A kennel owner has 164 ft of fencing to enclose a rectangular region. He wants to subdivide it into 3 sections of equal length. If the total area of the enclosed region is 576 square ft what are the dimensions.
I know that the answer is 18 ft by 32 ft or 64 ft by 9ft but not how to get it
Since there are three sections of equal length, there must be 4 lengths (2 outside and 2 inside) and the width on both sides.
4L + 2W = 164
2W = 164 - 4L
W = 82 -2L
L * W = 576
Substitute 82 - 2L for W in the second equation and solve for L. Put that value in the first equation and solve for W. Check by putting both values in the second equation.
I hope this helps. Thanks for asking.
Well, since you know the answer, it looks like you don't need my help. But, since I really want to lend a hand, I'll explain it to you anyway.
Let's call the length of one section "x." Since there are three sections, we can say that the total length of the fencing used for the sides is 3x. We also know that the remaining fencing is used for the ends of the rectangle, which makes it 164 - 3x.
To find the width of the rectangle, we divide the remaining fencing by 2 (because there are two ends). So, the width is (164 - 3x) / 2.
The area of a rectangle is calculated by multiplying the length by the width. So, we have:
Area = Length × Width
576 = x × ((164 - 3x) / 2)
Now, let's solve this equation:
1152 = x × (164 - 3x)
1152 = 164x - 3x^2
Rearranging it:
3x^2 - 164x + 1152 = 0
If we factorize this equation, we get:
(x - 18)(3x - 64) = 0
Solving for x, we find two possible values: x = 18 and x = 64/3. Since the kennel owner wants equal lengths, we can discard x = 64/3.
So, the length of one section is 18 ft. Plugging this value back into the equation for the width, we have:
Width = (164 - 3(18)) / 2
Width = 32 ft
Therefore, the dimensions of the rectangular region are 18 ft by 32 ft. Voila!
To find the dimensions of the rectangular region, we can set up the problem as follows:
Let's assume the length of the rectangular region is "x" feet.
The width of the rectangular region will then be (164 - 2x) / 3 feet, as we want to subdivide it into 3 equal sections.
Now, we can find the area of the rectangular region using the formula: Area = Length x Width.
Given that the total area is 576 square feet, we can write the equation as:
x * ((164 - 2x) / 3) = 576
To solve this equation, we can start by multiplying both sides by 3 to eliminate the fraction:
3x * ((164 - 2x) / 3) = 576 * 3
x * (164 - 2x) = 1728
Next, expand the equation:
164x - 2x^2 = 1728
Now, rearrange the equation in standard quadratic form:
2x^2 - 164x + 1728 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = -164, and c = 1728. Substituting these values into the quadratic formula, we get:
x = (-(-164) ± √((-164)^2 - 4*2*1728)) / (2*2)
x = (164 ± √(26896 - 13824)) / 4
x = (164 ± √(13072)) / 4
x = (164 ± 114.37) / 4
Now, let's calculate the two possible values for "x":
x1 = (164 + 114.37) / 4
x1 = 278.37 / 4
x1 = 69.59 feet
x2 = (164 - 114.37) / 4
x2 = 49.63 / 4
x2 = 12.41 feet
Since we're looking for positive values for length and width, we can discard x2.
Now, plug the value of x1 back into the equation for the width to find the dimensions:
Width = (164 - 2x1) / 3
Width = (164 - 2(69.59)) / 3
Width = (164 -139.18) / 3
Width = 24.82 / 3
Width = 8.27 feet
Therefore, the dimensions of the rectangular region that satisfy the given conditions are approximately 69.59 ft by 8.27 ft.
To find the dimensions of the rectangular region, we can use the given information about the total area and the amount of fencing available.
Let's assume the length of the rectangular region is L, and the width is W. We know that the total area is given as 576 square ft, so we can use the formula for the area of a rectangle:
Area = Length × Width
Substituting the given values, we have:
576 = L × W ...........(Equation 1)
We also know that the kennel owner wants to subdivide the fencing into 3 equal sections, so each section would have the same length. This means that the total length of the fencing would be 3 times the length of the rectangular region plus twice the width of the rectangular region (since there are two sides with the width, but only one side with the length).
The equation for the total length of fencing is:
Total Length of Fencing = 3L + 2W
Given that the kennel owner has 164 ft of fencing, we can write:
164 = 3L + 2W ...........(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (L and W) that we can solve simultaneously to find the dimensions of the rectangular region.
There are a few ways to solve this system of equations, such as substitution, elimination, or graphing. Let's use the substitution method in this case.
From Equation 2, we can isolate one of the variables. Let's solve for L:
3L = 164 - 2W
L = (164 - 2W)/3
Now, substitute this expression for L into Equation 1:
576 = [(164 - 2W)/3] × W
Next, multiply both sides of the equation by 3 to eliminate the fraction:
1728 = (164 - 2W) × W
Expand the equation:
1728 = 164W - 2W^2
Bring everything to one side of the equation:
2W^2 - 164W + 1728 = 0
This is a quadratic equation. To solve for W, we can use factoring, completing the square, or the quadratic formula. In this case, let's use factoring.
Factor the quadratic equation:
2(W - 32)(W - 27) = 0
Now, set each factor equal to zero and solve for W:
W - 32 = 0 or W - 27 = 0
Solving for W yields two possible values: W = 32 or W = 27.
If W = 32, substitute this value back into Equation 2:
164 = 3L + 2(32)
164 = 3L + 64
100 = 3L
L = 100/3
So, one possible set of dimensions is L = 100/3 ft and W = 32 ft.
If W = 27, substitute this value back into Equation 2:
164 = 3L + 2(27)
164 = 3L + 54
110 = 3L
L = 110/3
So, another possible set of dimensions is L = 110/3 ft and W = 27 ft.
Therefore, the two possible sets of dimensions for the rectangular region are:
1) L = 100/3 ft and W = 32 ft
2) L = 110/3 ft and W = 27 ft