Can you guys help me solve this question, please? Thank you so much. **Problem Statement:**You need to make a fence that borders a cliff. It must be rectangular in shape and you are asked to use the cliff as one of the sides (so only 3 sides of fencing). The fencing costs $5 per foot. If the area of the rectangular enclosure must be 10,000 square feet, what must the dimensions be in order to minimize the cost?**Explanation:**In this problem, we want to find the dimensions of the rectangle with one side as the cliff. The enclosure consists of a width and two lengths. To minimize the cost, we need to minimize the total length of fencing used, considering the cost per foot.**Approach to Solution:**1. **Define Variables:** - Let $ x $ be the width (the side perpendicular to the cliff). - Let $ y $ be the length (parallel to the cliff).2. **Area Constraint:** - The area of the rectangle is given by $ x \times y = 10,000 $.3. **Cost Function:** - The cost to fence the three sides is proportional to their length: $ C = 5 \times (2x + y) $. - Substitute $ y $ from the area constraint: $ y = \frac{10,000}{x} $. - Thus, the cost function becomes: $ C(x) = 5 \times (2x + \frac{10,000}{x}) $.4. **Optimization:** - Differentiate $ C(x) $ to find the critical points, then solve for $ x $. - Use these critical points to determine the dimension $ x $ that minimizes the fencing cost and compute $ y $ accordingly.

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Description: Can you guys help me solve this question, please? Thank you so much. **Problem Statement:**You need to make a fence that borders a cliff. It must be rectangular in shape and you are asked to use the cliff as one of the sides (so only 3 sides of fenci

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Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Can you guys help me solve this question, please? Thank you so much.

**Problem Statement:**

You need to make a fence that borders a cliff. It must be rectangular in shape and you are asked to use the cliff as one of the sides (so only 3 sides of fencing). The fencing costs $5 per foot. If the area of the rectangular enclosure must be 10,000 square feet, what must the dimensions be in order to minimize the cost?

**Explanation:**

In this problem, we want to find the dimensions of the rectangle with one side as the cliff. The enclosure consists of a width and two lengths. To minimize the cost, we need to minimize the total length of fencing used, considering the cost per foot.

**Approach to Solution:**

1. **Define Variables:**
- Let $ x $ be the width (the side perpendicular to the cliff).
- Let $ y $ be the length (parallel to the cliff).

2. **Area Constraint:**
- The area of the rectangle is given by $ x \times y = 10,000 $.

3. **Cost Function:**
- The cost to fence the three sides is proportional to their length: $ C = 5 \times (2x + y) $.
- Substitute $ y $ from the area constraint: $ y = \frac{10,000}{x} $.
- Thus, the cost function becomes: $ C(x) = 5 \times (2x + \frac{10,000}{x}) $.

4. **Optimization:**
- Differentiate $ C(x) $ to find the critical points, then solve for $ x $.
- Use these critical points to determine the dimension $ x $ that minimizes the fencing cost and compute $ y $ accordingly.

Expert Solution

Step 1

Let x feet be the length of the side of the rectangle that is parallel to the cliff and y feet be the length of the other side.

Area of the rectangular enclosure is 10,000 square feet.

$x y = 10000 y = \frac{10000}{x}$

Step 2

Cost of fencing is

$C = 5 [x + 2 y] C = 5 x + 10 y C = 5 x + \frac{100, 000}{x}$

Now, differentiate both sides with respect to x.

$C' (x) = 5 - \frac{100, 000}{x^{2}}$

Now, we solve the equation

$C' (x) = 0 5 - \frac{100, 000}{x^{2}} = 0 x^{2} = 20, 000 x = 100 \sqrt{2}$

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