The diagram shows a solid formed by joining a hemisphere, of radius r cm, to a cylinder, of radius r cm and height h cm. The total height of the solid is 18 cm and the surface area is 205π cm².

• Find the value of r and the value of h.

To solve the problem, we need to find the values of the radius \( r \) and the height \( h \) of the solid formed by a hemisphere and a cylinder. The total height of the solid is given as 18 cm, and the surface area is 205π cm².

Understanding the Components

The solid consists of two parts:

• A hemisphere with radius \( r \)
• A cylinder with radius \( r \) and height \( h \)

Height Relationship

The total height of the solid can be expressed as:

Height of the solid = Height of the cylinder + Radius of the hemisphere

This gives us the equation:

h + r = 18

Surface Area Calculation

The surface area of the solid includes the curved surface area of the cylinder, the curved surface area of the hemisphere, and the base area of the cylinder. The formula for the surface area is:

Surface Area = 2πrh + 2πr² + πr²

Combining the terms, we have:

Surface Area = 2πrh + 3πr²

Setting this equal to the given surface area:

2πrh + 3πr² = 205π

Dividing through by π simplifies to:

2rh + 3r² = 205

Solving the Equations

Now we have two equations:

• 1. \( h + r = 18 \)
• 2. \( 2rh + 3r² = 205 \)

From the first equation, we can express \( h \) in terms of \( r \):

h = 18 - r

Substituting this into the second equation gives:

2r(18 - r) + 3r² = 205

Expanding this results in:

36r - 2r² + 3r² = 205

Combining like terms leads to:

r² + 36r - 205 = 0

Finding the Roots

Using the quadratic formula \( r = \frac{-b \pm \sqrt{b² - 4ac}}{2a} \), where \( a = 1, b = 36, c = -205 \):

r = \frac{-36 \pm \sqrt{36² - 4(1)(-205)}}{2(1)}

Calculating the discriminant:

36² + 820 = 1296

Thus, we find:

r = \frac{-36 \pm 36}{2}

This gives us two potential solutions for \( r \): \( r = 0 \) or \( r = 5 \). Since \( r \) must be positive, we have:

r = 5 cm

Calculating Height

Substituting \( r \) back into the height equation:

h = 18 - 5 = 13 cm

Final Values

The values we found are:

• Radius (r): 5 cm
• Height (h): 13 cm