IDNLearn.com: Where curiosity meets clarity and questions find their answers. Our experts are available to provide in-depth and trustworthy answers to any questions you may have.
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com provides the answers you need. Thank you for visiting, and see you next time for more valuable insights.
