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Let's solve each question step-by-step:

### Question 20

A factory has \( n \) employees, \( w \) of whom are women. If 10 percent of the men employed are 60 years of age or older, how many men under 60 years of age are employed in this factory?

1. **Calculate the number of men:**

   The total number of employees is \( n \). The number of women is \( w \), so the number of men is:
   \[
   n - w
   \]

2. **Calculate the number of men aged 60 or older:**

   10% of the men are aged 60 or older:
   \[
   0.1(n - w)
   \]

3. **Calculate the number of men under 60:**

   The number of men under 60 is the total number of men minus those who are 60 or older:
   \[
   (n - w) - 0.1(n - w) = (n - w)(1 - 0.1) = (n - w) \times 0.9 = 0.9(n - w)
   \]

Thus, the correct answer is **(E) \(\frac{0.9(n - w)}{n}\)**.

### Question 21

Two parallel planes are 3 inches apart. If a sphere with a radius of 1 inch has its center equidistant from the two planes, in how many points does the sphere intersect the two planes?

1. **Understand the geometry:**

   - The sphere has a radius of 1 inch, and its center is exactly in the middle of the two planes, meaning it is 1.5 inches away from each plane.
   - Since the radius of the sphere is 1 inch, and the center is 1.5 inches away from each plane, the sphere does not reach either plane. The distance from the center of the sphere to each plane is greater than the radius.

2. **Intersection:**

   Since the sphere does not reach either plane, it does not intersect with either of the two planes.

Thus, the correct answer is **(A) None**.

### Question 22

Given \( f(x) = 2x^2 - bx + 10 \) and \( f(3) = 13 \), we are to find \( f(2) \).

1. **Use the given function and point:**

   Plug \( x = 3 \) into the function \( f(x) \):
   \[
   f(3) = 2(3)^2 - 3b + 10
   \]
   \[
   13 = 18 - 3b + 10
   \]

2. **Solve for \( b \):**

   Simplify the equation:
   \[
   13 = 28 - 3b
   \]
   \[
   3b = 28 - 13
   \]
   \[
   3b = 15
   \]
   \[
   b = 5
   \]

3. **Find \( f(2) \):**

   Now, substitute \( b = 5 \) and \( x = 2 \) into the function:
   \[
   f(2) = 2(2)^2 - 5 \cdot 2 + 10
   \]
   \[
   f(2) = 2 \times 4 - 10 + 10
   \]
   \[
   f(2) = 8 - 10 + 10
   \]
   \[
   f(2) = 8
   \]

Thus, the correct answer is **(D) 8**.

Would you like more details on any of these solutions? Here are some follow-up questions you might find interesting:

1. How do you find the number of intersections between different geometrical objects?
2. What is the process of solving quadratic equations with given conditions?
3. How does one manipulate percentages in algebraic equations?
4. What are some real-world applications of quadratic functions?
5. Can you explain how a sphere's position relative to planes affects their intersections?

**Tip:** Understanding the properties of geometric shapes, such as spheres and planes, is crucial in solving intersection problems.

Learn how to solve math problems involving factory employee demographics, sphere geometry with intersecting planes, and quadratic equations. Includes step-by-step solutions and explanations for each problem.

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