the area of a rectangular garden is 120 square meters. the width is 3 meters longer than 1/3 of the length. find the legth and the width of the garden.
length --- x
width ---- x/3 + 3
x(x/3 + 3) = 120
x^2/3 + 3x - 120 = 0
x^2 + 9x - 360 = 0
(x-15)(x+24) = 0
x = 15 or x = -24, but x > 0
the length is 15
width = 15/3 + 3 or 8
check (8)(15) = 120
3 more than 1/3 of 15 gives us 8
To find the length and width of the rectangular garden, we'll break down the problem step by step.
Let's start by assigning variables:
Let L be the length of the garden.
Let W be the width of the garden.
Given information:
The area of the garden is 120 square meters, so we have the equation: L * W = 120.
We are also told that the width is 3 meters longer than one-third (1/3) of the length. This can be written as:
W = (1/3)L + 3.
Now, let's substitute the value of W from the second equation into the first equation to get a single equation with one variable.
(L * W) = 120
(L * ((1/3)L + 3)) = 120
Now, let's solve this equation step by step:
Distribute the L:
(L/3)L + 3L = 120
Multiply everything by 3 to get rid of the fraction:
L^2 + 9L = 360
Rearrange the equation to make it a quadratic equation:
L^2 + 9L - 360 = 0
Next, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use factoring:
(L - 15)(L + 24) = 0
Hence, we have two possible solutions for the length:
L - 15 = 0 --> L = 15
L + 24 = 0 --> L = -24 (we can ignore this since length can't be negative)
Now, let's substitute the value of L = 15 into the equation for W:
W = (1/3)(15) + 3
W = 5 + 3
W = 8
So, the length of the garden is 15 meters, and the width is 8 meters.