Solution

\[ \begin{aligned} \text{No. of observations} &= x+3 \\[2pt] \text{Sum of observations} &= x^4-81 \\[6pt] \color{green}\text{Mean} &= \color{green}\frac{\text{Sum of observations}}{\text{No. of observations}} \\[6pt] &= \frac{x^4-81}{x+3} \\[6pt] &= \frac{(x^2)^2-(9)^2}{(x+3)} \\[6pt] &= \frac{(x^2 - 9)(x^2 + 9)}{(x+3)} \\[6pt] &= \frac{[(x)^2 - (3)^2](x^2 + 9)}{(x+3)} \\[6pt] &= \frac{(x -3)\cancel{(x+3)}(x^2 + 9)}{\cancel{(x+3)}} \\[6pt] \text{Mean} &= (x-3)(x^2+9) \end{aligned} \]

Answer Mean \(=\ \color{red}{(x -3)(x^2 + 9)}\)

Solution

\[ \begin{aligned} \color{green} \text{Area of a circle} &= \pi x^2+10\pi x+25\pi \\[6pt] \color{green}\pi r^2 &= \pi\,(x^2+10x+25) \\[6pt] \pi r^2 &= \pi\,[(x)^2+ 2(x)(5)+(5)^2] \\[6pt] \pi r^2 &= \pi\,(x+5)^2 \\[6pt] r^2 &= \frac{\cancel{\pi}(x+5)^2}{\cancel{\pi}}\\[6pt] r^2 &= (x+5)^2 \\[6pt] r &= \sqrt{(x+5)^2} \\[6pt] r &= x+5 \end{aligned} \]

Answer Radius of the circle \(=\ \color{red}{(x+5)}\) units