Question

example 3 use factoring to solve a real-world problem

1. a picture inside a frame has an area of \\(375\text{ cm}^2\\). what is the width of the frame?

Explanation:

Define the variables

Let \(x\) represent the width of the frame in centimeters.

Express dimensions of the picture

The outer dimensions of the framed picture are \(30\text{ cm}\) by \(20\text{ cm}\).
Since the frame has a uniform width of \(x\) on all sides:

• The width of the inner picture is \(20 - 2x\).
• The height of the inner picture is \(30 - 2x\).

Set up the area equation

The area of the inner picture is given as \(375\text{ cm}^2\).
Using the formula for the area of a rectangle:

$$(30 - 2x)(20 - 2x) = 375$$

Solve the quadratic equation

Expand the left side of the equation:

$$600 - 60x - 40x + 4x^2 = 375$$

$$4x^2 - 100x + 600 = 375$$

Subtract \(375\) from both sides to set the equation to zero:

$$4x^2 - 100x + 225 = 0$$

Factor and find the valid solution

Factor the quadratic equation:

$$(2x - 5)(2x - 45) = 0$$

This gives two potential solutions:

$$2x - 5 = 0 \implies x = 2.5$$

$$2x - 45 = 0 \implies x = 22.5$$

Since the width of the frame cannot be larger than half of the total width (\(20\text{ cm}\)), \(x = 22.5\) is extraneous.
Thus, the width of the frame is \(2.5\text{ cm}\).

Answer:

1. A picture inside a frame has an area of \(375\text{ cm}^2\). What is the width of the frame? <blank>\(2.5\text{ cm}\)</blank>