rubber - band - powered sled a sled powered by a wound rubber band moves along a track until friction and…

rubber - band - powered sled a sled powered by a wound rubber band moves along a track until friction and the unwinding of the rubber band gradually slow it to a stop. a speedometer in the sled monitors its speed, which is recorded at 3 - second intervals during the 27 - second run.\ntime (sec) speed (ft/sec)\n0 5.30\n3 5.25\n6 5.04\n9 4.71\n12 4.25\n15 3.66\n18 2.94\n21 2.09\n24 1.11\n27 0\n(a) give an upper estimate and a lower estimate for the distance traveled by the sled.\n(b) use the trapezoidal rule to estimate the distance traveled by the sled.

rubber - band - powered sled a sled powered by a wound rubber band moves along a track until friction and the unwinding of the rubber band gradually slow it to a stop. a speedometer in the sled monitors its speed, which is recorded at 3 - second intervals during the 27 - second run.\ntime (sec) speed (ft/sec)\n0 5.30\n3 5.25\n6 5.04\n9 4.71\n12 4.25\n15 3.66\n18 2.94\n21 2.09\n24 1.11\n27 0\n(a) give an upper estimate and a lower estimate for the distance traveled by the sled.\n(b) use the trapezoidal rule to estimate the distance traveled by the sled.

Answer

Explanation:

Step1: Define Δt

The time - interval $\Delta t = 3$ seconds.

Step2: Calculate upper estimate

For the upper estimate, we use the left - hand endpoints of each sub - interval. The upper estimate $U=\sum_{i = 0}^{9 - 1}v(t_i)\Delta t$. $U=3\times(5.30 + 5.25+5.04 + 4.71+4.25+3.66+2.94+2.09+1.11)$ $U = 3\times(34.35)=103.05$ feet.

Step3: Calculate lower estimate

For the lower estimate, we use the right - hand endpoints of each sub - interval. The lower estimate $L=\sum_{i = 1}^{9}v(t_i)\Delta t$. $L=3\times(5.25 + 5.04+4.71+4.25+3.66+2.94+2.09+1.11+0)$ $L = 3\times(29.05)=87.15$ feet.

Step4: Apply Trapezoidal Rule

The Trapezoidal Rule formula is $T=\frac{\Delta t}{2}\left[v(t_0)+2v(t_1)+2v(t_2)+\cdots+2v(t_{n - 1})+v(t_n)\right]$. Here, $\Delta t = 3$, $n = 9$, $t_0 = 0,t_1 = 3,\cdots,t_9=27$. $T=\frac{3}{2}\left[5.30+2\times(5.25 + 5.04+4.71+4.25+3.66+2.94+2.09+1.11)+0\right]$ $T=\frac{3}{2}\left[5.30+2\times29.05\right]$ $T=\frac{3}{2}(5.30 + 58.1)$ $T=\frac{3}{2}\times63.4$ $T = 95.1$ feet.

Answer:

(a) Upper estimate: $103.05$ feet, Lower estimate: $87.15$ feet (b) $95.1$ feet