Sam is going to plant a rectangular flower garden next to his house. There will be a fence on three sides of the garden( fencing is not necessary along the houe). The area of the garden is to be 1,000sq. ft., and he has 90 feet of fencing available. Determine the dimensions of the garden.
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.
To determine the dimensions of the garden, let's assume the length of the garden is L feet and the width is W feet.
Given:
The area of the garden is 1,000 sq. ft., so we have: L * W = 1000.
Also, there will be a fence on three sides of the garden, so we have:
2L + W = 90.
Now we can solve these two equations to find the values of L and W.
1. Rearrange the second equation: W = 90 - 2L.
2. Substitute the value of W from the second equation into the first equation:
L * (90 - 2L) = 1000.
3. Simplify the equation: 90L - 2L^2 = 1000.
4. Rearrange the equation and set it equal to zero: 2L^2 - 90L + 1000 = 0.
5. Solve the quadratic equation using factoring, completing the square, or quadratic formula. In this case, factoring can be used, which gives:
(L - 25)(2L - 40) = 0.
6. Set each factor equal to zero and solve for L:
L - 25 = 0 -> L = 25
2L - 40 = 0 -> L = 20.
Now we have two possible values for the length of the garden: L = 25 ft and L = 20 ft.
Substituting these values into the second equation, we can solve for the width:
When L = 25: 2(25) + W = 90 -> W = 90 - 50 = 40.
When L = 20: 2(20) + W = 90 -> W = 90 - 40 = 50.
Therefore, the dimensions of the garden can be either 25 ft by 40 ft or 20 ft by 50 ft.