(a) The area of each rectangular room is given by the formula:
Area = length x width
Since the area of each room is 240 m², and the length of one room is x m, we can write:
240 = x × width of the first room
Therefore, the width of the first room is:
width of the first room = 240 / x m
The length of the other room is 4 m longer than x, so we can write:
length of the second room = x + 4 m
And using the formula for the area of the second room, we have:
240 = (x + 4) × width of the second room
Therefore, the width of the second room is:
width of the second room = 240 / (x + 4) m
(b) If the widths of the rooms differ by 3 m, we can write:
width of the second room - width of the first room = 3
Substituting the expressions for the widths obtained in part (a), we get:
240 / (x + 4) - 240 / x = 3
Multiplying both sides by x(x+4), we get:
240x - 240(x + 4) = 3x(x + 4)
Simplifying and rearranging terms, we get:
x^2 + 4x - 320 = 0
(c) To solve the quadratic equation x^2 + 4x - 320 = 0, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = 4, and c = -320.
Substituting these values, we get:
x = (-4 ± sqrt(4^2 - 4(1)(-320))) / 2(1)
Simplifying the expression under the square root, we get:
x = (-4 ± sqrt(1296)) / 2
x = (-4 ± 36) / 2
Therefore, x = -20 or x = 16.
Since the length of the room cannot be negative, we reject the solution x = -20, and conclude that x = 16 m.
(d) Using the value of x obtained in part (c), we can find the dimensions of each room:
- The first room has length x = 16 m and width 240 / x ≈ 15 m.
- The second room has length x + 4 = 20 m and width 240 / (x + 4) ≈ 12 m.
Therefore, the perimeters of the rooms are:
- Perimeter of the first room = 2(length + width) = 2(16 + 15) = 62 m
- Perimeter of the second room = 2(length + width) = 2(20 + 12) = 64 m
The difference between the perimeters is:
64 - 62 = 2 m
Therefore, the difference between the perimeters of the rooms is 2 m.
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