Question

A landscaper has 32 square feet of mulch that they want to install in a yard. They need a rectangular shape, with the length to be 4 feet longer than the width, w.

a) Write the equation, in standard form, that would be used to determine the width of the rectangle

(standard form is ax^2+bx+c=0)


b) use the quadratic equation to solve for the width and length of the rectangle

Answer 1

a)
`w^(2)+4w-32=0` is the equation to determine the width of rectangle.

b) Length of rectangle is
`L = 8\:ft` and width of rectangle is
`W=4\:ft`

Explanation:

Part a:

Given that the landscaper is rectangular in shape. So to find the equation of width use area formula for rectangle.

Formula for area of rectangle,

`Area\:of\:rectangle=length* width`

To find the length and width, it is given that length is 4 feet longer than width that is,

`L = 4 + w`

Also, Area is
`32\:ft^(2)`.

Substituting the values,

`32=\left(4 + w\right)* width`

Using distributive property and simplifying,

`32=4w + w^(2)`

Subtracting 32 from both sides,

`0=4w + w^(2)-32`

Rewriting it in form of
`ax^(2)+bx+c=0`,

`w^(2)+4w-32=0`

So, the equation in standard form which is used to determine the width of rectangle is
`w^(2)+4w-32=0`

Part b:

To solve the above equation use the quadratic formula,

`x=(-b\pm √(b^2-4ac))/(2a)`

Rewriting it in terms of w,

`w=(-b\pm √(b^2-4ac))/(2a)`

where, a = 1, b = 8 and c = - 32.

Substituting the values,

`w=(-4\pm√(4^2-4\cdot \:1\left(-32\right)))/(2\cdot\:1)`

Simplifying,

`w=(-4\pm√(16-4left(-32\right)))/(2)`

Applying rule,
`-\left(-x\right)=x`

`w=(-4\pm√(16+128))/(2)`

`w=(-4\pm√(144))/(2)`

`w=(-4 \pm 12)/(2)`

Hence there are two values of x as follows,

`w=(-4 + 12)/(2)` and
`w=(-4 - 12)/(2)`

`w=(8)/(2)` and
`w=(-16)/(2)`

`w=4` and
`w=-8`

Since value of width cannot be negative so,
`w=4`

Now using
`L = 4 + w` to find the length.

`L = 4 + 4`

`L = 8`

Therefore, length and width of rectangle is
`L = 8` and
`W=4` respectively.