The length of a rectangular backyard is 7 feet less than four times its width. If the area of this garden is 15 ft2, find the length of the garden?
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w(4w-7) = 15
Now, 15=3x5, so try w=3. In that case, the length is 5.
Algebraically,
4w^2 - 7w - 15 = 0
(4w+5)(w-3) = 0
w = 3 or -5/4
To find the length of the garden, we can set up an equation based on the information given in the problem. Let's break down the problem and convert the given information into an algebraic expression:
1. Let's assume that the width of the rectangular backyard is represented by the variable "w".
2. According to the problem, the length of the backyard is 7 feet less than four times its width. So, the length can be expressed as "4w - 7".
Now, we can set up an equation based on the given information:
Area = Length * Width
The area is given as 15 square feet (ft2), so we have:
15 = (4w - 7) * w
To solve this equation and find the length of the garden, we will simplify and rearrange it:
15 = 4w^2 - 7w
Rearranging the equation:
4w^2 - 7w - 15 = 0
Now, we have a quadratic equation, which we can solve using factoring, completing the square, or the quadratic formula. Let's solve it by factoring:
(4w + 3)(w - 5) = 0
Setting each factor equal to zero:
4w + 3 = 0 or w - 5 = 0
Solving for "w" in each equation:
4w = -3 or w = 5
w = -3/4 or w = 5
Since the width cannot be negative, we discard the solution w = -3/4.
Therefore, the width of the rectangular backyard is 5 feet.
To find the length, substitute the value of "w" back into the expression:
Length = 4w - 7
Length = 4(5) - 7
Length = 20 - 7
Length = 13
Therefore, the length of the garden is 13 feet