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Answer :
To find out how far the snail traveled in a day, we need to evaluate this distance by using the velocity function given, which is [tex]\( v(t) = 4t + 1 \)[/tex] feet per hour. The total distance traveled is found by integrating the velocity function over the time interval of one day. Since one day has 24 hours, we'll calculate the definite integral of the velocity from [tex]\( t = 0 \)[/tex] to [tex]\( t = 24 \)[/tex].
Here’s how we can calculate it step-by-step:
1. Set Up the Problem: The velocity function is [tex]\( v(t) = 4t + 1 \)[/tex]. To find the distance, we need to integrate this function from 0 to 24.
2. Integrate the Function: The integral of [tex]\( v(t) \)[/tex] from 0 to 24 is written as:
[tex]\[
\int_{0}^{24} (4t + 1) \, dt
\][/tex]
3. Calculate the Indefinite Integral: The antiderivative of [tex]\( 4t + 1 \)[/tex] can be calculated as follows:
- The antiderivative of [tex]\( 4t \)[/tex] is [tex]\( 2t^2 \)[/tex].
- The antiderivative of [tex]\( 1 \)[/tex] is [tex]\( t \)[/tex].
Therefore, the antiderivative of [tex]\( 4t + 1 \)[/tex] is [tex]\( 2t^2 + t \)[/tex].
4. Evaluate the Definite Integral: Use the limits 0 and 24 to find:
[tex]\[
\left[ 2t^2 + t \right]_{0}^{24} = (2(24)^2 + 24) - (2(0)^2 + 0)
\][/tex]
5. Simplify the Expression:
- [tex]\( 2(24)^2 = 2 \times 576 = 1152 \)[/tex]
- Add 24: [tex]\( 1152 + 24 = 1176 \)[/tex]
6. Conclusion: The distance traveled by the snail in a day is 1176 feet.
Therefore, the correct answer is C) 1176 feet.
Here’s how we can calculate it step-by-step:
1. Set Up the Problem: The velocity function is [tex]\( v(t) = 4t + 1 \)[/tex]. To find the distance, we need to integrate this function from 0 to 24.
2. Integrate the Function: The integral of [tex]\( v(t) \)[/tex] from 0 to 24 is written as:
[tex]\[
\int_{0}^{24} (4t + 1) \, dt
\][/tex]
3. Calculate the Indefinite Integral: The antiderivative of [tex]\( 4t + 1 \)[/tex] can be calculated as follows:
- The antiderivative of [tex]\( 4t \)[/tex] is [tex]\( 2t^2 \)[/tex].
- The antiderivative of [tex]\( 1 \)[/tex] is [tex]\( t \)[/tex].
Therefore, the antiderivative of [tex]\( 4t + 1 \)[/tex] is [tex]\( 2t^2 + t \)[/tex].
4. Evaluate the Definite Integral: Use the limits 0 and 24 to find:
[tex]\[
\left[ 2t^2 + t \right]_{0}^{24} = (2(24)^2 + 24) - (2(0)^2 + 0)
\][/tex]
5. Simplify the Expression:
- [tex]\( 2(24)^2 = 2 \times 576 = 1152 \)[/tex]
- Add 24: [tex]\( 1152 + 24 = 1176 \)[/tex]
6. Conclusion: The distance traveled by the snail in a day is 1176 feet.
Therefore, the correct answer is C) 1176 feet.
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