Right cone from height, radius and slant relations

Text

In a right cone the height is $\dfrac{15}{8}$ of the base radius. Knowing that $\dfrac32$ of the slant height plus $\dfrac54$ of the base diameter measures $91$ cm, find the radius, height, slant height and the total surface area.

Solution

With $h=\dfrac{15}{8}r$, the slant height is $a=\sqrt{h^2+r^2}=\sqrt{\dfrac{225}{64}r^2+r^2}=\dfrac{17}{8}r$. The condition: $\frac32\cdot\frac{17}{8}r+\frac54\cdot2r=\frac{51}{16}r+\frac{40}{16}r=\frac{91}{16}r=91\ \Rightarrow\ r=16,\ h=30,\ a=34.$ Total surface: $S_t=\pi r(r+a)=\pi\cdot16\cdot50=800\pi\approx2513.27$ cm$^2$. $\boxed{r=16;\ h=30;\ a=34\ \text{cm};\ S_t=800\pi\approx2513.27\ \text{cm}^2}$