1. Mrs. Macdonald wants to put a uniform border of tulips around the outside of her vegetable garden. Her vegetable garden is 20m by 28m. This border will look best if the area of the tulips border equal to the area of the garden. How wide should the tulip border be (accurate to one decimal place)? Justify your answer algebraically.
If the border has width w, we need the total area to be twice the garden area:
(20+2w)(28+2w) = 2(20)(28)
Now just expand, collect terms, and solve for w.
96
To find the width of the tulip border, we need to calculate the total area of the garden and then find the width of the border such that its area is equal to the area of the garden.
First, let's find the area of the garden. The area of a rectangular garden can be determined by multiplying its length by its width. In this case, the garden is 20m by 28m, so the area is:
Area of the garden = 20m * 28m = 560 square meters.
Now, let's represent the width of the tulip border as "x". Since the border is uniform around the garden, we can subtract twice the width of the border from both the length and width of the garden to find the dimensions of the inside rectangle, which would represent the area of the tulips.
The length of the inside rectangle would be (20m - 2x), and the width would be (28m - 2x). The area of the inside rectangle, representing the area of the tulips, would be:
Area of the tulips = (20m - 2x) * (28m - 2x)
According to the problem, the area of the tulips should be equal to the area of the garden. Therefore, we can set up the following equation:
560 square meters = (20m - 2x) * (28m - 2x)
Now, let's solve this equation to find the value of "x", which represents the width of the tulip border.
Multiplying out the right side of the equation, we have:
560 = 560m^2 - 56xm - 40xm + 4x^2
Combining like terms, we get:
4x^2 - 96xm + 560m^2 - 560 = 0
To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, the equation does not appear to be easily factorable, so we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 4, b = -96m, and c = 560m^2 - 560. Substituting these values into the quadratic formula, we have:
x = (-(-96m) ± √((-96m)^2 - 4 * 4 * (560m^2 - 560))) / (2 * 4)
Simplifying further, we get:
x = (96m ± √(9216m^2 - 4 * 4 * (560m^2 - 560))) / 8
x = (96m ± √(9216m^2 - 4 * 4 * 560m^2 + 4 * 4 * 560)) / 8
x = (96m ± √(9216m^2 - 8960m^2 + 8960)) / 8
x = (96m ± √(256m^2 + 8960)) / 8
x = (96m ± √(256m^2 + 8960)) / 8
Now, we need to simplify the expression under the square root and calculate the two possible values of x:
x = (96m ± √(256m^2 + 8960)) / 8
Calculating the expression under the square root, we get:
256m^2 + 8960 = 256m^2 + 8960 = 9216m^2
Therefore, the expression simplifies to:
x = (96m ± √(9216m^2)) / 8
x = (96m ± 96m) / 8
Applying the positive and negative solutions, we have:
x = (96m + 96m) / 8 = 192m / 8 = 24m
x = (96m - 96m) / 8 = 0m / 8 = 0m
Since a border width of 0m would not exist, the only possible solution for the width of the tulip border is 24m.
Thus, the tulip border should be 24 meters wide to have an area equal to the area of the garden.