Volume and Surface Area of Cylinders, Cones, and Spheres

We will calculate the surface area of the given birdcage. To do so, we will first calculate the surface area of the cylindrical part and then the surface area of the hemispherical part.

The cylindrical part consists of a rectangular lateral face and a circular bottom base. The circular top base is missing. This means that we can write its surface area formula in the following way. SA_c=2π r h +π r^2 Next, we will consider the hemispherical part. Recall that the surface area of a hemisphere is half of the surface area of the sphere with the same radius. In our situation that means the bottom circular base is missing. SA_h=2π r^2 Let's now add these formulas up to find the total surface area of the birdcage. SA=2π r h +π r^2+2π r^2 ⇓ SA=2π r l +3π r^2 Finally, we can substitute r= 18 and h= 45 into the formula to calculate for the total surface area of the birdcage.

The surface area of the given birdcage is about 8143 square centimeters.

This time we will calculate the volume of this cage. To do so, we will calculate the sum of the volumes of the cylindrical part and the hemispherical part. V= V_c+ V_h Recall that the volume of a cylinder is the product of its circular base area and height. The volume of a hemisphere is half of the volume of a sphere with the same radius. V= π r^2 h+ 2/3π r^3 Let's now substitute r= 18 and h= 45 into this formula.

The volume of the birdcage is about 58 019 cubic centimeters.

First, note that the glass container is the shape of a cylinder. We can find its volume that is not occupied by the tennis balls by subtracting the their total volume from the volume of the cylinder. Begin by finding the volume of the container. Since there are three tennis balls on top of each other, the height of the cylinder will be 3 times 2r.

Recall that the volume of a cylinder is the product of the base area and its height. V_c=π r^2 h Let's substitute h= 6r into the formula.

Next, we will write the volume of three tennis balls. Note that a tennis ball is in the shape of a sphere. V_b=4/3π r^3 ⇓ V_(3b)= 3 * 4/3π r^3 = 4π r^3 Since there are three tennis balls, the total volume occupied is 4π r^3. By subtracting this volume from the volume of the cylinder, we can find the volume that is not occupied by the tennis balls. 6π r^3- 4π r^3=2π r^3 Note that the radius of a tennis ball is also equal to the radius of the cylindrical container. Therefore, we will substitute r= 3.4 into the formula.

We found that the volume that is not occupied by the tennis balls in the glass container is about 247 cubic centimeters.

Let's look closely at the two given cases.

• Case 1: Double the radius of a cone
• Case 2: Double the slant height of the cone

Understanding how these changes affect the surface area of the cone can help us determine which is increasing the surface area more than the other. Recall the formula for the surface area of a cone. SA=π r^2+π r l Note that a cone's surface area consists of a circular base and a lateral face.

Case 1: Double the Radius of a Cone

Let's substitute r= 2r into the original area formula.

Case 2: Double the Slant Height

We can substitute l= 2l into the area formula. SA=π r^2+π r( 2 l) ⇓ SA=π r^2+2 π r l

Which Increases the Surface Area More?

Let's see how the formula SA=π r^2+π r l has changed in both cases.

Both expressions include the term 2π r l. That means we ought to focus on the other term in this comparison. Since 4π r^2 is greater than π r^2, we can conclude that doubling the radius of the cone leads to a greater surface area. Let's find the difference!

The surface area of a cone with a doubled radius will be 3π r^2 greater the surface area of the cone with the doubled slant height.

We will calculate the surface area of the given chart. Notice that it is in the shape of a cylinder with a missing part. Therefore, its surface area consists of the 75 % of the surface area of full cylinder plus the area of two lateral rectangles.

Let's first calculate the areas of the rectangles. Notice that the radius is 15 centimeters and the height of the cylinder is 3 centimeters. Therefore, the length of each rectangle is 15 centimeters and the width is 3 centimeters. \begin{gathered} A_\text{Rectangle}=3 \cdot 15 =45 \end{gathered} The area of one rectangle is 45 square centimeters. Now, we will calculate the surface area of the full cylinder. Then, we will calculate its 75 %. Recall that the surface area of a cylinder consists of the areas of the base and top circles and the lateral face. SA=2π r h +2π r^2 We know that r= 15 and h= 3. Let's substitute these values into the formula.

The surface area of a full cylinder with the same radius and height is about 540π square centimeters. Now, we will calculate 75 % of this area.

Finally, we will add the areas of two rectangles to this area. 1272+2 *45 =1362 The total surface area of the given chart is about 1362 square centimeters.

This time we will calculate the volume of this partial chart. To do so, we will first calculate the volume of the full cylinder and then we will find 75 % of this volume. Recall that the volume of a cylinder is the product of the area of its base and its height. V=π r^2h We will substitute r= 15 and h= 3 into the above formula.

Next, we will calculate 75 % of the volume of the full cylinder.

The volume of the given partial chart is about 1590 cubic centimeters.