Question

The area of a rectangular rug is given by the trinomial r squared minus 4 r minus 77r2−4r−77. What are the possible dimensions of the​ rug? Use factoring.

Answer 1

R2-4r-77 = (r-11)(r+7)
r has to be greater than 11 in order for the formula to represent the area of a rug since a side length cannot be less than or equal to 0.

Answer 2

Answer:

$r>\frac{4 +\sqrt{23732} }{154} \approx1.03$

Step-by-step explanation:

The area of the rectangular rug is given by this equation:

$77r^2-4r-77$

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:

$77r^2-4r-77>0$

Let's find the roots of this equation using the quadratic formula:

$r=\frac{-b\pm \sqrt{b^2-4ac} }{2a} =\frac{-(-4)\pm\sqrt{(-4)^2-(4)(77)(-77)} }{2(77)} \\\\r=\frac{4 \pm \sqrt{23732} }{154} \\\\r_1=\frac{4 +\sqrt{23732} }{154} \approx1.03\\\\r_2=\frac{4 - \sqrt{23732} }{154} \approx-0.97$

Now, let's evaluate the area for $r>r_1$ for example r=1.05:

$A=77(1.05)^2-4(1.05)-77=3.6925>0$

The result is greater than zero, so for $r>r_1$ the values make sense.

Now let's evaluate the area for $r_2$ for example r=0.5 and r=-0.7:

$A=77(0.5)^2-4(0.5)-77=-59.75$

The result is less than zero, so for $r_2$ the values don't make sense.

Now, let's evaluate the area for $r$ for example r=-1:

$A=77(-1)^2-4(-1)-77=4>0$

The result is greater than zero, so for $r$ 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:

$r>\frac{4 +\sqrt{23732} }{154} \approx1.03$