Question

I need help. The answers has to be exact so I can’t use decimals

Answer 1

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:

$LSA=π(radius\times length)$

The radius will be = diameter/2 = 15cm/2 = 7.5 cm

To find legth, we use Pythagoras rule:

$\begin{gathered} l^2=h^2+r^2 \\ \\ l^2=15^2+7.5^2 \\ \\ l^2=225+56.25=281.25 \\ \\ l=\sqrt{281.25} \\ \\ l=16.77cm \end{gathered}$

Put π = 3.14, r = 7.5 cm, and l = 16.77 cm into the lateral surface area formula:

$\begin{gathered} LSA=3.14\times7.5cm\times16.77cm \\ \\ LSA=394.93\text{ }cm^2 \\ \\ LSA\approx395\text{ }cm^2 \end{gathered}$

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:

$TSA=\pi r^2+\pi rl$

put π = 3.14, r = 7.5 cm, and l = 16.77 cm

$\begin{gathered} TSA=(3.14\times(7.5cm)^2)+(3.14\times7.5cm\times16.77cm) \\ \\ TSA=176.63cm^2+394.93cm^2 \\ \\ TSA=571.56cm^2 \\ \\ TSA\approx572\text{ }cm^2 \end{gathered}$

The exact value of the total surface area = 572 cm²