Question 1073062
.
You just had a pool installed that has dimensions 10 feet by 24 feet by 4 feet. You want to install a cement patio all the way 
around the pool that extends the same distance around every side of the pool. A cement truck holds enough cement to cover 
810 square feet at a 4-inch depth. If you don't want to have to pay for a second truck, 
what is the widest you can make the patio around your new pool? 
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<pre>
The outer dimensions of the patio are (10+2x) feet wide and (24+2x) feet long.
The area of this rectangle is (10+2x)*(24+2x) square feet.
The area of the pool surface (which obviously is not covered by cement) is 24*10 square feet.

The difference of the areas is the area of the cement patio all the way around the pool. So, this area is

(10+2x)*(24+2x) - 10*24 = {{{4x^2 +68x + 10*24 - 10*24}}} = {{{4x^2 + 68x}}}.

So your equation is

{{{4x^2 + 68x}}} = {{{810}}},

{{{4x^2 + 68x - 810}}} = 0,

{{{x[1,2]}}} = {{{(-68 +- sqrt(68^2 - 4*4*(-810)))/(2*4)}}} = {{{(-68 +- 132.6)/8}}}.

The only positive root is x = {{{(-68+132.6)/8}}} = 8.1 ft.

<U>Answer</U>.  The width of the patio is approximately 8.1 ft all the way around the pool.

<U>Check</U>.   (10+2*8.1)*(24+2*8.1) - 10*24 = 813.24 ft^2 which is close enough.
</pre>


The solution by "josgaritmetic" is &nbsp;a) &nbsp;{{{highlight(WRONG)}}} &nbsp;and &nbsp;b) &nbsp;{{{highlight(ABSURD)}}} &nbsp;starting from his very first equation.



For many other similar solved problems see the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Problems-on-the-area-of-a-rectangle-surrounded-by-a-strip.lesson>Problems on the area and the dimensions of a rectangle surrounded by a strip</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Circular-pool-and-a-walkway-around.lesson>Problems on a circular pool and a walkway around it</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic 
"<U>Dimensions and the area of rectangles and circles and their elements</U>".