Claire is making a frame to display a drawing she made. The frame is 1/5 cm wide. The frame has a rectangular shape and an area of 560 cm². How tall is the frame?

Understand the Problem

The question is asking to find the height of a rectangular frame given its width, area, and the addition of the frame surrounding the drawing. We will need to apply the formula for the area of a rectangle and account for the width of the frame to solve for the height.

Answer

$$ h = \frac{A - 2wf - 4f^2}{2f} $$

Steps to Solve

Identify the given information

Let's denote the width of the drawing as $w$, the height of the drawing as $h$, the width of the frame as $f$, and the total area of the rectangle as $A$. We know the following from the problem statement:

Total area, $A = \text{(width of drawing + 2 * width of frame) * (height of drawing + 2 * width of frame)} = (w + 2f)(h + 2f)$

Set up the equation for the area

We can rewrite the area equation using the known total area: $$ A = (w + 2f)(h + 2f) $$

Expand the equation

Now we need to expand the expression to isolate $h$: $$ A = wh + 2wf + 2hf + 4f^2 $$

Rearranging the equation for height

To find the height $h$, we will rearrange the equation: $$ h = \frac{A - 2wf - 4f^2}{2f} $$

Plug in the known values

Now we can substitute the known values of $A$, $w$, and $f$ into the equation to find the height $h$.

The height of the rectangular frame is given by the formula: $$ h = \frac{A - 2wf - 4f^2}{2f} $$

More Information

This formula allows you to find the height of a rectangular drawing when you know the total area, the width of the drawing, and the width of the frame. The presence of the frame essentially adjusts the dimensions for the area calculation.

Tips

Forgetting to include both the width of the frame on each side of the drawing can lead to incorrect area calculations.
Confusing the frame dimensions with the dimensions of the drawing itself.

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