Question

an object is launched from ground level. the equation for the objects height is \\(y = -16x^2 + 48x\\). what values of \\(x\\) (time in seconds) is the object at or above a height of 32 meters?

Explanation:

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<reasoning>

Set up the inequality

Using the Quadratic Inequalities knowledge point
\[
-16x^2 + 48x \ge 32
\]

Rearrange into standard form

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &-16x^2 + 48x - 32 \ge 0\\ &x^2 - 3x + 2 \le 0 \end{aligned}$$

\]

Find the critical boundary points

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &(x - 1)(x - 2) = 0\\ &x = 1 \quad \text{or} \quad x = 2 \end{aligned}$$

\]

Determine the solution interval

Using the Quadratic Inequalities knowledge point
\[
1 \le x \le 2
\]
</reasoning>

<answer>
\(1 \le x \le 2\)
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-16xx + 48*x",
"latex": "y = -16x^2 + 48x"
},
0,
3
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = -16x^2 + 48x",
"withLabel": true
}
},
{
"type": "line",
"params": [
[0, 32],
[3, 32]
],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 1.5,
"dash": 2,
"name": "y = 32",
"withLabel": true
}
},
{
"type": "point",
"params": [
[1, 32]
],
"properties": {
"name": "(1, 32)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [
[2, 32]
],
"properties": {
"name": "(2, 32)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "segment",
"params": [
[1, 0],
[2, 0]
],
"properties": {
"strokeColor": "#55DDF2",
"strokeWidth": 4,
"name": "1 <= x <= 2",
"withLabel": false
}
}
]
}
</plot>

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Answer:

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</pre_analysis>

<reasoning>

Set up the inequality

Using the Quadratic Inequalities knowledge point
\[
-16x^2 + 48x \ge 32
\]

Rearrange into standard form

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &-16x^2 + 48x - 32 \ge 0\\ &x^2 - 3x + 2 \le 0 \end{aligned}$$

\]

Find the critical boundary points

Using the Solving Quadratic Equations knowledge point
\[

$$\begin{aligned} &(x - 1)(x - 2) = 0\\ &x = 1 \quad \text{or} \quad x = 2 \end{aligned}$$

\]

Determine the solution interval

Using the Quadratic Inequalities knowledge point
\[
1 \le x \le 2
\]
</reasoning>

<answer>
\(1 \le x \le 2\)
</answer>

<plot>
{
"elements": [
{
"type": "functiongraph",
"params": [
{
"js": "-16xx + 48*x",
"latex": "y = -16x^2 + 48x"
},
0,
3
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2,
"name": "y = -16x^2 + 48x",
"withLabel": true
}
},
{
"type": "line",
"params": [
[0, 32],
[3, 32]
],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 1.5,
"dash": 2,
"name": "y = 32",
"withLabel": true
}
},
{
"type": "point",
"params": [
[1, 32]
],
"properties": {
"name": "(1, 32)",
"color": "#5583F2",
"size": 3,
"withLabel": true
}
},
{
"type": "point",
"params": [
[2, 32]
],
"properties": {
"name": "(2, 32)",
"color": "#5583F2",
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"withLabel": true
}
},
{
"type": "segment",
"params": [
[1, 0],
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],
"properties": {
"strokeColor": "#55DDF2",
"strokeWidth": 4,
"name": "1 <= x <= 2",
"withLabel": false
}
}
]
}
</plot>

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