the length of a rectangular garden exceeds its width by 8 feet. if each side of the garden is increased by 2 feet, the area of the garden will be increased by 60 square feet. find the dimensions of the original garden
original:
width -- x
length -- x+8
area = x(x+8)
new width = x+2
new length = x+10
new area = (x+2)(x+10)
(x+2)(x+10) - x(x+8) = 60
x^2 + 12x + 20 - x^2 - 8x = 60
2x = 40
x = 10
original garden is 10 ft by 18 ft
check:
original area = 10(18) = 180
new area = 12(20) = 240
increase in area = 240-180 = 60
all looks good.
To find the dimensions of the original garden, we can set up a system of equations based on the given information.
Let's denote the width of the original garden as "w" (in feet) and the length as "l" (in feet).
From the first sentence, we know that the length exceeds the width by 8 feet, which can be expressed as:
l = w + 8
The second sentence states that if each side of the garden is increased by 2 feet, the area will increase by 60 square feet. The area of a rectangle is given by the formula A = length × width, so we have:
(l + 2)(w + 2) = lw + 60
To solve this system of equations, we can substitute the value of l from the first equation into the second equation:
(w + 8 + 2)(w + 2) = w(w + 8) + 60
(w + 10)(w + 2) = w^2 + 8w + 60
Expanding both sides gives:
w^2 + 12w + 20 = w^2 + 8w + 60
Rearranging the terms:
w^2 - w^2 + 12w - 8w = 60 - 20
4w = 40
Dividing both sides by 4:
w = 10
Now that we have the value of w, we can substitute it back into the first equation to find l:
l = w + 8
l = 10 + 8
l = 18
Therefore, the original garden has dimensions of 10 feet by 18 feet.