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Sagot :
To determine which quadratic function [tex]\( h(t) \)[/tex] models the height of the ball, we need to evaluate each given quadratic function at [tex]\( t = 1 \)[/tex] and [tex]\( t = 4 \)[/tex] and check if both evaluations yield a height of 10 feet. The given functions are:
1. [tex]\( h(t) = 4(t+4)^2 + 46 \)[/tex]
2. [tex]\( h(t) = -4(t-4)^2 + 46 \)[/tex]
3. [tex]\( h(t) = 2(t+3)^2 + 46 \)[/tex]
4. [tex]\( h(t) = -2(t-3)^2 + 46 \)[/tex]
Let's evaluate these functions step by step.
### 1. Evaluating [tex]\( h(t) = 4(t+4)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = 4(1+4)^2 + 46 \][/tex]
[tex]\[ = 4(5)^2 + 46 \][/tex]
[tex]\[ = 4(25) + 46 \][/tex]
[tex]\[ = 100 + 46 \][/tex]
[tex]\[ = 146 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = 4(4+4)^2 + 46 \][/tex]
[tex]\[ = 4(8)^2 + 46 \][/tex]
[tex]\[ = 4(64) + 46 \][/tex]
[tex]\[ = 256 + 46 \][/tex]
[tex]\[ = 302 \][/tex]
So, this model does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### 2. Evaluating [tex]\( h(t) = -4(t-4)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = -4(1-4)^2 + 46 \][/tex]
[tex]\[ = -4(-3)^2 + 46 \][/tex]
[tex]\[ = -4(9) + 46 \][/tex]
[tex]\[ = -36 + 46 \][/tex]
[tex]\[ = 10 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = -4(4-4)^2 + 46 \][/tex]
[tex]\[ = -4(0)^2 + 46 \][/tex]
[tex]\[ = 0 + 46 \][/tex]
[tex]\[ = 46 \][/tex]
Again, this model does not fit since [tex]\( h(4) \neq 10 \)[/tex].
### 3. Evaluating [tex]\( h(t) = 2(t+3)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = 2(1+3)^2 + 46 \][/tex]
[tex]\[ = 2(4)^2 + 46 \][/tex]
[tex]\[ = 2(16) + 46 \][/tex]
[tex]\[ = 32 + 46 \][/tex]
[tex]\[ = 78 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = 2(4+3)^2 + 46 \][/tex]
[tex]\[ = 2(7)^2 + 46 \][/tex]
[tex]\[ = 2(49) + 46 \][/tex]
[tex]\[ = 98 + 46 \][/tex]
[tex]\[ = 144 \][/tex]
This model also does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### 4. Evaluating [tex]\( h(t) = -2(t-3)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = -2(1-3)^2 + 46 \][/tex]
[tex]\[ = -2(-2)^2 + 46 \][/tex]
[tex]\[ = -2(4) + 46 \][/tex]
[tex]\[ = -8 + 46 \][/tex]
[tex]\[ = 38 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = -2(4-3)^2 + 46 \][/tex]
[tex]\[ = -2(1)^2 + 46 \][/tex]
[tex]\[ = -2(1) + 46 \][/tex]
[tex]\[ = -2 + 46 \][/tex]
[tex]\[ = 44 \][/tex]
This model also does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### Conclusion
After evaluating all given functions, none of them satisfies the condition that both [tex]\( h(1) \)[/tex] and [tex]\( h(4) \)[/tex] equal 10. Therefore, there is no quadratic function among the given options that accurately models the height of the ball at the specified times.
1. [tex]\( h(t) = 4(t+4)^2 + 46 \)[/tex]
2. [tex]\( h(t) = -4(t-4)^2 + 46 \)[/tex]
3. [tex]\( h(t) = 2(t+3)^2 + 46 \)[/tex]
4. [tex]\( h(t) = -2(t-3)^2 + 46 \)[/tex]
Let's evaluate these functions step by step.
### 1. Evaluating [tex]\( h(t) = 4(t+4)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = 4(1+4)^2 + 46 \][/tex]
[tex]\[ = 4(5)^2 + 46 \][/tex]
[tex]\[ = 4(25) + 46 \][/tex]
[tex]\[ = 100 + 46 \][/tex]
[tex]\[ = 146 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = 4(4+4)^2 + 46 \][/tex]
[tex]\[ = 4(8)^2 + 46 \][/tex]
[tex]\[ = 4(64) + 46 \][/tex]
[tex]\[ = 256 + 46 \][/tex]
[tex]\[ = 302 \][/tex]
So, this model does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### 2. Evaluating [tex]\( h(t) = -4(t-4)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = -4(1-4)^2 + 46 \][/tex]
[tex]\[ = -4(-3)^2 + 46 \][/tex]
[tex]\[ = -4(9) + 46 \][/tex]
[tex]\[ = -36 + 46 \][/tex]
[tex]\[ = 10 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = -4(4-4)^2 + 46 \][/tex]
[tex]\[ = -4(0)^2 + 46 \][/tex]
[tex]\[ = 0 + 46 \][/tex]
[tex]\[ = 46 \][/tex]
Again, this model does not fit since [tex]\( h(4) \neq 10 \)[/tex].
### 3. Evaluating [tex]\( h(t) = 2(t+3)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = 2(1+3)^2 + 46 \][/tex]
[tex]\[ = 2(4)^2 + 46 \][/tex]
[tex]\[ = 2(16) + 46 \][/tex]
[tex]\[ = 32 + 46 \][/tex]
[tex]\[ = 78 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = 2(4+3)^2 + 46 \][/tex]
[tex]\[ = 2(7)^2 + 46 \][/tex]
[tex]\[ = 2(49) + 46 \][/tex]
[tex]\[ = 98 + 46 \][/tex]
[tex]\[ = 144 \][/tex]
This model also does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### 4. Evaluating [tex]\( h(t) = -2(t-3)^2 + 46 \)[/tex]
#### For [tex]\( t = 1 \)[/tex]:
[tex]\[ h(1) = -2(1-3)^2 + 46 \][/tex]
[tex]\[ = -2(-2)^2 + 46 \][/tex]
[tex]\[ = -2(4) + 46 \][/tex]
[tex]\[ = -8 + 46 \][/tex]
[tex]\[ = 38 \][/tex]
#### For [tex]\( t = 4 \)[/tex]:
[tex]\[ h(4) = -2(4-3)^2 + 46 \][/tex]
[tex]\[ = -2(1)^2 + 46 \][/tex]
[tex]\[ = -2(1) + 46 \][/tex]
[tex]\[ = -2 + 46 \][/tex]
[tex]\[ = 44 \][/tex]
This model also does not fit since neither [tex]\( h(1) \)[/tex] nor [tex]\( h(4) \)[/tex] equals 10.
### Conclusion
After evaluating all given functions, none of them satisfies the condition that both [tex]\( h(1) \)[/tex] and [tex]\( h(4) \)[/tex] equal 10. Therefore, there is no quadratic function among the given options that accurately models the height of the ball at the specified times.
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