https://file.aiquickdraw.com/mathbot/file/1726850807323a1l1z5cj.jpg

In the image, there are two tasks that need to be solved, both involving volumes of solids of revolution:

### Task 1:
The first task asks for the design of a cylinder with a given volume of \( V = 2309.1 \, \text{cm}^3 \) and a circular base with an area of \( A = 153.94 \, \text{cm}^2 \). The problem asks to calculate the dimensions (specifically the height and radius) and to draw the generatrix of the cylinder.

To solve this:
- The formula for the volume of a cylinder is:
  \[
  V = A \cdot h = \pi r^2 \cdot h
  \]
  Where \( r \) is the radius of the base and \( h \) is the height.

1. Using the area of the base:
   \[
   A = \pi r^2 = 153.94 \, \text{cm}^2
   \]
   Solve for \( r \).

2. Once the radius \( r \) is determined, use the volume formula:
   \[
   V = \pi r^2 \cdot h
   \]
   to solve for the height \( h \).

### Task 2:
The second task asks to design a sphere that can generate the semicircle that has the same area as the top of the cylinder from the first task. It also asks for the volume of this sphere. 

The volume of a sphere is given by:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
Here, the radius of the sphere should be equal to the radius of the base of the cylinder calculated in Task 1.

Let's go ahead and calculate the radius of the base and the height of the cylinder from Task 1.

I'll start by calculating the radius of the cylinder.The radius of the cylinder's base is approximately \( r = 7.00 \, \text{cm} \).

Now, I'll use this radius to calculate the height \( h \) of the cylinder using the volume formula:

\[
V = \pi r^2 h
\]

Solving for \( h \):

\[
h = \frac{V}{\pi r^2}
\]

Let's calculate the height next.The height of the cylinder is \( h = 15.0 \, \text{cm} \).

Now, let's move to **Task 2**, where we need to calculate the volume of the sphere that generates a semicircle with the same radius as the base of the cylinder. The formula for the volume of the sphere is:

\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]

I'll calculate the volume of the sphere using the same radius \( r = 7.00 \, \text{cm} \).The volume of the sphere is approximately \( V_{\text{sphere}} = 1436.78 \, \text{cm}^3 \).

### Summary:
1. **Cylinder**:
   - Radius: \( r = 7.00 \, \text{cm} \)
   - Height: \( h = 15.0 \, \text{cm} \)
   
2. **Sphere**:
   - Volume: \( V_{\text{sphere}} = 1436.78 \, \text{cm}^3 \)

Would you like a detailed explanation of any part of the solution, or perhaps help drawing the generatrix for these solids?

Here are 5 related questions for practice:
1. What would happen to the height of the cylinder if the base area were doubled?
2. How would the volume of the sphere change if the radius increased by 1 cm?
3. What is the surface area of the cylinder with the given dimensions?
4. How does the volume of a hemisphere compare to that of a full sphere with the same radius?
5. What is the ratio of the volume of the cylinder to the sphere?

**Tip:** When dealing with solids of revolution, understanding the relationship between base area and height (for cylinders) and radius (for spheres) is key to solving complex geometric problems.

Un ingeniero industrial necesita diseñar un cilindro con un volumen de 2 309.1 cm³ y base circular de 153.94 cm². Calcula las dimensiones y dibuja la generatriz que genera este sólido de revolución. Esta vez, el ingeniero necesita diseñar una esfera que tenga como generatriz el semicírculo que tiene la tapa del cilindro anterior. Calcula el volumen del sólido de revolución que genera y dibuja su generatriz.

Learn how to calculate the dimensions of a cylinder with a volume of 2 309.1 cm³ and a base area of 153.94 cm². The problem also involves finding the volume of a sphere with the same base radius. This exercise covers geometric concepts of solid revolutions.

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