A model rocket has upward velocity seconds after launch. Use a Riemann sum with n = 10 to estimate how high the rocket is 2 seconds after launch.
A)184.8 feet
B)158.4 feet
C)136.8 feet
D)115.2 feet
E)240 feet

Question

A model rocket has upward velocity  seconds after launch. Use a Riemann sum with n = 10 to estimate how high the rocket is 2 seconds after launch.
A)184.8 feet 
B)158.4 feet 
C)136.8 feet 
D)115.2 feet 
E)240 feet

Quizplus · Accepted Answer

Using a Riemann sum with n=10, we can estimate the rocket's height at 2 seconds after launch as follows:
$\Delta t = \frac{11-2}{10} = 0.9$ (the time interval for each subinterval)
$v(t_i) = 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2.9i$ (the velocity at the start of each subinterval, where $i$ represents the subinterval number)
The approximate height at each subinterval is given by $h(t_i) \approx h(t_{i-1}) + v(t_{i-1}) \Delta t$. We start with $h(2)=0$ and use this formula to estimate the height after each subinterval:
$h(2.9) \approx 0 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 9.9ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(3.8) \approx 9.9ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 19.8ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(4.7) \approx 19.8ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 29.7ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(5.6) \approx 29.7ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 39.6ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(6.5) \approx 39.6ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 49.5ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(7.4) \approx 49.5ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 59.4ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(8.3) \approx 59.4ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 69.3ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(9.2) \approx 69.3ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 79.2ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(10.1) \approx 79.2ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 89.1ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
$h(11) \approx 89.1ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 + 11ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2 \cdot 0.9 = 99ebfe6a_2ce3_14ff_96cc_a3a6c031cb6a_TBX8650_2$
So the rocket's height 2 seconds after launch is approximately $h(2)=136.8$ feet. Therefore, the answer is (C) 136.8 feet.

A Model Rocket Has Upward Velocity Seconds After Launch