Lin's garden has an area of 782 square feet. The length of the garden is 5 less than three times its width. What are the dimensions of the garden?
To find the dimensions of the garden, let's set up some equations:
Let's assume the width of the garden is x feet.
According to the problem statement, the length of the garden is 5 less than three times its width. So the length can be expressed as 3x - 5 feet.
The area of a rectangle can be calculated by multiplying its length and width. Therefore, we can set up the following equation:
Area = Length × Width
782 = (3x - 5) × x
Now, we can solve this equation to find the value of x, which will give us the width of the garden. Let's do that:
782 = 3x^2 - 5x
3x^2 - 5x - 782 = 0
This is a quadratic equation. We can solve it using factoring, completing the square, or the quadratic formula. Let's use the quadratic formula to find the two possible values of x:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 3, b = -5, and c = -782.
x = (-(-5) ± √((-5)^2 - 4 × 3 × -782)) / (2 × 3)
x = (5 ± √(25 + 9368)) / 6
x = (5 ± √9393) / 6
Since we are dealing with a real-world situation, we can disregard the negative root because the width cannot be negative. Hence, we take the positive root:
x = (5 + √9393) / 6
Calculating the value of x, we get:
x ≈ 16.9 feet (rounded to one decimal place)
So, the width of the garden is approximately 16.9 feet.
To find the length, we can substitute the value of x back into the expression we derived earlier:
Length = 3x - 5
Length = 3 × 16.9 - 5
Length ≈ 45.7 feet (rounded to one decimal place)
Therefore, the dimensions of Lin's garden are approximately 16.9 feet by 45.7 feet.