The velocity v(t) in the table is increasing,0<t<12 . Find an upper estimate for the total distance traveled using
(a) n=4
(b) n=2
Which of the two answers is more accurate?
t 0 3 6 9 12
v(t) 23 27 30 33 37
To find an upper estimate for the total distance traveled, we can use the trapezoidal rule to approximate the area under the velocity-time curve.
(a) For n = 4, we divide the interval [0, 12] into 4 subintervals of equal width. The width of each subinterval would be (12 - 0) / 4 = 3.
Using the trapezoidal rule, the upper estimate for the total distance traveled would be:
Distance = (1/2) * [v(0) + 2v(3) + 2v(6) + 2v(9) + v(12)] * width
Distance = (1/2) * [23 + 2*27 + 2*30 + 2*33 + 37] * 3
Distance = (1/2) * [23 + 54 + 60 + 66 + 37] * 3
Distance = (1/2) * [240] * 3
Distance = 360 units
(b) For n = 2, we divide the interval [0, 12] into 2 subintervals of equal width. The width of each subinterval would be (12 - 0) / 2 = 6.
Using the trapezoidal rule, the upper estimate for the total distance traveled would be:
Distance = (1/2) * [v(0) + 2v(6) + v(12)] * width
Distance = (1/2) * [23 + 2*30 + 37] * 6
Distance = (1/2) * [23 + 60 + 37] * 6
Distance = (1/2) * [120] * 6
Distance = 360 units
Both (a) and (b) yield the same upper estimate for the total distance traveled, which is 360 units. Therefore, both answers are equally accurate.
To find an upper estimate for the total distance traveled, we can use the trapezoidal rule. The formula for estimating the distance traveled is given by:
Total distance ≈ Δt × (0.5 × v(0) + v(Δt) + v(2Δt) + ... + v((n-1)Δt) + 0.5 × v(nΔt))
Where Δt is the time interval between measurements and n is the number of intervals.
(a) For n = 4:
Δt = (12-0)/4 = 3
Total distance ≈ 3 × (0.5 × 23 + 27 + 30 + 33 + 0.5 × 37)
≈ 3 × (11.5 + 27 + 30 + 33 + 18.5)
≈ 3 × 120
≈ 360
(b) For n = 2:
Δt = (12-0)/2 = 6
Total distance ≈ 6 × (0.5 × 23 + 30 + 0.5 × 37)
≈ 6 × (11.5 + 30 + 18.5)
≈ 6 × 60
≈ 360
In this case, both answers are the same, so the upper estimate for the total distance traveled using n = 4 and n = 2 is the same. Therefore, both estimates are equally accurate.