Answer – The radius of the given cylinder can be found using the formula for the surface area of a cylinder $\mathit{A}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{2}\mathbf{\pi }\mathit{r}\mathit{h}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{2}\mathbf{\pi }{\mathit{r}}^{\mathbf{2}}$ as $\mathbf{4}\mathbf{ }\mathit{c}\mathit{m}$.

Explanation:

When given the height and surface area of a cylinder, we can use the below formula to find its radius:

$A = 2\mathrm{\pi }rh + 2\mathrm{\pi }{r}^2$

This is the formula for the surface area of a cylinder, where A is the surface area, r is the radius, and h is the height of the cylinder.

From the question, we know that $A = 200\mathrm{\pi } c{m}^2$ and $h = 21 cm$. We can substitute these values in the above formula and solve for the radius r of the cylinder.

$\Rightarrow 200\pi = 2\pi r \left(21\right) + 2\pi r^2$

To simplify the equation further, we divide both sides by $2\pi$:

$200\pi = 2\pi r \left(21\right) + 2\pi r^2$

$÷2\pi ÷ 2\pi$

$\Rightarrow 100 = 21r + r^2$

Now, we subtract both sides by 100 to get a quadratic equation in r:

$100 = 21r + r^2$

$–100 –100$

$\Rightarrow r^2 + 21r – 100 = 0$

Finally, we solve this quadratic equation in r by factoring as shown below:

$\Rightarrow r^2 + 25r –4r – 100 = 0$

$\Rightarrow r (r + 25) – 4 (r + 25) = 0$

$\Rightarrow (r – 4) (r + 25) = 0$

$\Rightarrow r – 4 = 0$ or $r + 25 = 0$

$\Rightarrow r = 4$ or $r = –25$

But since r represents the radius of the cylinder, it cannot be negative.

Therefore, the radius of the cylinder is $4 cm$.

We can also verify our answer by substituting the value of r in the surface area equation:

$A = 2\mathrm{\pi }rh + 2\mathrm{\pi }{r}^2$

$A = 2\pi \left(4\right) \left(21\right) + 2\pi {\left(4\right)}^2$

$A = 168\pi + 32\pi = 200\pi c{m}^2$

Thus, the radius is, in fact, $4 cm$.

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