you are putting in a 20ftx30ft rectangular pool. your pool must be surrounded by a concrete sidewalk of uniform width and 3in thick.
write an expression in terms of x to represent the volume of concrete contained in the sidewalk. explain how you got your expression.
If x is the width in feet, the volume in cubic feet is
[(30 + 2x)(20 + 2x)- 30*20]*(1/4)
That expression can be simplified somewhat by canceling out the 600's.
To represent the volume of concrete contained in the sidewalk, we need to subtract the inner rectangular area (representing the pool) from the outer rectangular area (representing the total area including the sidewalk).
Given that the pool is 20ft x 30ft, we can calculate the area using the formula: length x width = area. So, the pool area is 20ft x 30ft = 600 square feet.
The total area including the sidewalk is the area of the larger rectangle that the pool and the sidewalk together form. To find this area, we need to add twice the width of the sidewalk to the length and width of the pool. Therefore, the dimensions of the larger rectangle are (20ft + 2x) x (30ft + 2x), where x represents the width of the sidewalk.
Calculating the area of the larger rectangle gives us (20ft + 2x) x (30ft + 2x) = (600ft² + 40x + 60x + 4x²) = (600ft² + 100x + 4x²).
Now, to find the volume of the concrete contained in the sidewalk, we multiply this area by the thickness of the sidewalk. Since the thickness is given as 3 inches, we need to convert it to feet by dividing by 12 (as there are 12 inches in a foot). Therefore, the volume of the concrete is (600ft² + 100x + 4x²) x (3/12ft) = (600/12ft³ + 100x/12ft² + 4x²/4ft²) = (50ft³ + (25/3)x ft² + x² ft²).
Thus, the expression in terms of x to represent the volume of concrete contained in the sidewalk is: 50ft³ + (25/3)x ft² + x² ft².