10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9516df6d-0e85-45d8-afb0-281c80450159-28_387_474_340_324}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9516df6d-0e85-45d8-afb0-281c80450159-28_448_716_315_1023}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 2 shows a picture of a plant pot.
The plant pot has
- a flat circular base of radius 10 cm
- a height of 15 cm
Figure 3 shows a sketch of the curve \(C\) with parametric equations
$$x = 10 + 15 t - 5 t ^ { 3 } \quad y = 15 t ^ { 2 } \quad 0 \leqslant t \leqslant 1$$
The curved inner surface of the plant pot is modelled by the surface of revolution formed by rotating curve \(C\) through \(2 \pi\) radians about the \(y\)-axis.
- Show that, according to the model, the area of the curved inner surface of the plant pot is given by
$$150 \pi \int _ { 0 } ^ { 1 } \left( 2 + 3 t + 2 t ^ { 2 } + 2 t ^ { 3 } - t ^ { 5 } \right) \mathrm { d } t$$
- Determine, according to the model, the total area of the inner surface of the plant pot.
Each plant pot will be painted with one coat of paint, both inside and outside. The paint in one tin will cover an area of \(12 \mathrm {~m} ^ { 2 }\)
- Use the answer to part (b) to estimate how many plant pots can be painted using one tin of paint.
- Give a reason why the model might not give an accurate answer to part (c).