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Sagot :
To solve the problem, let's determine which of the given functions correctly models the situation where a baseball is hit from an initial height of 3 feet and reaches a maximum height of 403 feet.
The general form of the quadratic function to model the height [tex]\( h(t) \)[/tex] of a projectile over time [tex]\( t \)[/tex] is typically given as:
[tex]\[ h(t) = -16(t - h)^2 + k \][/tex]
where:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola.
- In this context, [tex]\( h \)[/tex] is the time at which the maximum height is reached.
- [tex]\( k \)[/tex] is the maximum height.
Here's a detailed step-by-step solution:
1. Analyze the maximum height and time:
- We know the maximum height is [tex]\( 403 \)[/tex] feet.
- The initial height is [tex]\( 3 \)[/tex] feet.
- The maximum height is reached at some time [tex]\( t \)[/tex].
2. Understand the vertex form:
- The vertex form of the quadratic function in context is [tex]\( h(t) = -16(t - h)^2 + 403 \)[/tex] since 403 is the maximum height.
3. Checking the given options:
- Option A: [tex]\( h(t) = -16(t - 403)^2 + 3 \)[/tex]
- This suggests the maximum height is [tex]\( 3 \)[/tex] feet, which is incorrect.
- Option B: [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex]
- Substituting [tex]\( t = 5 \)[/tex]:
[tex]\[ h(5) = -16(5 - 5)^2 + 403 = 403 \][/tex]
- Substituting [tex]\( t = 0 \)[/tex] to check the initial height:
[tex]\[ h(0) = -16(0 - 5)^2 + 403 = -16(25) + 403 = -400 + 403 = 3 \][/tex]
- Both the initial height and maximum height checks out. So this could be the correct model.
- Option C: [tex]\( h(t) = -16(t - 3)^2 + 403 \)[/tex]
- Substituting [tex]\( t = 3 \)[/tex]:
[tex]\[ h(3) = -16(3 - 3)^2 + 403 = 403 \][/tex]
- Substituting [tex]\( t = 0 \)[/tex]:
[tex]\[ h(0) = -16(0 - 3)^2 + 403 = -16(9) + 403 = -144 + 403 = 259 \][/tex]
- Initial height is 259 feet, not 3 feet. This option is incorrect.
- Option D: [tex]\( h(t) = -16(t - 5)^2 + 3 \)[/tex]
- Substituting [tex]\( t = 5 \)[/tex]:
[tex]\[ h(5) = -16(5 - 5)^2 + 3 = 3 \][/tex]
- This suggests the maximum height is 3 feet, which is incorrect.
Given the analysis above, the function [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex] correctly models the situation with an initial height of [tex]\( 3 \)[/tex] feet and a maximum height of [tex]\( 403 \)[/tex] feet.
Thus, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
The general form of the quadratic function to model the height [tex]\( h(t) \)[/tex] of a projectile over time [tex]\( t \)[/tex] is typically given as:
[tex]\[ h(t) = -16(t - h)^2 + k \][/tex]
where:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola.
- In this context, [tex]\( h \)[/tex] is the time at which the maximum height is reached.
- [tex]\( k \)[/tex] is the maximum height.
Here's a detailed step-by-step solution:
1. Analyze the maximum height and time:
- We know the maximum height is [tex]\( 403 \)[/tex] feet.
- The initial height is [tex]\( 3 \)[/tex] feet.
- The maximum height is reached at some time [tex]\( t \)[/tex].
2. Understand the vertex form:
- The vertex form of the quadratic function in context is [tex]\( h(t) = -16(t - h)^2 + 403 \)[/tex] since 403 is the maximum height.
3. Checking the given options:
- Option A: [tex]\( h(t) = -16(t - 403)^2 + 3 \)[/tex]
- This suggests the maximum height is [tex]\( 3 \)[/tex] feet, which is incorrect.
- Option B: [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex]
- Substituting [tex]\( t = 5 \)[/tex]:
[tex]\[ h(5) = -16(5 - 5)^2 + 403 = 403 \][/tex]
- Substituting [tex]\( t = 0 \)[/tex] to check the initial height:
[tex]\[ h(0) = -16(0 - 5)^2 + 403 = -16(25) + 403 = -400 + 403 = 3 \][/tex]
- Both the initial height and maximum height checks out. So this could be the correct model.
- Option C: [tex]\( h(t) = -16(t - 3)^2 + 403 \)[/tex]
- Substituting [tex]\( t = 3 \)[/tex]:
[tex]\[ h(3) = -16(3 - 3)^2 + 403 = 403 \][/tex]
- Substituting [tex]\( t = 0 \)[/tex]:
[tex]\[ h(0) = -16(0 - 3)^2 + 403 = -16(9) + 403 = -144 + 403 = 259 \][/tex]
- Initial height is 259 feet, not 3 feet. This option is incorrect.
- Option D: [tex]\( h(t) = -16(t - 5)^2 + 3 \)[/tex]
- Substituting [tex]\( t = 5 \)[/tex]:
[tex]\[ h(5) = -16(5 - 5)^2 + 3 = 3 \][/tex]
- This suggests the maximum height is 3 feet, which is incorrect.
Given the analysis above, the function [tex]\( h(t) = -16(t - 5)^2 + 403 \)[/tex] correctly models the situation with an initial height of [tex]\( 3 \)[/tex] feet and a maximum height of [tex]\( 403 \)[/tex] feet.
Thus, the correct answer is:
[tex]\[ \boxed{\text{B}} \][/tex]
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