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Answer :
- Substitute $t=5$ into the height function $f(t) = -16t^2 + 160t$.
- Calculate $f(5) = -16(5^2) + 160(5)$.
- Simplify the expression: $f(5) = -400 + 800$.
- The rocket's height after 5 seconds is $\boxed{400}$.
### Explanation
1. Understanding the Problem
We are given the function $f(t) = -16t^2 + 160t$ which describes the height of a toy rocket $t$ seconds after launch. We want to find the height of the rocket after 5 seconds, which means we need to evaluate $f(5)$.
2. Substituting t=5
To find the height after 5 seconds, we substitute $t=5$ into the function: $$f(5) = -16(5)^2 + 160(5)$$.
3. Calculating f(5)
Now, we calculate the value: $$f(5) = -16(25) + 160(5) = -400 + 800 = 400$$.
4. Final Answer
Therefore, the rocket's height after 5 seconds is 400.
### Examples
Understanding the trajectory of rockets is crucial in space exploration and satellite deployment. The height function we used is a simplified model, but it demonstrates how mathematical functions can predict the position of objects over time. In real-world scenarios, engineers use more complex models to account for factors like air resistance and gravity variations, ensuring accurate navigation and mission success. By studying these models, we can optimize rocket launches, predict landing sites, and even design safer and more efficient transportation systems.
- Calculate $f(5) = -16(5^2) + 160(5)$.
- Simplify the expression: $f(5) = -400 + 800$.
- The rocket's height after 5 seconds is $\boxed{400}$.
### Explanation
1. Understanding the Problem
We are given the function $f(t) = -16t^2 + 160t$ which describes the height of a toy rocket $t$ seconds after launch. We want to find the height of the rocket after 5 seconds, which means we need to evaluate $f(5)$.
2. Substituting t=5
To find the height after 5 seconds, we substitute $t=5$ into the function: $$f(5) = -16(5)^2 + 160(5)$$.
3. Calculating f(5)
Now, we calculate the value: $$f(5) = -16(25) + 160(5) = -400 + 800 = 400$$.
4. Final Answer
Therefore, the rocket's height after 5 seconds is 400.
### Examples
Understanding the trajectory of rockets is crucial in space exploration and satellite deployment. The height function we used is a simplified model, but it demonstrates how mathematical functions can predict the position of objects over time. In real-world scenarios, engineers use more complex models to account for factors like air resistance and gravity variations, ensuring accurate navigation and mission success. By studying these models, we can optimize rocket launches, predict landing sites, and even design safer and more efficient transportation systems.
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