Answer:
[tex]r>\frac{4 +\sqrt{23732} }{154} \approx1.03\\[/tex]
Step-by-step explanation:
The area of the rectangular rug is given by this equation:
[tex]77r^2-4r-77[/tex]
The area must be a number greater than 0 since a negative area, or an area equal to 0 wouldn't have much sense. So:
[tex]77r^2-4r-77>0[/tex]
Let's find the roots of this equation using the quadratic formula:
[tex]r=\frac{-b\pm \sqrt{b^2-4ac} }{2a} =\frac{-(-4)\pm\sqrt{(-4)^2-(4)(77)(-77)} }{2(77)} \\\\r=\frac{4 \pm \sqrt{23732} }{154} \\\\r_1=\frac{4 +\sqrt{23732} }{154} \approx1.03\\\\r_2=\frac{4 - \sqrt{23732} }{154} \approx-0.97[/tex]
Now, let's evaluate the area for [tex]r>r_1[/tex] for example r=1.05:
[tex]A=77(1.05)^2-4(1.05)-77=3.6925>0[/tex]
The result is greater than zero, so for [tex]r>r_1[/tex] the values make sense.
Now let's evaluate the area for [tex]r_2<r<r_1[/tex] for example r=0.5 and r=-0.7:
[tex]A=77(0.5)^2-4(0.5)-77=-59.75<0\\\\A=77(-0.7)^2-4(-0.7)-77=-36.47<0[/tex]
The result is less than zero, so for [tex]r_2<r<r_1[/tex] the values don't make sense.
Now, let's evaluate the area for [tex]r<r_2[/tex] for example r=-1:
[tex]A=77(-1)^2-4(-1)-77=4>0[/tex]
The result is greater than zero, so for [tex]r<r_2[/tex] the values make sense
However, since negative values of r wouldn't make much sense (I never heard about of -8 inches for example) the possible dimensions of the rug are just:
[tex]r>\frac{4 +\sqrt{23732} }{154} \approx1.03\\[/tex]
