Julius has a garden that has dimensions of 12 ft by 20 ft. Julius needs more room to plant some peppers, and determines the garden will need a new area of 360 square feet. How much should he increase the length and width by, if
3.4
12N * 20N = 360
240N^2 - 360 = 0
2N^2 - 3 = 0
2N^2 = 3
N^2 = 1.5
N = 1.225 = 122.5%
%Increase = 122.5% - 100% = 22.5% = The
amount each dimension is increased.
Length = 20 * 1.225 = 24.5 Ft.
Width = 12 * 1.225 = 14.7 Ft.
Area = 24.5 * 14.7 = 360.15 Ft^2.
We can solve this problem by using the concept of area.
The current area of the garden is 12 ft * 20 ft = 240 square feet.
To find the new dimensions, we need to determine the change in area.
The change in area is 360 square feet - 240 square feet = 120 square feet.
Let's assume the increase in length is x feet and the increase in width is y feet.
The new length will be 12 ft + x feet, and the new width will be 20 ft + y feet.
The new area can be calculated by multiplying the new length by the new width:
(12 ft + x feet) * (20 ft + y feet) = 360 square feet + 240 square feet.
Expanding this equation gives us:
240 + 12x + 20y + xy = 600.
Simplifying further allows us to rewrite the equation as:
xy + 12x + 20y = 360.
To find the values of x and y, we need to factorize the left side of the equation:
xy + 12x + 20y = (x + 20)(y + 12).
Setting this equal to 360:
(x + 20)(y + 12) = 360.
Now we can find the factors of 360, which are:
1 * 360, 2 * 180, 3 * 120, 4 * 90, 5 * 72, 6 * 60, 8 * 45, 9 * 40, 10 * 36, 12 * 30, 15 * 24, 18 * 20.
From these factors, we can see that x + 20 = 18 and y + 12 = 20 will give us a solution.
Therefore, x = 18 - 20 = -2 and y = 20 - 12 = 8.
Since we are dealing with dimensions, we consider only positive values for x and y.
Therefore, Julius would need to increase the length of the garden by 2 feet and the width by 8 feet.
To determine how much Julius should increase the length and width of his garden, we need to find out how much additional area he needs to achieve a total of 360 square feet.
First, let's calculate the current area of Julius's garden. The area of a rectangle is calculated by multiplying its length by its width. So, the current area is:
Area = length × width
= 12 ft × 20 ft
= 240 square feet
Now, let's find out how much additional area Julius needs:
Additional area = Total desired area - Current area
= 360 square feet - 240 square feet
= 120 square feet
To maintain the rectangular shape of the garden, we need to divide the additional area equally between the length and the width. Since the length and width are the same, we can divide the additional area by 2 to get the increase in each dimension:
Increase in each dimension = (Additional area)/2
= 120 square feet / 2
= 60 square feet
Therefore, Julius should increase both the length and width of his garden by 60 square feet to achieve the desired additional area of 360 square feet.