Volume of a container?
A cubic shipping container had a volume of a^3 cubic meters. The height was decreased by a whole number of meters and the width was increased by a whole number of meters so that the volume of the container is now a^3+2a^2 � 3a cubic meters. By how many meters were the height and width changed?
Please help!
To find the change in height and width of the container, we need to compare the initial volume (a^3 cubic meters) with the final volume (a^3 + 2a^2 - 3a cubic meters).
Let's set up an equation to represent this:
(a - x)(a + y) = a^3 + 2a^2 - 3a
where x represents the decrease in height and y represents the increase in width.
Expanding the left side of the equation using the FOIL method, we get:
a^2 + ay - ax - xy = a^3 + 2a^2 - 3a
Comparing the coefficients of like terms on both sides of the equation, we can form two equations:
a^2 - xy = a^3 (Equation 1)
ay - ax = 2a^2 - 3a (Equation 2)
Simplifying Equation 1, we have:
-xy = a^3 - a^2
Multiplying both sides by -1, we get:
xy = a^2 - a^3 (Equation 3)
Simplifying Equation 2, we have:
a(y - x) = a(2a - 3)
Dividing by "a," we get:
y - x = 2a - 3 (Equation 4)
Now we have a system of two equations (Equations 3 and 4) in two variables (x and y).
To solve the system, we can substitute Equation 3 into Equation 4:
(a^2 - a^3) = 2a - 3
Rearranging the equation, we get:
a^3 + 2a - a^2 - 3 = 0
This is a cubic equation in the variable "a." To find the value of "a" and then determine the changes in height and width, we can use numerical or graphical methods (such as using a graphing calculator or software).