide.

Step 1: Understand the problem

The problem states that Ali places a rectangular sheet (picture) measuring 20 in. by 15 in. on a mat. The mat shows a uniformly wide strip around the picture, and the total perimeter of this mat is 102 in. We need to find the width of the strip.

Step 2: Define variables

Let the width of the strip be represented by \( x \). This means the overall dimensions of the mat are \((20 + 2x)\) inches for the width and \((15 + 2x)\) inches for the height. The added \(2x\) accounts for the strip on both sides of the width and height.

Step 3: Set up the perimeter equation

The formula for the perimeter \( P \) of a rectangle is given by \( P = 2(l + w) \), where \( l \) and \( w \) are the length and width of the rectangle. Here, the perimeter equation becomes:\[102 = 2((20 + 2x) + (15 + 2x))\]

Step 4: Simplify the equation

First, simplify the expression inside the bracket:\((20 + 2x) + (15 + 2x) = 35 + 4x\). Then substitute this back into the perimeter formula:\[102 = 2(35 + 4x)\]

Step 5: Solve for the width of the strip \( x \)

Simplify and solve the equation:1. Distribute the 2: \[ 102 = 70 + 8x \]2. Subtract 70 from both sides: \[ 32 = 8x \]3. Divide both sides by 8: \[ x = 4 \] Thus, the width of the strip is 4 inches.

Key Concepts

Perimeter Calculation

When calculating the perimeter of a rectangle, you're essentially measuring the total distance around the outer edge of the shape. The formula used is a straightforward one:

• Perimeter, denoted as \( P \), is calculated by \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width.
• In the problem, the mat has a perimeter of 102 inches.

We first need to determine the dimensions of the mat by considering the strip around the picture. The dimensions of the mat are affected by the width of the strip \( x \) on all sides, adding \( 2x \) to both the length and the width of the original picture.
By using the perimeter equation with these dimensions, we can isolate and calculate the width of the strip. This helpful approach allows us to break down word problems into manageable mathematical equations.

Rectangular Geometry

Understanding the basics of rectangular geometry can simplify how we approach problems like this one. A rectangle is characterized by its opposite sides being equal in length and its four right angles.

• In the case of the exercise, the painting with the mat forms a larger rectangle around the painting.
• The mat around the painting is uniform, meaning the added strip is even along all the sides.

This understanding helps us realize why we add \( 2x \) to both the width and the length. The uniform strip surrounds the original rectangle (the painting), enhancing the dimensions consistently on all four sides.
By visualizing this, we're able to set proper dimensions for our perimeter calculation, which will subsequently enable us to solve correctly for the unknown variable. Grasping these key geometric principles ensures a solid foundation for tackling more complex problems involving rectangles.

Variable Representation

In algebra, variables are symbols we use to represent unknown values. In the word problem given, we need to find the width of the mat strip, which we represent using a variable:

• The width of the strip is represented by \( x \).
• Using \( x \), we express the total width and height of the mat as \( (20 + 2x) \) and \( (15 + 2x) \), respectively.

Expressing unknown dimensions with variables is crucial:

• It lets us set up equations to solve for these unknowns efficiently.
• It provides clarity and structure to complex word problems by breaking them down into identifiable parts.

Properly defining and using variables allows us to convert a word problem into a mathematical one, making it easier to manipulate and solve. Through solving, we determine \( x = 4 \), indicating that the strip is 4 inches wide.