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Optimization Problems - Worksheet 4 - 12th Grade

Medium

Use derivative signs and stated geometric constraints to interpret functions and rates.

medium: Practice the same exercise types with a fresh numbered set.

12 problems · 30-35 minutes · Skills: derivatives, optimization, problem solving

About This Worksheet

This medium-level worksheet contains 12 problems designed for 12th Grade students practicing optimization problems. Medium worksheets introduce larger numbers and varied question formats to deepen understanding and build fluency.

Manipulate algebraic expressions and understand functions, limits and derivative notation.

  • Use derivative signs and stated geometric constraints to interpret functions and rates.
  • Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: A rectangle has fixed perimeter 12 m and positive side lengths. What is the largest possible area?
Solution Steps:
  1. Let one side be x, so the other is 6 − x.
  2. A(x) = x(6 − x), whose derivative is 6 − 2x.
  3. The maximum occurs at x = 3, a square.
Answer: 9 m²
Tip: Check your result using the method in the example before moving to the next question.
1
A rectangle has fixed perimeter 8 m and positive side lengths. What is the largest possible area?
2
A rectangle has fixed perimeter 20 m and positive side lengths. What is the largest possible area?
3
A rectangle has fixed perimeter 4 m and positive side lengths. What is the largest possible area?
4
A rectangle has fixed perimeter 16 m and positive side lengths. What is the largest possible area?
5
A rectangle has fixed perimeter 12 m and positive side lengths. What is the largest possible area?
6
A rectangle has fixed perimeter 24 m and positive side lengths. What is the largest possible area?
7
A rectangle has fixed perimeter 8 m and positive side lengths. What is the largest possible area?
8
A rectangle has fixed perimeter 16 m and positive side lengths. What is the largest possible area?
9
A rectangle has fixed perimeter 20 m and positive side lengths. What is the largest possible area?
10
A rectangle has fixed perimeter 16 m and positive side lengths. What is the largest possible area?
11
A rectangle has fixed perimeter 20 m and positive side lengths. What is the largest possible area?
12
A rectangle has fixed perimeter 12 m and positive side lengths. What is the largest possible area?

Answer Key

1. 4 m²2. 25 m²3. 1 m²4. 16 m²5. 9 m²6. 36 m²7. 4 m²8. 16 m²9. 25 m²10. 16 m²11. 25 m²12. 9 m²

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What skills does this applications of derivatives worksheet cover?

This worksheet focuses on derivatives, optimization, problem solving. It is designed for 12th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

Can I use this worksheet in my classroom?

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How many problems are on this worksheet?

This worksheet contains 12 problems. It typically takes 30-35 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.