Suppose you have a piece of cardboard 32 inches by 48 inches and cut equal sized squares from each corner and fold up the sides of the cardboard to form a box with an open top. What length of x maximizes the volume of the box?

Volume = lwh

l = 48 - 2x
w = 32 - 2x
h = x

V= x(48 - 2x)(32 - 2x)

V= x(1536 - 96x - 64x + 4x^2)

V = 1536x - 160x^2 + 4x^3

V' = 1536 - 320x + 12x^2

set V' = 0 and solve for x.

From quadratic formula, x =  6.28 inches

y=mx + b is equation of a line. The slope of the line is "m" and the y-intercept is "b". We can get the equation of the line given the slope and y-intercept or given two points (x1, y1) and (x2, y2). If given two points, use the formula (y2 - y1)/(x2 -x1) to find the slope. Then use the slope and one of the coordinates give to find "b". Then you have the slope and y-intercept needed to write the equation.

Division with remainders is simply a division problem in which one number does not divide into the other number evenly. For example, 2 divides into 24 evenly 12 times. But 2 divides into 25, 12 times with 1 left over, which is the remainder. The answer to 25 divided by 2 would be 12 R1 or 12 1/2.

Suppose you have two triangles, ABC and DEF and want to determine if they are congruent. The Side Angle Side theorem (known as SAS) says that if two sides and the included angle of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. So if side AB is length 5 and AC is length 6 with angle A of 50 degrees, and side DE is also 5 and DF is 6 with angle D of 50 degrees, the triangles are congruent by the SAS theorem.

Algebra is a type of mathematics where symbols are used to represent amounts that are unknown. These
unknown quantities are generally combined with mathematical operations (addition, subtraction, multiplication,division, square root, cube root, exponents, etc) to form statements that describe the relationship of things that change over a period of time. These statements are expressed using equations, expressions and terms. Problems can be solved by translating words into algebraic equations. The description of a problem using an equation and other mathematical concepts is known as a mathematical model. A mathematical model can be used to solve numerous types of problems in every day life

We use the change of base formula to solve logs that are not base 10. The log button on a calculator solves for base 10 logs. So if you have a problem such as log (base 5) 8, you change that to base 10, which would be log8/log5. This can be simply solved on a calculator as 1.29.