Question

a school is building a rectangular playground with dimensions of 50 meters by 30 meters. one side of the playground will have an opening for a gate that is half the length of the side. the perimeter of the playground will be fenced except this opening. the cost of fencing is $10 per meter. to minimize the cost, which side should have the opening and how much will the total cost of fencing be?

a 50 - meter side should have the opening. the total cost will be $1,350.

a 30 - meter side should have the opening. the total cost will be $650.

a 30 - meter side should have the opening. the total cost will be $1,150.

a 50 - meter side should have the opening. the total cost will be $550.

Explanation:

Step1: Calculate the perimeter of the rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\), where \(l = 50\) meters and \(w=30\) meters.
\(P=2(50 + 30)=2\times80 = 160\) meters.

Step2: Calculate the length of fencing when the opening is on the 50 - meter side

The opening length is \(\frac{50}{2}=25\) meters.
The length of fencing \(L_1=160 - 25=135\) meters.
The cost \(C_1=10\times135 = 1350\) dollars.

Step3: Calculate the length of fencing when the opening is on the 30 - meter side

The opening length is \(\frac{30}{2}=15\) meters.
The length of fencing \(L_2=160 - 15 = 145\) meters.
The cost \(C_2=10\times145=1450\) dollars.

Answer:

A 50 - meter side should have the opening. The total cost will be \(\$1,350\). So the answer is the first option.