Jim has 400 ft of fence material and needs to enclose an area of 10,000 square feet for his garden as the diagram depicts. (The length of the house is 62 ft and the fence extends the same distance on either side of the house.) What must the dimensions of the rectangular enclosed region be? That is, what are the values of x and y? round answers to the nearest hundredth of a foot.
To find the dimensions of the rectangular enclosed region, we can start by drawing a diagram of the garden.
Let's label the length of the rectangular region as x and the width as y. We are given that the total area of the garden is 10,000 square feet. Since the formula for the area of a rectangle is A = length × width, we can create an equation:
10,000 = x * y
Next, we need to consider the fence material. The fence extends the same distance on either side of the house, so the length of the fence will be the sum of the length of the house (62 ft) and twice the width of the rectangular region (2y). Therefore, the total length of the fence required is:
Total length of fence = 62 + 2y
Since Jim has 400 ft of fence material, we can create another equation:
Total length of fence = 400
Thus, we have another equation:
62 + 2y = 400
Now we have two equations:
1. x * y = 10,000
2. 62 + 2y = 400
We can solve this system of equations to find the values of x and y. Let's solve equation 2 for y:
2y = 400 - 62
2y = 338
y = 338 / 2
y ≈ 169
Now that we have the value of y, we can substitute it back into equation 1 to solve for x:
x * 169 = 10,000
x = 10,000 / 169
x ≈ 59.17
Therefore, the dimensions of the rectangular enclosed region should be approximately x = 59.17 ft and y = 169 ft (rounded to the nearest hundredth of a foot).