Adeveloper wants to enclose a rectangular grassy lot that borders a city street for parking. if the developer has 220 feet of fencing and does not fence the side along the street, what is the largest area that can be enclosed?

* The largest area can be enclosed is 6050 feet²

Step-by-step explanation:

* Lets explain the situation to solve the problem

- There is a rectangular parking

- The parking will surrounded by fencing from three sides only

- The length of fencing is 220 feet

- Lets consider the width of the rectangle is x and the length of it is y

- The side along the street will not fence

* Lets put all of these data in equation

∵ The width of the parking is x

∵ The length of the parking is y

- He will not fence the side along the street

∴ The perimeter of the parking = x + y + x

∴ The perimeter of the parking = 2x + y

- The length of the fencing = the perimeter of the park

∵ The length of the fencing = 220 feet

∵ The perimeter of the parking = 2x + y

∴ 2x + y = 220 ⇒ (1)

- Lets find the area of the parking

∵ The area of any rectangle is length × width

∵ The width of the rectangle is x

∵ The length of the rectangle is y

∴ The area of the parking (A) = x × y

∴ The area of the parking = xy ⇒ (2)

- Lets find the value of y from equation (1) and substitute this value

 in equation (2)

∵ 2x + y = 220 ⇒ subtract 2x from both sides

∴ y = 220 - 2x

- Substitute this value in equation (2)

∵ A = xy

∴ A = x(220 - 2x) ⇒ open the bracket

∴ A = 220x - 2x²

- To find the largest area differentiate the area with respect to x

  and equate the result by 0 to find x which gives the largest area

∵ A = 220x - 2x²

- Lets remember the differentiation rules

# If y = a x^n, where a is the coefficient of x then dy/dx = (an) x^(n-1)

# If y = ax, then dy/dx = a

# If y = a, where a is constant then dy/dx = 0

∴ dA/dx = 220 - 2(2) x^(2-1)

∴ dA/dx = 220 - 4x

- Put dA/dx = 0 ⇒ for largest area

∵ dA/dx = 0

∴ 220 - 4x = 0 ⇒ add 4x to both sides

∴ 220 = 4x ⇒ divide both sides by 4

∴ 55 = x

* The width of the parking is 55 feet

- Substitute this value of x in the equation of the area to find the

 largest area

∵ A = 220x - 2x²

∵ x = 55

∴ A = 220(55) - 2(55)² = 12100 - 6050 = 6050 feet²

* The largest area can be enclosed is 6050 feet²

4400

step-by-step explanation:

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