4. A right trapezoid has bases of lengths $10 \mathrm{~cm}$ and $2 \mathrm{~cm}$ and an area of $90 \mathrm{~cm}^{2}$. If we rotate it around the longer base, calculate the surface area and volume of the resulting solid.
4. From the formula for the area of a trapezoid, we calculate the height of the trapezoid $v=$ $\frac{2 \cdot S}{a+c}=15 \mathrm{~cm}$. We also calculate the height of the cone $v_{s}=a-c=8 \mathrm{~cm}$. We determine that $v_{v}=c$ and that the radii of the cylinder and the cone are equal to the height of the trapezoid. We calculate the volume of the composite body $V=$ $V_{s}+V_{v}=\pi r^{2}\left(v_{v}+\frac{v_{s}}{3}\right)=1050 \pi \mathrm{cm}^{3}$. We calculate the surface area of the composite body $P=S+S_{p l_{v}}+S_{p l_{s}}=\pi\left(r^{2}+2 r v_{v}+r_{s}\right)=$ $540 \pi \mathrm{cm}^{2}$

Calculation of the height of the trapezoid $v=15 \mathrm{~cm}$ .1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-12.jpg?height=477&width=483&top_left_y=221&top_left_x=1366)

Calculation of the slant height $s=\sqrt{r^{2}+v_{s}^{2}}=17 \mathrm{~cm} \ldots \ldots \ldots .1$ point Calculation of the volume $V=V_{s}+V_{v}=\pi r^{2}\left(v_{v}+\frac{v_{s}}{3}\right)=1050 \pi \mathrm{cm}^{3}$ $1+1^{*}$ point Calculation of the surface area $P=S+S_{p l_{v}}+S_{p l_{s}}=\pi\left(r^{2}+2 r v_{v}+r_{s}\right)=540 \pi \mathrm{cm}^{2}$ $1+1^{*}$ point NOTE: For an incorrect unit, we deduct 1 point.