Ti'a rectangular gaden has perimeter of 20 meters. She wants its area to be at least 24 square metesr. Find the interval for the possible length of Tia's garden.
if the garden is x by y meters,
x+y = 10
and xy >= 24
so,
x(10-x) >= 24
x^2 - 10x + 24 <= 0
(x-4)(x-6) <= 0
since the parabola open up, the graph is below the x-axis between the roots.
the roots are at x=4 and x=6
So, we need x <= 4 <= 6
Since usually the length is greater than the width, that means we should choose
√24 <= x <= 6
You are correct
ur awesome
To find the interval for the possible length of Tia's garden, we can use the given information about the perimeter and the minimum area.
Let's assume the length of the garden is "l" meters and the width is "w" meters.
The perimeter of a rectangle is calculated by adding the lengths of all its sides. So we have:
Perimeter = 2l + 2w = 20 meters.
From this equation, we can solve for one of the variables in terms of the other. Let's solve for "l":
2l + 2w = 20
2l = 20 - 2w
l = (20 - 2w) / 2
l = 10 - w
Now, we know that the area of a rectangle is given by the formula:
Area = length × width = l × w.
We are given that the area should be at least 24 square meters. So we have:
Area ≥ 24
l × w ≥ 24
Substituting the value of "l" we found earlier:
(10 - w) × w ≥ 24
Expanding and rearranging the equation:
10w - w^2 ≥ 24
w^2 - 10w + 24 ≤ 0
To find the interval for possible values of "w," we need to solve this quadratic inequality. We can factor it to simplify:
(w - 4)(w - 6) ≤ 0
The solutions to this inequality are w ≤ 4 or w ≥ 6.
Since "w" represents the width of the garden, it cannot be negative, so w ≥ 0. Therefore, the possible values for the width are:
0 ≤ w ≤ 4 or w ≥ 6.
Now, we can substitute these values into the equation for finding "l" we derived earlier:
0 ≤ w ≤ 4: l = 10 - w ≥ 10 - 4 = 6
w ≥ 6: l = 10 - w
Therefore, the interval for the possible length of Tia's garden is:
6 ≤ l ≤ 10.
In summary, the possible length of Tia's garden falls within the interval 6 to 10 meters.