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Answer :
To find out how far the snail traveled in a day, we need to calculate the distance covered over a 24-hour period using the given velocity function [tex]\( v(t) = 4t + 1 \)[/tex].
1. Understanding the Problem:
- The velocity function [tex]\( v(t) = 4t + 1 \)[/tex] tells us the velocity of the snail in feet per hour at any time [tex]\( t \)[/tex].
- To find the total distance traveled over a period of time, we need to integrate the velocity function over that time interval.
2. Setting Up the Integral:
- We want to find the distance traveled from [tex]\( t = 0 \)[/tex] to [tex]\( t = 24 \)[/tex] hours.
- The total distance [tex]\( D \)[/tex] is the integral of the velocity function over this period:
[tex]\[
D = \int_0^{24} (4t + 1) \, dt
\][/tex]
3. Calculating the Integral:
- Integrate the velocity function term by term:
- [tex]\(\int 4t \, dt = 2t^2\)[/tex]
- [tex]\(\int 1 \, dt = t\)[/tex]
- So, the integral becomes:
[tex]\[
D = \left[ 2t^2 + t \right]_0^{24}
\][/tex]
4. Evaluating the Integral:
- Substitute the upper limit (24) and the lower limit (0) into the integrated expression:
[tex]\[
D = \left( 2(24)^2 + 24 \right) - \left( 2(0)^2 + 0 \right)
\][/tex]
- Calculate each term:
[tex]\[
2(24)^2 = 2 \times 576 = 1152
\][/tex]
[tex]\[
2(0)^2 = 0
\][/tex]
[tex]\[
D = 1152 + 24 = 1176
\][/tex]
5. Conclusion:
- The snail traveled exactly [tex]\( 1176 \)[/tex] feet in 24 hours.
So, the correct answer is C) 1176 feet.
1. Understanding the Problem:
- The velocity function [tex]\( v(t) = 4t + 1 \)[/tex] tells us the velocity of the snail in feet per hour at any time [tex]\( t \)[/tex].
- To find the total distance traveled over a period of time, we need to integrate the velocity function over that time interval.
2. Setting Up the Integral:
- We want to find the distance traveled from [tex]\( t = 0 \)[/tex] to [tex]\( t = 24 \)[/tex] hours.
- The total distance [tex]\( D \)[/tex] is the integral of the velocity function over this period:
[tex]\[
D = \int_0^{24} (4t + 1) \, dt
\][/tex]
3. Calculating the Integral:
- Integrate the velocity function term by term:
- [tex]\(\int 4t \, dt = 2t^2\)[/tex]
- [tex]\(\int 1 \, dt = t\)[/tex]
- So, the integral becomes:
[tex]\[
D = \left[ 2t^2 + t \right]_0^{24}
\][/tex]
4. Evaluating the Integral:
- Substitute the upper limit (24) and the lower limit (0) into the integrated expression:
[tex]\[
D = \left( 2(24)^2 + 24 \right) - \left( 2(0)^2 + 0 \right)
\][/tex]
- Calculate each term:
[tex]\[
2(24)^2 = 2 \times 576 = 1152
\][/tex]
[tex]\[
2(0)^2 = 0
\][/tex]
[tex]\[
D = 1152 + 24 = 1176
\][/tex]
5. Conclusion:
- The snail traveled exactly [tex]\( 1176 \)[/tex] feet in 24 hours.
So, the correct answer is C) 1176 feet.
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