Karen makes pottery on a pottery wheel. Today she is making vases that are in the shape of a circular cylinder that is open at the top, that is, it has only one base. The base has a radius of 4.5 centimeters and 0.75 centimeters thick. The lateral surface of the cylinder will be 0.4 centimeters thick. She uses 206 cubic centimeters of clay for each vase.

a. How much clay is used for the base of the vase to the nearest tenth.
b. How much clay will be used for the lateral surface of the vase to the nearest tenth
c. How tall will the vase be to the nearest tenth?
c. What will be the lateral surface to the nearest tenth. ( Use the value of the height of the vase found in part c.

The volume of clay is for a cylinder of radius r is

pi r^2(0.75) + pi(r^2-(r-.4)^2)h = 206
Since we have r=4.5, h = 14.6 cm

The base is easy, and just subtract that from 206 to get the lateral volume.

The lateral surface is just 2pi r h

To find the answer to this question, we need to calculate various measurements related to the vase. Let's go through each part step-by-step.

a. How much clay is used for the base of the vase to the nearest tenth?

To find the clay used for the base, we need to calculate the volume of the base. The base is a circular disk, which can be represented by the formula:

Volume of the base = π * radius^2 * thickness

Given:
Radius of the base (r) = 4.5 cm
Thickness of the base (t) = 0.75 cm

Plugging these values into the formula:

Volume of the base = π * (4.5 cm)^2 * 0.75 cm

Calculating this value gives us the volume of the base. Since we need to round to the nearest tenth, we will round the answer to one decimal place.

b. How much clay will be used for the lateral surface of the vase to the nearest tenth?

To find the clay used for the lateral surface, we need to calculate the volume of the cylindrical part of the vase. The formula for the volume of a cylinder is:

Volume of the cylinder = π * radius^2 * height

Given:
Radius of the base (r) = 4.5 cm
Thickness of the base (t) = 0.4 cm

Since the total height of the vase is not given explicitly, we'll calculate it in part c. But for now, we can find the lateral surface area by subtracting the thickness of the base from the total height of the vase.

c. How tall will the vase be to the nearest tenth?

To find the height of the vase, we need to subtract the thickness of the base and the thickness of the lateral surface from the total height of the vase. The total height (h) of the vase can be represented as:

Total height of the vase = height of the cylinder + thickness of the base + thickness of the lateral surface

We know the thickness of the base (t1 = 0.75 cm) and the thickness of the lateral surface (t2 = 0.4 cm). We can substitute these values into the equation and solve for the total height of the vase.

c. What will be the lateral surface to the nearest tenth?

Once we have calculated the height of the vase, we can find the lateral surface area. The formula for the lateral surface area of a cylinder is:

Lateral surface area = 2 * π * radius * height

Using the height value from part c and the radius:

Lateral surface area = 2 * π * 4.5 cm * height

Substituting the values, we can calculate the lateral surface area rounded to the nearest tenth.

With these steps, you can find the answers to each part of the question and have a comprehensive understanding of how to calculate them.