A new fence is constructed on three sides of a
housing block with area 240 m2.
The fourth side facing the road is left open.
If 52 m of fencing is used, what are the
dimensions of the block?
let the side parallel to the open side by y m
let each of the other two sides be x m
2x + y = 52
y = 52-2x
area = xy
x(52 - 2x) = 240
-2x^2 + 52x - 240 = 0
x^2 - 26x + 120 = 0
(x-20)(x-6) = 0
x = 20 or x = 6
if x = 20, then y = 52-40 = 12
if x = 6, then y = 52-12 = 40
make a sketch to show that both answers are valid
Well, it sounds like the housing block is really trying to keep its options open with that open fourth side! As for the dimensions, let's break it down.
Since the block is enclosed on three sides, we can assume that two sides have the same length, while the other two sides have different lengths. Let's call the length of the common side "x" and the length of the other two sides "y" and "z".
So, we have two equal sides (x) and two different sides (y and z). The perimeter of the block would be x + x + y + z, which should equal 52 meters.
Now, we know that the area of the housing block is 240 m². Since the block is rectangular in shape, we can use the formula: Area = Length × Width.
Substituting the values we have, we get:
240 = x × y
Now, knowing that the perimeter is 52, we can make another equation:
52 = 2x + y + z
Now, we have two equations:
1) 240 = x × y
2) 52 = 2x + y + z
Solving these equations will give us the values of x, y, and z, which would be the dimensions of the block.
But you know what? Solving equations can sometimes be a tall order, just like finding the perfect dimensions for a housing block. Let me grab my calculator and get back to you with the dimensions in a jiffy!
To find the dimensions of the housing block, we first need to determine the perimeter of the block.
Let's assume the length of the block is L and the width is W.
We know that the area of the block is 240 m^2, so we can write:
L * W = 240
Now, let's calculate the perimeter of the block by considering the three sides with the fence:
Perimeter = 2L + W
We are given that a total of 52 m of fencing is used, so we can write:
2L + W = 52
Now we have a system of two equations:
L * W = 240
2L + W = 52
We can solve these equations simultaneously to find the values of L and W.
Using substitution method, we can solve for W in terms of L:
W = 52 - 2L
Substituting this into the first equation:
L * (52-2L) = 240
Rearranging and simplifying:
52L - 2L^2 = 240
2L^2 - 52L + 240 = 0
Dividing the equation by 2:
L^2 - 26L + 120 = 0
Now, we can either factor this quadratic equation or use the quadratic formula. In this case, we can factor it by finding two numbers that multiply to give 120 and add up to -26. These numbers are -10 and -12:
(L - 10)(L - 12) = 0
Setting each factor equal to zero:
L - 10 = 0 or L - 12 = 0
Solving for L:
L = 10 or L = 12
Now let's find the corresponding values of W:
For L = 10:
W = 52 - 2L
W = 52 - 2(10)
W = 52 - 20
W = 32
For L = 12:
W = 52 - 2L
W = 52 - 2(12)
W = 52 - 24
W = 28
Therefore, the two possible dimensions of the housing block are:
1. Length = 10 m and Width = 32 m
2. Length = 12 m and Width = 28 m
To find the dimensions of the block, we need to determine the length and width of the block.
Let:
Length of the block = L
Width of the block = W
We are given that the area of the block is 240 m^2. We can use this information to set up an equation:
L * W = 240
Now, let's consider the fencing. We are told that a fence is constructed on three sides of the block, and the fourth side facing the road is left open. The total length of the fencing used is 52 m.
Since the opposite sides of a rectangle are equal, we can calculate the length of the fencing needed for the three sides of the block:
2L + W = 52
To solve this system of equations, we can use substitution or elimination. We'll use substitution here:
We can rearrange the second equation to solve for W:
W = 52 - 2L
Now substitute this value of W into the first equation:
L * (52 - 2L) = 240
Simplify the equation:
52L - 2L^2 = 240
Rearrange the equation in standard form:
2L^2 - 52L + 240 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. In this case, factoring is more convenient:
2L^2 - 40L - 12L + 240 = 0
2L(L - 20) - 12(L - 20) = 0
(2L - 12)(L - 20) = 0
Now set each factor equal to zero and solve for L:
2L - 12 = 0 or L - 20 = 0
2L = 12 or L = 20
L = 6 or L = 20
Since the dimensions cannot be negative, we can discard L = 6.
So, the length of the block is L = 20.
Now substitute this value of L into one of the equations to solve for W:
2(20) + W = 52
40 + W = 52
W = 52 - 40
W = 12
Therefore, the dimensions of the block are 20 meters by 12 meters.