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Answer :
To determine the domain for the function [tex]\( h(t) = -16t^2 + 64t \)[/tex] in the given context, we need to consider what the function represents: the height of the rocket above the ground over time.
The height [tex]\( h(t) \)[/tex] must be non-negative because the rocket cannot have a negative height above the ground. Therefore, we need to solve the inequality:
[tex]\[ -16t^2 + 64t \geq 0. \][/tex]
Let's solve this step-by-step:
1. Factor the expression:
The equation can be factored as:
[tex]\[ -16t(t - 4) \geq 0. \][/tex]
2. Find the critical points:
Set the expression equal to zero to find the critical points:
[tex]\[ -16t(t - 4) = 0. \][/tex]
Solving this gives:
[tex]\[ t = 0 \quad \text{or} \quad t = 4. \][/tex]
3. Determine the intervals and test them:
The critical points divide the number line into the intervals [tex]\((-\infty, 0)\)[/tex], [tex]\((0, 4)\)[/tex], and [tex]\((4, \infty)\)[/tex].
- Interval [tex]\((-\infty, 0)\)[/tex]: Choose a test point, say [tex]\( t = -1 \)[/tex].
[tex]\[
-16(-1)(-1 - 4) = -16 \cdot (-1) \cdot (-5) = -80,
\][/tex]
which is less than 0, so this interval does not satisfy the inequality.
- Interval [tex]\((0, 4)\)[/tex]: Choose a test point, say [tex]\( t = 2 \)[/tex].
[tex]\[
-16 \cdot 2 \cdot (2 - 4) = -16 \cdot 2 \cdot (-2) = 64,
\][/tex]
which is greater than 0, so this interval satisfies the inequality.
- Interval [tex]\((4, \infty)\)[/tex]: Choose a test point, say [tex]\( t = 5 \)[/tex].
[tex]\[
-16 \cdot 5 \cdot (5 - 4) = -16 \cdot 5 \cdot 1 = -80,
\][/tex]
which is less than 0, so this interval does not satisfy the inequality.
4. Include the endpoints:
Since the inequality is non-strict ([tex]\(\geq\)[/tex]), we include the endpoints where the expression is zero:
- At [tex]\( t = 0 \)[/tex], [tex]\(-16 \cdot 0 \cdot (0 - 4) = 0\)[/tex].
- At [tex]\( t = 4 \)[/tex], [tex]\(-16 \cdot 4 \cdot (4 - 4) = 0\)[/tex].
Given our testing, the rocket is above ground (or at ground level) when [tex]\( t \)[/tex] is in the interval [tex]\([0, 4]\)[/tex].
Therefore, the domain of the function [tex]\( h(t) \)[/tex] in this context is [tex]\( [0, 4] \)[/tex].
The height [tex]\( h(t) \)[/tex] must be non-negative because the rocket cannot have a negative height above the ground. Therefore, we need to solve the inequality:
[tex]\[ -16t^2 + 64t \geq 0. \][/tex]
Let's solve this step-by-step:
1. Factor the expression:
The equation can be factored as:
[tex]\[ -16t(t - 4) \geq 0. \][/tex]
2. Find the critical points:
Set the expression equal to zero to find the critical points:
[tex]\[ -16t(t - 4) = 0. \][/tex]
Solving this gives:
[tex]\[ t = 0 \quad \text{or} \quad t = 4. \][/tex]
3. Determine the intervals and test them:
The critical points divide the number line into the intervals [tex]\((-\infty, 0)\)[/tex], [tex]\((0, 4)\)[/tex], and [tex]\((4, \infty)\)[/tex].
- Interval [tex]\((-\infty, 0)\)[/tex]: Choose a test point, say [tex]\( t = -1 \)[/tex].
[tex]\[
-16(-1)(-1 - 4) = -16 \cdot (-1) \cdot (-5) = -80,
\][/tex]
which is less than 0, so this interval does not satisfy the inequality.
- Interval [tex]\((0, 4)\)[/tex]: Choose a test point, say [tex]\( t = 2 \)[/tex].
[tex]\[
-16 \cdot 2 \cdot (2 - 4) = -16 \cdot 2 \cdot (-2) = 64,
\][/tex]
which is greater than 0, so this interval satisfies the inequality.
- Interval [tex]\((4, \infty)\)[/tex]: Choose a test point, say [tex]\( t = 5 \)[/tex].
[tex]\[
-16 \cdot 5 \cdot (5 - 4) = -16 \cdot 5 \cdot 1 = -80,
\][/tex]
which is less than 0, so this interval does not satisfy the inequality.
4. Include the endpoints:
Since the inequality is non-strict ([tex]\(\geq\)[/tex]), we include the endpoints where the expression is zero:
- At [tex]\( t = 0 \)[/tex], [tex]\(-16 \cdot 0 \cdot (0 - 4) = 0\)[/tex].
- At [tex]\( t = 4 \)[/tex], [tex]\(-16 \cdot 4 \cdot (4 - 4) = 0\)[/tex].
Given our testing, the rocket is above ground (or at ground level) when [tex]\( t \)[/tex] is in the interval [tex]\([0, 4]\)[/tex].
Therefore, the domain of the function [tex]\( h(t) \)[/tex] in this context is [tex]\( [0, 4] \)[/tex].
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