we need to fence off a rectangular section of land in order to make some huge play pens. All four sides will be fenced and there will also be 2 parallel fence lines inside the perimeter in order to make 3 identical fields. If we have 2600 meters of fencing, we need to determine the dimensions for the entire fenced area that will produce an area of 100 000m^2
John wants to build a fence around his yard. The yard is 10 ft long and 18 ft wide. How much fence does he need?
To find the dimensions of the fenced area, we can follow these steps:
Step 1: Assign variables to the unknowns.
Let's say the width of the rectangular section is 'w' (in meters) and the length is 'l' (in meters).
Step 2: Write equations based on the given information.
We know that the entire fenced area will be divided into three identical fields. This means that each individual field will have the same dimensions.
The perimeter of one field consists of two lengths and two widths (since all four sides will be fenced). Therefore, the perimeter of one field can be calculated as:
Perimeter = 2l + 2w
Also, the total length of the fence used will be equal to the three perimeters of the fields, which is given as 2600 meters:
Total Fence Length = 3 * Perimeter
Step 3: Express the equations in terms of one variable.
Since we want to find the dimensions of the fenced area that will produce an area of 100,000 m², we can express 'l' in terms of 'w' using the area formula.
Area of one field = l * w
Area of one field = 100,000 / 3 (since there are three identical fields)
Step 4: Solve the equations simultaneously.
Now, we can substitute the expressions for perimeter and area into the equation for Total Fence Length:
2600 = 3 * (2l + 2w)
And substitute the expression for area into the equation for Area of one field:
100,000 / 3 = l * w
Solving these two equations will give us the dimensions of the fenced area.
Step 5: Calculate the dimensions.
Let's solve the equations:
From the equation Total Fence Length = 3 * (2l + 2w), we can simplify:
2600 = 6l + 6w
From the equation Area of one field = 100,000 / 3, we can simplify:
33,333.33 = l * w
We can rearrange the first equation to solve for l:
l = (2600 - 6w) / 6
Now substitute this expression for l in the second equation:
33,333.33 = (2600 - 6w) / 6 * w
Solving this equation will give us the value of 'w,' which is the width of the fenced area. Then we can substitute that value back into the equation for 'l' to find its value.
By solving these equations, you will determine the dimensions of the rectangular section needed to create the three play pens with an area of 100,000m² and a total fence length of 2600m.