| Date | November 2016 | Marks available | 3 | Reference code | 16N.2.sl.TZ0.6 |
| Level | SL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 6 | Adapted from | N/A |
Question
All lengths in this question are in metres.
Let f(x)=−0.8x2+0.5, for −0.5⩽. Mark uses f(x) as a model to create a barrel. The region enclosed by the graph of f, the x-axis, the line x = - 0.5 and the line x = 0.5 is rotated 360° about the x-axis. This is shown in the following diagram.
Use the model to find the volume of the barrel.
The empty barrel is being filled with water. The volume V{\text{ }}{{\text{m}}^3} of water in the barrel after t minutes is given by V = 0.8(1 - {{\text{e}}^{ - 0.1t}}). How long will it take for the barrel to be half-full?
Markscheme
attempt to substitute correct limits or the function into the formula involving
{y^2}
eg\,\,\,\,\,\pi \int_{ - 0.5}^{0.5} {{y^2}{\text{d}}x,{\text{ }}\pi \int {{{( - 0.8{x^2} + 0.5)}^2}{\text{d}}x} }
0.601091
volume = 0.601{\text{ }}({{\text{m}}^3}) A2 N3
[3 marks]
attempt to equate half their volume to V (M1)
eg\,\,\,\,\,0.30055 = 0.8(1 - {{\text{e}}^{ - 0.1t}}), graph
4.71104
4.71 (minutes) A2 N3
[3 marks]
