A rectangular field is 90 meters long and 69 meters wide. Give the length and width of another rectangular field that has the same perimeter but a smaller area.
To find the dimensions of another rectangular field with the same perimeter but a smaller area, we need to decrease the length and increase the width.
Let's say the length of the new field is L meters and the width is W meters.
The perimeter of the given rectangular field is 2 × (length + width) = 2 × (90 + 69) = 2 × 159 = 318 meters.
The perimeter of the new rectangular field should be the same, so 2 × (L + W) = 318.
Let's solve this equation for L:
2L + 2W = 318
2L = 318 - 2W
L = (318 - 2W)/2
L = 159 - W
Now, let's find the area of the new rectangular field.
Area = length × width
Area = L × W
Area = (159 - W) × W
Area = 159W - W^2
Since we want the area of the new field to be smaller than the given rectangular field, we need to find a width (W) that gives a smaller value than the given area (90 × 69 = 6210 square meters).
6210 > 159W - W^2
Let's solve this inequality and find the possible range of values for W:
W^2 - 159W + 6210 < 0
This inequality can be factored as:
(W - 30)(W - 129) < 0
The product of two factors is negative when one factor is positive and the other is negative. This means that either:
-30 < W < 129
So, the width of the new rectangular field should be within the range -30 < W < 129.
Since the width cannot be negative, we can conclude that the width of the new rectangular field should be less than 129 meters.
Therefore, the length and width of another rectangular field that has the same perimeter but a smaller area are:
Length < 159 - Width
Width < 129 meters.