Problem 29 - Algebra and Trigonometry

Question

The rectangular painting in the figure shown measures 12 inches by 16 inches and is surrounded by a frame of uniform width around the four edges. The perimeter of the rectangle formed by the painting and its frame is 72 inches. Determine the width of the frame.

Step-by-Step Solution

Answer

The width of the frame is \( x = \frac{72}{8} - 7 = 2 \) inches.

Step 1: Understand the Problem

The painting is rectangular and measures 12 inches by 16 inches. It is surrounded by a frame of uniform width. The entire piece of art (painting and frame) has a perimeter of 72 inches. The goal is to find out the width of the frame.

Step 2: Set up the Equation

The perimeter of the entire artwork can be found using the formula for the perimeter of a rectangle, which is \( P = 2(l + w) \), where \( l \) and \( w \) are the length and width of the rectangle respectively. Since the frame is of uniform width all around the picture, it adds twice its width to the length and the width of the painting. Thus, the length becomes \( 16 + 2x \) and the width becomes \( 12 + 2x \), where \( x \) is the width of the frame. Therefore, the equation becomes \( P = 2[(16 + 2x) + (12 + 2x)] \). Given that \( P = 72 \) inches, we substitute to solve for \( x \).

Step 3: Solve the Equation

Substitute P into the equation: \( 72 = 2[(16 + 2x) + (12 + 2x)] \). Simplify the equation: \( 72 = 2[16+12+4x] = 2[28+4x] = 2*4*[7+x] = 8*(7+x) \). Solve for \( x \) to give: \( x = \frac{72}{8} - 7 \).

Step 4: Evaluate the Solution

Find the value of \( x \) from the last step. Remember \( x \) represents the uniform frame width.