Question

a area lateral de um cilindro é 324πcm2. sabendo que o raio da base é a metade da altura, calcule sua area total

Answer 1

 Dados:

$\displaystyle \mathsf{S_l = 324 \pi \: cm^2 }\\ \displaystyle \mathsf{r = \frac{h}{2}}\\ \displaystyle \mathsf{S_t = ? }$

Cálculo:

Calculando a altura:

$\displaystyle \mathsf{S_l = 2\pi r.h}\\ \displaystyle \mathsf{324 \pi = 2.\pi.\frac{h}{2}.h}\\ \\ \displaystyle \mathsf{324\pi = \frac{2\pi h^2}{2}}\\ \\ \displaystyle \mathsf{324\pi = \pi h^2}\\ \\ \displaystyle \mathsf{h^2 = \frac{324\pi}{\pi}}\\ \displaystyle \mathsf{h^2 = 324}\\ \displaystyle \mathsf{h = \sqrt{324}}\\ \displaystyle \mathsf{h = 18 \: cm}$

Calculando o raio:

$\displaystyle \mathsf{r = \frac{h}{2}}\\ \\ \displaystyle \mathsf{r = \frac{18}{2}}\\ \\ \displaystyle \mathsf{r = 9 \: cm}$

Calculando a Área total:

$\displaystyle \mathsf{S_t = 2\pi r(r + h)}\\ \displaystyle \mathsf{S_t = 2.\pi .9(9 + 18)}\\ \displaystyle \mathsf{S_t = 18 \pi (27)}\\ \displaystyle \mathsf{S_t = 486 \pi \: cm^2}$

Resposta: 486π cm²