(a)
To find: The equation of the volume.
(a)
Answer to Problem 50RE
The required expression for volume is
Explanation of Solution
Given information:
The piece of cardboard is
Calculation:
The height of the prism can be x, which is the labeled length of the rectangle next to the base and lid.
In order to write a formula in terms of the variable x, it is required to write expressions for the length, width, and height of the box.
It can be said that the length is the bottom of the base the prism. To find this value, we can subtract
The width is the side of the base of the prism. To find this value, we can subtract
The volume of the rectangular box is given by:
Substitute the value of length width and height gives:
This is the required expression for volume.
(b)
To find: The domain of
(b)
Answer to Problem 50RE
The required domain is
Explanation of Solution
Given information:
The required expression for volume is
Calculation:
Let the height of the prism be
In order to write a formula in terms of the variable
It can be said that the length is the bottom of the base of the prism. To find this value, we can subtract x and x from 10 and divide this by 2, since there are two equal rectangles (base and lid).
The width is the side of the base of the prism. To find this value, we can subtract
The domain of the volume is where
Combining these requirements the domain is
(c)
To find: Use the graphical method to find the maximum value of
(c)
Answer to Problem 50RE
It is observed from the graph that the maximum volume is 66.02 is at
Explanation of Solution
Given information:
The volume function is
Calculation:
In order to draw the graph of volume function substitute different values of
Let
The point will be
Similarly, more point can be found and plot the on the graph. The graph is given below:
It is observed from the graph that the maximum volume is 66.02 is at
(c)
To find: confirm your result in part (c) analytically.
(c)
Answer to Problem 50RE
It is found analytically that the maximum volume is 66.02 is at
Explanation of Solution
Given information:
Volume function is
Calculation:
In order to solve the equation
In order to find the maximum value, it is required to find the critical points.
In order to find critical points differentiate volume function with respect to
Now substitute
Use quadratic formula
Discard the largest value because it is not in the domain
Perform second order derivative test to check whether the volume is maximum.
Now differentiate again
Now find maximum value of volume by substituting
This proves that the volume is maximum at
Chapter 4 Solutions
CALCULUS:GRAPHICAL,...,AP ED.-W/ACCESS
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