Question

a picture that measures 10 cm by 5 cm is to be surrounded by a matte before being framed. the width of the matte is to be the same on all sides of the picture. the area of the matte is to be twice the area of the picture. what is the width of the matte?

Explanation:

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<reasoning>

Define variables and find the picture area

Let \(x\) be the uniform width of the matte in centimeters, where \(x > 0\).
The dimensions of the picture are \(10\text{ cm}\) by \(5\text{ cm}\).
The area of the picture is:
\[
A_{\text{picture}} = 10 \times 5 = 50\text{ cm}^2
\]

Set up the total area equation

The matte surrounds the picture on all four sides, adding \(2x\) to both the length and the width.
The dimensions of the combined picture and matte are \((10 + 2x)\) and \((5 + 2x)\).
The area of the matte is given as twice the area of the picture:
\[
A_{\text{matte}} = 2 \times A_{\text{picture}} = 2 \times 50 = 100\text{ cm}^2
\]
The total area of the combined region is:
\[
A_{\text{total}} = A_{\text{picture}} + A_{\text{matte}} = 50 + 100 = 150\text{ cm}^2
\]
Therefore, we can write the equation:
\[
(10 + 2x)(5 + 2x) = 150
\]

Expand and simplify the quadratic equation

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &50 + 20x + 10x + 4x^2 = 150\\ &4x^2 + 30x + 50 = 150\\ &4x^2 + 30x - 100 = 0\\ &2x^2 + 15x - 50 = 0 \end{aligned}$$

\]

Solve the quadratic equation

Using the Quadratic Formula knowledge point
\[

$$\begin{aligned} &x = \frac{-15 \pm \sqrt{15^2 - 4(2)(-50)}}{2(2)}\\ &x = \frac{-15 \pm \sqrt{225 + 400}}{4}\\ &x = \frac{-15 \pm \sqrt{625}}{4}\\ &x = \frac{-15 \pm 25}{4} \end{aligned}$$

\]

Determine the valid width

Since the width must be positive, we discard the negative root:
\[
x = \frac{-15 + 25}{4} = \frac{10}{4} = 2.5\text{ cm}
\]
</reasoning>

<answer>
\(2.5\text{ cm}\)
</answer>

<plot>
{
"elements": [
{
"type": "rectangle",
"params": [[-5, -2.5], [5, 2.5]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2,
"fillColor": "rgba(88, 60, 135, 0.15)",
"name": "Picture"
}
},
{
"type": "rectangle",
"params": [[-7.5, -5], [7.5, 5]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"fillColor": "rgba(140, 85, 242, 0.25)",
"name": "Matte"
}
},
{
"type": "arrow",
"params": [[5, 0], [7.5, 0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"name": "x"
}
},
{
"type": "text",
"params": [6.25, 0.5, "\\[x\\]"],
"properties": {
"color": "#F2557F"
}
},
{
"type": "text",
"params": [0, 0, "\\[10 \times 5\\]"],
"properties": {
"color": "#583C87"
}
}
],
"timestamps": [0.1]
}
</plot>

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"Algebra",
"Solving Quadratic Equations"
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Answer:

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<reasoning>

Define variables and find the picture area

Let \(x\) be the uniform width of the matte in centimeters, where \(x > 0\).
The dimensions of the picture are \(10\text{ cm}\) by \(5\text{ cm}\).
The area of the picture is:
\[
A_{\text{picture}} = 10 \times 5 = 50\text{ cm}^2
\]

Set up the total area equation

The matte surrounds the picture on all four sides, adding \(2x\) to both the length and the width.
The dimensions of the combined picture and matte are \((10 + 2x)\) and \((5 + 2x)\).
The area of the matte is given as twice the area of the picture:
\[
A_{\text{matte}} = 2 \times A_{\text{picture}} = 2 \times 50 = 100\text{ cm}^2
\]
The total area of the combined region is:
\[
A_{\text{total}} = A_{\text{picture}} + A_{\text{matte}} = 50 + 100 = 150\text{ cm}^2
\]
Therefore, we can write the equation:
\[
(10 + 2x)(5 + 2x) = 150
\]

Expand and simplify the quadratic equation

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &50 + 20x + 10x + 4x^2 = 150\\ &4x^2 + 30x + 50 = 150\\ &4x^2 + 30x - 100 = 0\\ &2x^2 + 15x - 50 = 0 \end{aligned}$$

\]

Solve the quadratic equation

Using the Quadratic Formula knowledge point
\[

$$\begin{aligned} &x = \frac{-15 \pm \sqrt{15^2 - 4(2)(-50)}}{2(2)}\\ &x = \frac{-15 \pm \sqrt{225 + 400}}{4}\\ &x = \frac{-15 \pm \sqrt{625}}{4}\\ &x = \frac{-15 \pm 25}{4} \end{aligned}$$

\]

Determine the valid width

Since the width must be positive, we discard the negative root:
\[
x = \frac{-15 + 25}{4} = \frac{10}{4} = 2.5\text{ cm}
\]
</reasoning>

<answer>
\(2.5\text{ cm}\)
</answer>

<plot>
{
"elements": [
{
"type": "rectangle",
"params": [[-5, -2.5], [5, 2.5]],
"properties": {
"strokeColor": "#583C87",
"strokeWidth": 2,
"fillColor": "rgba(88, 60, 135, 0.15)",
"name": "Picture"
}
},
{
"type": "rectangle",
"params": [[-7.5, -5], [7.5, 5]],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"fillColor": "rgba(140, 85, 242, 0.25)",
"name": "Matte"
}
},
{
"type": "arrow",
"params": [[5, 0], [7.5, 0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 2,
"name": "x"
}
},
{
"type": "text",
"params": [6.25, 0.5, "\\[x\\]"],
"properties": {
"color": "#F2557F"
}
},
{
"type": "text",
"params": [0, 0, "\\[10 \times 5\\]"],
"properties": {
"color": "#583C87"
}
}
],
"timestamps": [0.1]
}
</plot>

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