a gardener is planning to make a rectangular garden with an area of 80ft^2.she has 12 yards of fencing to put around the perimeter of the garden.what should the dimension of the garden be ?
For a given perimeter, that largest area area of a rectangle is obtained when the rectangle is a square.
so if each side is x ft
x^2 = 80
x = √80 = 4√5
She needs 16√5 ft or appr 35.8 ft of fencing.
With only 12 feet she cannot have a rectangular field of 80 ft
or
with 12 ft of fencing , she can build a 3by3 square which will hold 9 square ft, not the 80 she wants.
Your question is flawed.
She has 12 yards, or 36 feet of fencing.
That will enclose 9x9 = 81 ft^2, so by moving away from a square, she can use the whole 36 ft to enclose only 80 ft^2.
80 = 16*5 = 4*(4*5) so if the garden is 8x10 feet, its perimeter is 36 and its area is 80
I violated one of the first rules I taught my students years ago ...
Read the question carefully, and read it more than once.
To find the dimensions of the garden, we need to set up an equation using the given information.
Let's assume the length of the rectangular garden is "l" and the width is "w".
We know that the area of a rectangle is given by the formula: Area = length * width.
So, in this case, the area of the garden is given as 80ft². Therefore, we have:
l * w = 80
Next, we are given that the gardener has 12 yards of fencing. A rectangle has two lengths and two widths, and the total length of the fencing required would be the sum of all these sides.
The formula for the perimeter of a rectangle is: P = 2(length + width)
In this case, the perimeter is given as 12 yards. We need to convert yards to feet since the area is given in square feet.
1 yard = 3 feet
So, 12 yards = 12 * 3 = 36 feet.
Therefore, we have:
2(l + w) = 36
Now, we have a system of two equations:
l * w = 80 (Equation 1)
2(l + w) = 36 (Equation 2)
We can solve this system of equations to find the dimensions of the garden.
One way to solve this is by substitution:
1. Solve Equation 2 for l:
l = (36 - 2w) / 2
2. Substitute the value of l in Equation 1:
((36 - 2w) / 2) * w = 80
Simplify the equation and solve for w:
(36 - 2w) * w = 160
36w - 2w² = 160
2w² - 36w + 160 = 0
Now, we can either factor this quadratic equation or use the quadratic formula to find the values of w. After finding the value(s) of w, substitute it back into Equation 2 to find the corresponding value(s) of l.
The dimensions of the rectangular garden will be the values of length (l) and width (w) obtained from solving the equations.