Answer:
* 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
