Answer
The exact value of the lateral surface area = 395 cm²
The exact value of the total surface area = 572 cm²
Explanation
The solid shape is a cone with a height of 15 cm and the diameter of the base of the solid shape is 15 cm.
The Lateral Surface Area of the Solid Shape:
The formula to calculate the lateral surface area (LSA) of a cone is given by:
[tex]LSA=π(radius\times length)[/tex]The radius will be = diameter/2 = 15cm/2 = 7.5 cm
To find legth, we use Pythagoras rule:
[tex]\begin{gathered} l^2=h^2+r^2 \\ \\ l^2=15^2+7.5^2 \\ \\ l^2=225+56.25=281.25 \\ \\ l=\sqrt{281.25} \\ \\ l=16.77cm \end{gathered}[/tex]Put π = 3.14, r = 7.5 cm, and l = 16.77 cm into the lateral surface area formula:
[tex]\begin{gathered} LSA=3.14\times7.5cm\times16.77cm \\ \\ LSA=394.93\text{ }cm^2 \\ \\ LSA\approx395\text{ }cm^2 \end{gathered}[/tex]Therefore, The exact value of the lateral surface area = 395 cm²
Total Surface Area of the Solid Shape:
To find the exact value for the total surface area of the solid shape, we use the total surface area (TSA) formula of a cone which is:
[tex]TSA=\pi r^2+\pi rl[/tex]put π = 3.14, r = 7.5 cm, and l = 16.77 cm
[tex]\begin{gathered} TSA=(3.14\times(7.5cm)^2)+(3.14\times7.5cm\times16.77cm) \\ \\ TSA=176.63cm^2+394.93cm^2 \\ \\ TSA=571.56cm^2 \\ \\ TSA\approx572\text{ }cm^2 \end{gathered}[/tex]The exact value of the total surface area = 572 cm²
