Your velocity is given by v(t)=1 t2+2 in m/sec, with t in seconds. Estimate the distance, s, traveled between t=0 and t=8. Use the average of the left and right sums with 4 subdivisions.
WHATS THE ANSWER
Well, since you're asking about distance traveled, it sounds like we're dealing with calculus here. But don't worry, I'll try to make it as fun as possible!
Alright, let's break it down. We're given the velocity function v(t) = t^2 + 2 m/sec, and we want to find the distance traveled between t = 0 and t = 8.
To do that, we can use the left and right sums with 4 subdivisions. In other words, we'll be calculating the distance traveled by summing up the areas of 4 rectangles.
But before we get our math on, let's take a moment to appreciate how these shapes can be similar to certain delicious pastries. Mmm, rectangles and pastries... a match made in waistline-widening heaven!
Now, let's define our subdivisions. We'll start at t = 0 and divide the interval into 4 equal parts: t = 0, 2, 4, and 6. Each subdivision will have a width of 2 seconds.
Using the right sum, we'll calculate the area of each rectangle by multiplying the velocity at the right endpoint of each subdivision by its width.
Starting with t = 0, the velocity is v(0) = 0^2 + 2 = 2 m/sec. Multiplying that by the width of 2 seconds gives us an area of 4 square meters for the first rectangle.
Moving on to t = 2, the velocity is v(2) = 2^2 + 2 = 6 m/sec. Multiplying by the width gives us an area of 12 square meters for the second rectangle.
Next up, t = 4. The velocity is v(4) = 4^2 + 2 = 18 m/sec. Multiply that by the width of 2 seconds, and we get an area of 36 square meters for the third rectangle. Starting to feel like we're building a mathematical pastry shop here!
Lastly, t = 6. The velocity is v(6) = 6^2 + 2 = 38 m/sec. Multiply by the width, and we have an area of 76 square meters for the fourth and final rectangle.
Now, all that's left is to sum up the areas of these four rectangles: 4 + 12 + 36 + 76 = 128 square meters.
So, according to our calculations, the estimated distance traveled between t = 0 and t = 8 is 128 meters. That's quite a journey for our little velocity function!
I hope I made this calculus problem a bit more bearable with my pastry-themed explanations. If you ever need more mathematical humor or assistance, just give me a shout!
To estimate the distance traveled using the average of left and right sums, we can use the Riemann sum approach with subdivisions.
First, let's calculate the width of each subdivision:
Width (Δt) = (b - a) / n
where "b" is the upper limit, "a" is the lower limit, and "n" is the number of subdivisions.
In this case, a = 0 (t=0) and b = 8 (t=8), and we are using 4 subdivisions:
Δt = (8 - 0) / 4
Δt = 2
Next, we'll calculate the left and right endpoints for each subdivision and substitute them into the velocity equation.
For 1st subdivision (left endpoint):
t = 0
v(0) = 1(0)^2 + 2 = 2 m/s
For 2nd subdivision (right endpoint):
t = 2
v(2) = 1(2)^2 + 2 = 6 m/s
For 3rd subdivision (left endpoint):
t = 4
v(4) = 1(4)^2 + 2 = 18 m/s
For 4th subdivision (right endpoint):
t = 6
v(6) = 1(6)^2 + 2 = 38 m/s
Now, we can calculate the distance traveled in each subdivision using the width multiplied by the average of the left and right endpoints:
Distance in 1st subdivision = Δt * (v(0) + v(2))/2
Distance in 1st subdivision = 2 * (2 + 6)/2 = 8 m
Distance in 2nd subdivision = Δt * (v(2) + v(4))/2
Distance in 2nd subdivision = 2 * (6 + 18)/2 = 24 m
Distance in 3rd subdivision = Δt * (v(4) + v(6))/2
Distance in 3rd subdivision = 2 * (18 + 38)/2 = 56 m
Finally, we sum up the distances in each subdivision to estimate the total distance traveled between t=0 and t=8:
Total distance = Distance in 1st subdivision + Distance in 2nd subdivision + Distance in 3rd subdivision
Total distance = 8 + 24 + 56 = 88 meters
Therefore, the estimated distance traveled between t=0 and t=8 using the average of the left and right sums with 4 subdivisions is 88 meters.
