Edgar is building a sandbox. He bordered the sides of the sandbox using wood planks measuring 8x+10 feet
A.what are the possible measurements of the length and width of the sandbox?
B.Use the properties of numbers to justify that these are really possible measurements of the dimensions of the sandbox
C.which among these possible measurements gives the largest area? Justify your answer
To find the possible measurements of the length and width of the sandbox, we need to determine the factors of the expression 8x+10.
A. Factors of 8x+10:
8x+10 = 2(4x+5)
So, the possible measurements of the sandbox can be written as a product of 2 and (4x+5), where x is any real number.
B. The properties of numbers that justify these possible measurements are the associative property of multiplication and the commutative property of multiplication. The length and width of the sandbox can be expressed as 2(4x+5) or (4x+5)2. These expressions can be rearranged and multiplied without changing the result.
C. To find the largest possible area, we need to multiply the length and width of the sandbox. Let's call the length 2(4x+5) and the width (4x+5)2.
Area = Length x Width
= 2(4x+5) * (4x+5)2
= 8x(4x+5) + 5(4x+5)
= 32x^2 + 40x + 20x + 25
= 32x^2 + 60x + 25
The area, given by the expression 32x^2 + 60x + 25, is a quadratic equation. Using calculus, we can find the maximum value of this quadratic function. However, since we know that the coefficient of x^2 is positive (32 > 0), this quadratic function will be a U-shaped parabola with a minimum value. Therefore, the largest possible area occurs at the vertex of the parabola.
The x-coordinate of the vertex can be found using the formula x = -b/2a, where a = 32, and b = 60.
x = -60/2(32)
x = -60/64
x = -15/16
Now, substitute x = -15/16 into the quadratic function to find the maximum area:
Area = 32(-15/16)^2 + 60(-15/16) + 25
Area = 32(225/256) - 900/16 + 25
Area = (7200/256) - 900/16 + 25
Area = 2825/32
Therefore, the largest possible area for the sandbox is given by the expression 2825/32 square feet when the length is 2(4x+5) and the width is (4x+5)2, with x = -15/16.