A rectangular package to be sent by a delivery service can have a maximum combined length (y) and girth (perimeter of its cross section) of 300 inches. Assume that the width and height are the same (x). Find the equation for the volume of the box in terms of x alone as an expanded polynomial.
Well, this is quite the geometry problem! Let's see if we can solve it with a little humor.
To find the equation for the volume of the box in terms of x alone, we first need to understand the dimensions. Since the width and height are the same (x), and the length and girth have a maximum combined value of 300 inches, we can express the length (y) and the girth (p) in terms of x.
Now, let's calculate the girth. Since the cross-section is rectangular, the girth is equal to twice the width (x) plus twice the height (x). That gives us a girth of 2x + 2x = 4x.
Moving on to the length, the combined length and girth is given as 300 inches. So, we can set up the equation y + p = 300. Substituting the values, we have y + 4x = 300.
To find the volume, we multiply the length, width, and height together. The width (x) is equal to the height, so the volume equation becomes V = x * x * (y - 4x).
Now, let's expand this polynomial and solve for the volume:
V = x * x * (y - 4x)
V = x^3 + (-4x^2)*x
V = x^3 - 4x^3
V = -3x^3
And there you have it! The equation for the volume of the rectangular package is -3x^3.
Now, don't get too caught up in the negative sign. Just because it's negative doesn't mean the package is a negative experience! It's just a way of representing the polynomial equation. Keep that smile on your face and keep calculating!
To find the equation for the volume of the box in terms of x alone, we need to express the length (y) and girth (perimeter of the cross section) in terms of x.
The length of the rectangular package (y) can be expressed as:
y = 300 - 4x
The perimeter of the cross section (girth) can be expressed as:
2(x + y) = 300
Simplifying the equation:
2(x + y) = 300
2x + 2y = 300
Now substitute the value of y from the first equation:
2x + 2(300 - 4x) = 300
2x + 600 - 8x = 300
-6x = -300
x = 50
Now that we have the value of x, we can find the value of y:
y = 300 - 4x
y = 300 - 4(50)
y = 300 - 200
y = 100
The volume of the rectangular package can be calculated as:
Volume = length x width x height
Volume = y * x * x
Volume = 100 * 50 * 50
Volume = 250,000
Therefore, the equation for the volume of the box in terms of x alone is:
Volume = 250,000
To find the equation for the volume of the box in terms of x alone, we need to understand the relationship between the dimensions of the box.
Let's assume that the width and height of the box are both x inches.
The formula for the girth of a rectangular cross section is given by:
Girth = 2(height + width)
Since the width and height are both x, the girth formula becomes:
Girth = 2(x + x) = 4x
According to the problem, the maximum combined length and girth of the box is 300 inches. Therefore, we can write the equation:
Length + Girth = 300
Substituting the formulas for the length and girth:
Length + 4x = 300
Simplifying the equation:
Length = 300 - 4x
The volume of a rectangular box is given by the formula:
Volume = Length × Width × Height
Since the width and height are both x, the formula for the volume becomes:
Volume = (300 - 4x) × x × x
Expanding the equation:
Volume = 300x^2 - 4x^3
Thus, the equation for the volume of the box in terms of x alone is:
Volume = 300x^2 - 4x^3.