Conic Sections

To solve the problem, we need to determine the length of the edge of a new cube formed by melting three cubes with side lengths , , and . The volume of the new cube will be equal to the sum of the volumes of the three original cubes.

Step 1: Calculate the volume of each original cube.
The volume of a cube with side length is given by:

For the first cube ( ):

For the second cube ( ):

For the third cube ( ):

Step 2: Find the total volume of the three cubes.
The total volume is the sum of the volumes of the three cubes:

Step 3: Determine the side length of the new cube.
Let the side length of the new cube be . The volume of the new cube is also given by . Since the volume of the new cube is equal to the total volume of the three original cubes:

Solve for by taking the cube root of both sides:

Final Answer:
The length of the edge of the new cube is .

To solve the problem, we need to determine the capacity (volume) of a reservoir shaped like a frustum of a right circular cone. The given dimensions are:

Diameter at the top:
Diameter at the bottom:
Depth (height) of the frustum:

Step 1: Formula for the Volume of a Frustum of a Cone The volume of a frustum of a cone is given by: where:
is the radius of the larger base (top),
is the radius of the smaller base (bottom),
is the height (depth) of the frustum.

Step 2: Calculate the Radii The radii are half of the respective diameters: - Radius at the top ( ): - Radius at the bottom ( ): Step 3: Substitute Known Values into the Volume Formula We are given: - , - , - . Substitute these values into the formula: Step 4: Simplify the Expression First, calculate the terms inside the parentheses: Now substitute back: Step 5: Approximate the Volume Using : Step 6: Match with the Given Options The closest value to among the options is .

Final Answer:

To solve the problem, we need to determine the ratio of the vertical heights of two cones given that the ratio of their volumes is and the ratio of the radii of their bases is .

Step 1: Formula for the Volume of a Cone The volume of a cone is given by: where: - is the radius of the base, - is the height (or vertical height) of the cone. Step 2: Express the Volumes of the Two Cones Let the radii of the bases of the two cones be and , and their heights be and . The volumes of the two cones are: Step 3: Given Ratios We are given: 1. The ratio of the volumes: 2. The ratio of the radii: Step 4: Substitute the Ratios into the Volume Formula Using the volume formulas, we can write: Given that , we have: Step 5: Use the Ratio of Radii From the given ratio of radii, , we can square both sides to find the ratio of the squares of the radii: Thus: Step 6: Solve for the Ratio of Heights Substitute into the equation : Simplify: Multiply both sides by : Final Answer: The ratio of the vertical heights of the two cones is:

To solve the problem, we need to find the ratio of volumes of a right circular cylinder and a cone given the ratios of their radii and heights.

1. Volume Formulas:

The volume of a cylinder is given by:

The volume of a cone is given by:

2. Given Ratios:

Ratio of radii:

Ratio of heights:

3. Expressing Volumes in Terms of Ratios:

Let , , ,

Then:

4. Finding the Ratio:

Final Answer:
The ratio of their volumes is