A rectangular package to be sent by a delivery service can have a maximum combined length and girth (perimeter of a cross section) of 120 inches.
a. Show that the volume of the package is V(x)= 4xsquared(30-x)
b. Use your graphing calculator to find the dimensions of the package that maximize the volume. What is the maximum volume?
The formula V=4xsquared(30-x) is correct if the cross section is a square with side equal to x.
I do not have a graphing calculator.
You could find out the maximum volume by calculating with the above formula using various values of x. Try for x between 15 and 25.
-Equation 1: 4x+y=120 --> y = 120 - 4x
-Equation 2: x^2(y)=Volume
-Combine equations: x^2(120-4x) = volume
-Distribute: -4x^3 + 120x^2
Those are the initial steps
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.
tite
a. To find the volume of the package, we can use the formula: V = lwh, where l represents the length, w represents the width, and h represents the height of the package.
Since the maximum combined length and girth of the package is 120 inches, we can express this constraint as: 2l + 2w ≤ 120, which simplifies to l + w ≤ 60.
To find the volume in terms of a single variable, we can solve the constraint equation for one of the variables and substitute it into the volume formula.
Solving the constraint equation for l, we have: l ≤ 60 - w.
Substituting this into the volume formula, we get:
V(x) = lwh = (60 - w)wh = 4x²(30 - x), where x represents the width of the package.
b. To find the dimensions of the package that maximize the volume, we can graph the function V(x) = 4x²(30 - x) on a graphing calculator and look for the highest point on the graph.
Using a graphing calculator, we can plot the function and find the maximum volume.
Maximizing the volume, we find that the maximum volume is V = 21600 cubic inches.
To find the dimensions at this maximum volume, we substitute the value of x that corresponds to the maximum into the expressions for length and width.
By analyzing the graph, we can see that the dimensions that maximize the volume are approximately: length = 20 inches, width = 40 inches, and height = 20 inches.
To find the volume of the rectangular package and solve the problem, we need to follow these steps:
a. First, let's define the dimensions of the rectangular package. Let the length be x, width be y, and height be z.
Since the package is rectangular, the perimeter of a cross-section (girth) is given by:
Girth = 2(x + y)
According to the problem, the maximum combined length and girth is 120 inches. This can be represented by the equation:
Length + Girth = 120
Substituting the values for Length and Girth, we get:
x + 2(x + y) = 120
Simplifying the equation:
x + 2x + 2y = 120
3x + 2y = 120
2y = 120 - 3x
y = (120 - 3x)/2
Now, to find the volume of the package, we multiply the length, width, and height:
Volume = x * y * z
Substituting the value of y, we get:
Volume = x * [(120 - 3x)/2] * z
Volume = (x/2) * (120 - 3x) * z
Volume = 4x^2 * (30 - x) [Rearranging terms]
Therefore, the volume of the package is V(x) = 4x^2(30 - x).
b. To find the dimensions that maximize the volume, we need to find the value of x that maximizes the function V(x) = 4x^2(30 - x).
Using a graphing calculator, follow these steps:
1. Enter the equation of the volume function as V(x) = 4x^2(30 - x).
2. Graph the equation.
3. Determine the maximum point on the graph, which represents the maximum volume.
4. Read the corresponding x-value from the graph, which gives the dimensions that maximize the volume.
5. Calculate the maximum volume by plugging the x-value into the volume equation V(x).
By following these steps, you will be able to find the dimensions of the package that maximize the volume and determine the maximum volume.