Sharons rectangular vegteable garden is 20 feet by 30 feet. In addition mulching her garden, she wants to put mulch around the outside of her garden in a uniform width. If she has enough mulch to cover an area of 936 sqaure feet, how wide should the mulch border be?
(20+2w)(30+2w) = 936
w = 3
(20+2x)(30+2x) = 936
600 + 100 x + 4 x^2 = 936
4 x^2 + 100 x - 336 = 0
x^2 + 25 x - 84 = 0
(x - 3 )(x + 28) = 0
x = 3 feet
To find out the width of the mulch border, we need to subtract the area of Sharon's vegetable garden from the total area covered by the mulch.
The area of Sharon's vegetable garden is the length multiplied by the width, which is 20 feet multiplied by 30 feet, giving us a total area of 600 square feet.
So, the area covered by the mulch border is the total area minus the area of the vegetable garden: 936 square feet - 600 square feet = 336 square feet.
Since the mulch border has a uniform width, we can calculate its dimensions by simply multiplying the width by the length.
Let's denote the width of the mulch border as x.
The formula to calculate the area of a rectangle is length multiplied by width.
So, the equation for the area of the mulch border is x multiplied by (20 + 2x) (considering the width for both sides of the rectangle).
Now we set up the equation:
x * (20 + 2x) = 336
To solve this equation, we can simplify it as follows:
20x + 2x^2 = 336
Rearranging the equation, we get:
2x^2 + 20x - 336 = 0
Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use factoring to solve it.
First, divide the equation by 2 to simplify it:
x^2 + 10x - 168 = 0
Next, we factor the quadratic expression:
(x + 14)(x - 12) = 0
Set each factor to zero and solve for x:
x + 14 = 0 or x - 12 = 0
Solving these equations, we get:
x = -14 or x = 12
Since x represents the width, it cannot be negative. Thus, the width of the mulch border should be 12 feet.
Therefore, the mulch border should be 12 feet wide around the outside of Sharon's vegetable garden.