A solid cylinder has a radius of 10 mm and a height of 45 mm - Leaving Cert Mathematics - Question Question 1 - 2013

The volume of a cylinder is calculated using the formula:

$V = \pi r^2 h$

Substituting the known values:

• Radius, $r = 10 \text{ mm}$
• Height, $h = 45 \text{ mm}$

We get:

$V = \pi (10)^2 (45) = \pi (100)(45) = 4500\pi \text{ mm}^3$

Since the sphere has the same volume as the cylinder, we use the volume of the sphere formula:

$\frac{4}{3} \pi r^3 = 4500\pi$

Equating the volumes:

$\frac{4}{3} r^3 = 4500\implies r^3 = 4500 \times \frac{3}{4} = 3375$

Solving for $r$:

$r = \sqrt[3]{3375} = 15 \text{ mm}$

Now, using this radius, we can find the surface area (A) of the sphere:

$A = 4\pi r^2 = 4\pi (15)^2 = 4\pi (225) = 900\pi \text{ mm}^2$