Answer
(a) Exact lateral area:
[tex]3\sqrt{34}\pi\text{ m}^2[/tex]Approximate lateral area:
[tex]54.96\text{ m}^2[/tex](b) Exact total area:
[tex]\begin{equation*} 9\pi+3\sqrt{34}\pi\text{ m}^2{} \end{equation*}[/tex]Approximate total area:
[tex]\begin{equation*} 83.23\text{ m}^2 \end{equation*}[/tex](c) Exact volume:
[tex]\begin{equation*} 15\pi\text{ m}^3 \end{equation*}[/tex]
Approximate volume:
[tex]\begin{equation*} 47.12\text{ m}^3 \end{equation*}[/tex]Step-by-step explanation
Lateral area of a cone formula
[tex]LA=\pi r{\sqrt{h^2+r^2}}[/tex]where r is the radius and h is the height of the cone.
Substituting h = 5 m, and r = 3 m, we get:
[tex]\begin{gathered} LA=\pi\cdot3\cdot\sqrt{5^2+3^2} \\ LA=3\sqrt{34}\pi\text{ m}^2 \\ LA\approx54.96\text{ m}^2 \end{gathered}[/tex]Total area of a cone formula
[tex]TA=\pi r^2+LA[/tex]Substituting with the previous result and r = 3 m, we get:
[tex]\begin{gathered} TA=\pi\cdot3^2+3\sqrt{34}\pi \\ TA=9\pi+3\sqrt{34}\pi\text{ m}^2{} \\ TA\approx83.23\text{ m}^2 \end{gathered}[/tex]Volume of a cone formula
[tex]V=\frac{1}{3}\pi r^2h[/tex]Substituting h = 5 m, and r = 3 m, we get:
[tex]\begin{gathered} V=\frac{1}{3}\cdot\pi\cdot3^2\cdot5 \\ V=15\pi\text{ m}^3 \\ V\approx47.12\text{ m}^3 \end{gathered}[/tex]