Answer :
To find out how far the snail traveled in a day, we need to calculate the total distance covered over 24 hours. The velocity of the snail is given by the function [tex]\( v(t) = 4t + 1 \)[/tex] feet per hour, where [tex]\( t \)[/tex] represents time in hours.
To determine the distance traveled, we integrate the velocity function over the 24-hour period from [tex]\( t = 0 \)[/tex] to [tex]\( t = 24 \)[/tex]. Integration is a process that helps us find the area under the curve of the velocity function, which represents the total distance traveled.
The integral of the velocity function [tex]\( v(t) = 4t + 1 \)[/tex] gives us the following:
[tex]\[
\text{Distance} = \int_{0}^{24} (4t + 1) \, dt
\][/tex]
To integrate [tex]\( 4t + 1 \)[/tex]:
1. Integrate the term [tex]\( 4t \)[/tex]: [tex]\(\frac{4t^2}{2} = 2t^2\)[/tex]
2. Integrate the term [tex]\( 1 \)[/tex]: [tex]\(1t\)[/tex]
Combining these, the integral of the function is:
[tex]\[
2t^2 + t
\][/tex]
Now, we evaluate this from 0 to 24:
[tex]\[
\left[ 2(24)^2 + 24 \right] - \left[ 2(0)^2 + 0 \right]
\][/tex]
Calculate each part:
[tex]\[
2(24)^2 + 24 = 2 \times 576 + 24 = 1152 + 24 = 1176
\][/tex]
So, the total distance the snail traveled in 24 hours is 1176 feet.
Therefore, the correct answer is:
C ) 1176 feet
To determine the distance traveled, we integrate the velocity function over the 24-hour period from [tex]\( t = 0 \)[/tex] to [tex]\( t = 24 \)[/tex]. Integration is a process that helps us find the area under the curve of the velocity function, which represents the total distance traveled.
The integral of the velocity function [tex]\( v(t) = 4t + 1 \)[/tex] gives us the following:
[tex]\[
\text{Distance} = \int_{0}^{24} (4t + 1) \, dt
\][/tex]
To integrate [tex]\( 4t + 1 \)[/tex]:
1. Integrate the term [tex]\( 4t \)[/tex]: [tex]\(\frac{4t^2}{2} = 2t^2\)[/tex]
2. Integrate the term [tex]\( 1 \)[/tex]: [tex]\(1t\)[/tex]
Combining these, the integral of the function is:
[tex]\[
2t^2 + t
\][/tex]
Now, we evaluate this from 0 to 24:
[tex]\[
\left[ 2(24)^2 + 24 \right] - \left[ 2(0)^2 + 0 \right]
\][/tex]
Calculate each part:
[tex]\[
2(24)^2 + 24 = 2 \times 576 + 24 = 1152 + 24 = 1176
\][/tex]
So, the total distance the snail traveled in 24 hours is 1176 feet.
Therefore, the correct answer is:
C ) 1176 feet
