Respuesta :
Answer:
a) [tex]w^{2}+4w-32=0[/tex] is the equation to determine the width of rectangle.
b) Length of rectangle is [tex]L = 8\:ft[/tex] and width of rectangle is [tex]W=4\:ft[/tex]
Step-by-step 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,
[tex]Area\:of\:rectangle=length\times width[/tex]
To find the length and width, it is given that length is 4 feet longer than width that is,
[tex]L = 4 + w[/tex]
Also, Area is [tex] 32\:ft^{2}[/tex].
Substituting the values,
[tex]32=\left(4 + w\right)\times width[/tex]
Using distributive property and simplifying,
[tex]32=4w + w^{2}[/tex]
Subtracting 32 from both sides,
[tex]0=4w + w^{2}-32[/tex]
Rewriting it in form of [tex]ax^{2}+bx+c=0[/tex],
[tex]w^{2}+4w-32=0[/tex]
So, the equation in standard form which is used to determine the width of rectangle is [tex]w^{2}+4w-32=0[/tex]
Part b:
To solve the above equation use the quadratic formula,
[tex]x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
Rewriting it in terms of w,
[tex]w=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
where, a = 1, b = 8 and c = - 32.
Substituting the values,
[tex]w=\dfrac{-4\pm\sqrt{4^2-4\cdot \:1\left(-32\right)}}{2\cdot\:1}[/tex]
Simplifying,
[tex]w=\dfrac{-4\pm\sqrt{16-4left(-32\right)}}{2}[/tex]
Applying rule, [tex]-\left(-x\right)=x[/tex]
[tex]w=\dfrac{-4\pm\sqrt{16+128}}{2}[/tex]
[tex]w=\dfrac{-4\pm\sqrt{144}}{2}[/tex]
[tex]w=\dfrac{-4 \pm 12}{2} [/tex]
Hence there are two values of x as follows,
[tex]w=\dfrac{-4 + 12}{2}[/tex] and [tex]w=\dfrac{-4 - 12}{2}[/tex]
[tex]w=\dfrac{8}{2}[/tex] and [tex]w=\dfrac{-16}{2}[/tex]
[tex]w=4[/tex] and [tex]w=-8[/tex]
Since value of width cannot be negative so, [tex]w=4[/tex]
Now using [tex]L = 4 + w[/tex] to find the length.
[tex]L = 4 + 4[/tex]
[tex]L = 8[/tex]
Therefore, length and width of rectangle is [tex]L = 8[/tex] and [tex]W=4[/tex] respectively.
