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Answer :
Sure! Let's go through the problem step-by-step to understand why a specific function represents the height of a T-shirt thrown into the air.
We're given a selection of functions that all describe parabolic motion. The parabolic equations given are in vertex form:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
Where:
- [tex]\( a \)[/tex] indicates the direction and steepness of the parabola.
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
In this context involving parabolic motion, imagine a T-shirt being thrown into the air and then coming down. This kind of motion is known as projectile motion, which is characterized by a parabola opening downwards due to gravity.
Step-by-step explanation:
1. Direction of Parabola:
- A downward-opening parabola, typical in projectile motion (like throwing a T-shirt upwards), has a negative "a" value. The functions provided all have a negative coefficient [tex]\(-16\)[/tex], which is correct for a parabola that opens downwards.
2. Vertex Form and Impact on Choice:
- The height of the T-shirt depends on both time, [tex]\( t \)[/tex], and the maximum height it reaches, which is the vertex of the parabola.
- In the equation [tex]\( f(t) = -16(t - h)^2 + k \)[/tex], the vertex [tex]\((h, k)\)[/tex] also represents the time [tex]\(h\)[/tex] at which the maximum height [tex]\(k\)[/tex] is reached.
3. Selecting the Correct Function:
- Among the given functions, choose the one whose vertex suggests a realistic scenario. Since there is a peak height involved and we usually start looking at time from [tex]\(t = 0\)[/tex], having the vertex close to this can make sense.
- Consider the function: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]. The vertex form suggests that:
- The vertex is [tex]\((1, 24)\)[/tex].
- This means at [tex]\(t = 1\)[/tex] second, the T-shirt reaches its highest point (24 units high).
Therefore, the function that best represents the projectile motion of the T-shirt, with an initial upward motion peaking at 1 second and a maximum height of 24, is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
This equation appropriately describes the T-shirt's path based on the indicated motion dynamics in the problem context.
We're given a selection of functions that all describe parabolic motion. The parabolic equations given are in vertex form:
[tex]\[ f(t) = a(t - h)^2 + k \][/tex]
Where:
- [tex]\( a \)[/tex] indicates the direction and steepness of the parabola.
- [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
In this context involving parabolic motion, imagine a T-shirt being thrown into the air and then coming down. This kind of motion is known as projectile motion, which is characterized by a parabola opening downwards due to gravity.
Step-by-step explanation:
1. Direction of Parabola:
- A downward-opening parabola, typical in projectile motion (like throwing a T-shirt upwards), has a negative "a" value. The functions provided all have a negative coefficient [tex]\(-16\)[/tex], which is correct for a parabola that opens downwards.
2. Vertex Form and Impact on Choice:
- The height of the T-shirt depends on both time, [tex]\( t \)[/tex], and the maximum height it reaches, which is the vertex of the parabola.
- In the equation [tex]\( f(t) = -16(t - h)^2 + k \)[/tex], the vertex [tex]\((h, k)\)[/tex] also represents the time [tex]\(h\)[/tex] at which the maximum height [tex]\(k\)[/tex] is reached.
3. Selecting the Correct Function:
- Among the given functions, choose the one whose vertex suggests a realistic scenario. Since there is a peak height involved and we usually start looking at time from [tex]\(t = 0\)[/tex], having the vertex close to this can make sense.
- Consider the function: [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]. The vertex form suggests that:
- The vertex is [tex]\((1, 24)\)[/tex].
- This means at [tex]\(t = 1\)[/tex] second, the T-shirt reaches its highest point (24 units high).
Therefore, the function that best represents the projectile motion of the T-shirt, with an initial upward motion peaking at 1 second and a maximum height of 24, is:
[tex]\[ f(t) = -16(t-1)^2 + 24 \][/tex]
This equation appropriately describes the T-shirt's path based on the indicated motion dynamics in the problem context.
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