To determine the dimensions of the garden, we need to set up an equation using the given information.
Let's assume the length of the garden is L feet and the width of the garden is W feet.
We know that the area of the garden is 1,000 square feet, so we have the equation: L * W = 1,000.
We also know that there will be a fence on three sides of the garden, and the total length of the fencing available is 90 feet.
Since there are three sides with fences, the perimeter of the garden will be 90 feet. The perimeter of a rectangle is given by the equation: 2L + W = 90.
Now we have a system of two equations:
L * W = 1,000 (equation 1)
2L + W = 90 (equation 2)
We can solve this system of equations to find the dimensions of the garden.
Let's solve equation 2 for W:
W = 90 - 2L.
Now substitute this value of W into equation 1:
L * (90 - 2L) = 1,000.
Expanding and rearranging the equation, we get:
90L - 2L^2 = 1,000.
Rearranging further, we obtain a quadratic equation:
2L^2 - 90L + 1,000 = 0.
To solve this equation, we can either factor it or use the quadratic formula. However, in this case, factoring is not possible, so let's use the quadratic formula:
L = (-b ± √(b^2 - 4ac)) / 2a,
where a = 2, b = -90, and c = 1,000.
Calculating the values:
L = (-(-90) ± √((-90)^2 - 4 * 2 * 1,000)) / (2 * 2)
= (90 ± √(8,100 - 8,000)) / 4
= (90 ± √100) / 4.
Simplifying further, we have:
L = (90 ± 10) / 4.
Now we have two possible values for L:
L1 = (90 + 10) / 4 = 25,
L2 = (90 - 10) / 4 = 20.
Substituting these values of L into equation 2, we can solve for W:
For L = 25:
2(25) + W = 90,
50 + W = 90,
W = 40.
For L = 20:
2(20) + W = 90,
40 + W = 90,
W = 50.
Therefore, the possible dimensions of the garden are:
L = 25 feet, W = 40 feet, or
L = 20 feet, W = 50 feet.