Question

a rectangular painting measures 10 inches by 19 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 82 inches. determine the width of the frame. the width of the frame is inch(es).

Explanation:

Step1: Find the dimensions of the outer - rectangle

The length of the outer - rectangle is $19 + 2x$ inches and the width is $10+2x$ inches, where $x$ is the width of the frame.

Step2: Use the perimeter formula

The perimeter formula for a rectangle is $P = 2(l + w)$. Here, $P=82$ inches, $l = 19 + 2x$, and $w = 10 + 2x$. So, $82=2((19 + 2x)+(10 + 2x))$.

Step3: Simplify the equation

First, simplify the expression inside the parentheses: $(19 + 2x)+(10 + 2x)=29 + 4x$. Then the equation becomes $82 = 2(29 + 4x)$. Distribute the 2: $82=58 + 8x$.

Step4: Solve for $x$

Subtract 58 from both sides of the equation: $82−58=8x$, which gives $24 = 8x$. Divide both sides by 8: $x = 3$.

Answer:

3