Optimization of Fencing Cost for a Rectangular Field

The video tutorial explains how to minimize the cost of fencing a rectangular field with an area of 800 square feet. It involves setting up a cost function based on different fencing costs for opposite sides, using calculus to find the dimensions that minimize the cost, and verifying the solution using the second derivative test. The optimal dimensions are approximately 25.3 feet by 31.6 feet, resulting in a total cost of $505.80.

What is the primary goal when enclosing the rectangular field?

• Minimize the cost of the fencing

• Maximize the length of the longer sides

• Maximize the area of the field

• Equalize the length of all sides

How much does it cost per foot to fence the shorter sides of the field?

• $6.00

• $5.00

• $4.00

• $3.00

What is the equation for the total cost of the fencing?

• 8X + 10Y

• 10X + 8Y

• 4X + 5Y

• 5X + 4Y

What is the constraint equation for the area of the field?

• X + Y = 800

• X * Y = 800

• X - Y = 800

• X / Y = 800

How is the cost function rewritten in terms of one variable?

• C = 5X + 4(800/Y)

• C = 8X + 10(800/Y)

• C = 10X + 8(800/X)

• C = 10X + 8Y

What is the first step in finding the minimum cost?

• Multiply the cost by the area

• Set the cost function equal to zero

• Find the derivative of the cost function

• Calculate the area of the field

What is the critical number found for X that minimizes the cost?

• 35.1 feet

• 30.7 feet

• 25.3 feet

• 20.5 feet

What does a positive second derivative indicate about the cost function?

• The function is concave down

• The function is concave up

• The function has no critical points

• The function is linear

What is the approximate length of the longer side of the field?

• 20.3 feet

• 35.6 feet

• 25.3 feet

• 31.6 feet

What is the total cost of the fencing for the field?

• $400.50

• $505.80

• $600.30

• $700.20