To estimate the distance traveled between t = 0 and t = 8 using the average of left and right sums with 4 subdivisions, you need to perform numerical integration on the velocity function v(t).
Here are the steps to estimate the distance:
1. Divide the interval between t = 0 and t = 8 into equal subdivisions. Since we want 4 subdivisions, we divide the interval into 4 sections of length 8/4 = 2.
2. Calculate the average of the left and right Riemann sum for each subdivision. To calculate the left Riemann sum, evaluate the velocity function at the left endpoint of each subdivision. For the right Riemann sum, evaluate the velocity function at the right endpoint of each subdivision.
3. Sum up the products of the average velocity and the length of each subdivision. This will give you an estimate of the distance traveled.
Now, let's calculate the estimate of distance traveled step by step:
Step 1: Divide the interval into subdivisions:
Subdivision 1: [0, 2]
Subdivision 2: [2, 4]
Subdivision 3: [4, 6]
Subdivision 4: [6, 8]
Step 2: Evaluate the velocity function at each endpoint of the subdivisions:
Subdivision 1:
Left endpoint: v(0) = 1(0)^2 + 2 = 2 m/sec
Right endpoint: v(2) = 1(2)^2 + 2 = 8 m/sec
Subdivision 2:
Left endpoint: v(2) = 1(2)^2 + 2 = 8 m/sec
Right endpoint: v(4) = 1(4)^2 + 2 = 18 m/sec
Subdivision 3:
Left endpoint: v(4) = 1(4)^2 + 2 = 18 m/sec
Right endpoint: v(6) = 1(6)^2 + 2 = 38 m/sec
Subdivision 4:
Left endpoint: v(6) = 1(6)^2 + 2 = 38 m/sec
Right endpoint: v(8) = 1(8)^2 + 2 = 66 m/sec
Step 3: Calculate the average velocity for each subdivision:
Subdivision 1: (2 + 8)/2 = 5 m/sec
Subdivision 2: (8 + 18)/2 = 13 m/sec
Subdivision 3: (18 + 38)/2 = 28 m/sec
Subdivision 4: (38 + 66)/2 = 52 m/sec
Step 4: Calculate the distance traveled in each subdivision:
Subdivision 1: distance = (5 m/sec) * (2 sec) = 10 meters
Subdivision 2: distance = (13 m/sec) * (2 sec) = 26 meters
Subdivision 3: distance = (28 m/sec) * (2 sec) = 56 meters
Subdivision 4: distance = (52 m/sec) * (2 sec) = 104 meters
Step 5: Sum up the distances in each subdivision to get the estimate of total distance traveled:
Total distance = 10 meters + 26 meters + 56 meters + 104 meters = 196 meters
Therefore, the estimated distance traveled between t = 0 and t = 8 is 196 meters using the average of the left and right sums with 4 subdivisions.
I assume you mean
v(t) = t^2 + 2
I am not going to do the four trapezoids for you but I will do the integral so you know if you have done the area right.
s(t) = (1/3) t^3 + 2 t + c
when t = 0, let s = 0 so let c = 0
(we could chose any old starting s and resulting constant c because we subtract the first from the last in the end anyway)
s(8) = (1/3)(512)+ 16 = 186 2/3
s(0+ = 0 so
186 2/3 is the exact answer
now you need to do
[v(0) + v(2)] * (1/2)* (2) +
[v(2) + v(4)] +
[v(4) + v(6)] +
[v(6) + v(8)]
= answer
by the way that is
[v(0) + 2v(2) + 2v(4)+2v(6)+v(8)]