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Answer :
To solve this problem, we need to determine which quadratic function represents the height of a T-shirt as a function of time, [tex]\( t \)[/tex], when thrown in the air and described by a parabolic path.
The functions given are in the vertex form of a parabola:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
When analyzing projectile motion, the vertex [tex]\( (h, k) \)[/tex] represents the highest point of the path with:
- [tex]\( h \)[/tex] being the time when the maximum height is reached.
- [tex]\( k \)[/tex] as the maximum height of the object.
Since all functions have a negative coefficient in front of the squared term [tex]\(-16\)[/tex], they all represent parabolas that open downwards. Now, let's evaluate each option based on the vertex characteristics:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- This means at [tex]\( t = 1 \)[/tex] second, the T-shirt reaches its highest point at 24 units of height.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- This means at [tex]\( t = -1 \)[/tex] second, which isn't realistic in this physical context, as time cannot be negative, the T-shirt would reach its highest point at 24 units of height.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- This means at [tex]\( t = 1 \)[/tex] second, the T-shirt reaches its highest point at -24 units of height. A negative height doesn't make sense in this context.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- This suggests the T-shirt reaches its highest point at -24 units of height at time [tex]\( t = -1 \)[/tex] second. Both a negative time and height are not realistic.
Given these analyses, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is the most plausible choice because it gives a realistic time [tex]\( t = 1 \)[/tex] second for reaching a positive maximum height of 24. Thus, this function best describes the height of the T-shirt as a function of time in this context.
The functions given are in the vertex form of a parabola:
[tex]\[ f(t) = a(t-h)^2 + k \][/tex]
Here, [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
When analyzing projectile motion, the vertex [tex]\( (h, k) \)[/tex] represents the highest point of the path with:
- [tex]\( h \)[/tex] being the time when the maximum height is reached.
- [tex]\( k \)[/tex] as the maximum height of the object.
Since all functions have a negative coefficient in front of the squared term [tex]\(-16\)[/tex], they all represent parabolas that open downwards. Now, let's evaluate each option based on the vertex characteristics:
1. [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (1, 24) \)[/tex]
- This means at [tex]\( t = 1 \)[/tex] second, the T-shirt reaches its highest point at 24 units of height.
2. [tex]\( f(t) = -16(t+1)^2 + 24 \)[/tex]
- Vertex: [tex]\( (-1, 24) \)[/tex]
- This means at [tex]\( t = -1 \)[/tex] second, which isn't realistic in this physical context, as time cannot be negative, the T-shirt would reach its highest point at 24 units of height.
3. [tex]\( f(t) = -16(t-1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (1, -24) \)[/tex]
- This means at [tex]\( t = 1 \)[/tex] second, the T-shirt reaches its highest point at -24 units of height. A negative height doesn't make sense in this context.
4. [tex]\( f(t) = -16(t+1)^2 - 24 \)[/tex]
- Vertex: [tex]\( (-1, -24) \)[/tex]
- This suggests the T-shirt reaches its highest point at -24 units of height at time [tex]\( t = -1 \)[/tex] second. Both a negative time and height are not realistic.
Given these analyses, the function [tex]\( f(t) = -16(t-1)^2 + 24 \)[/tex] is the most plausible choice because it gives a realistic time [tex]\( t = 1 \)[/tex] second for reaching a positive maximum height of 24. Thus, this function best describes the height of the T-shirt as a function of time in this context.
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