SOLUTION: 3. A piece of cardboard measuring 10 inches by 15 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let x

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Question 114309: 3. A piece of cardboard measuring 10 inches by 15 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches.
a) Draw a diagram



b) What are the restrictions on x?
c) Determine a function V that gives the volume of the box.

d) Draw a graph the function and find the value of x that produces the maximum volume. What is the maximum volume of the box?




e) When will the volume of the box be greater than 80 cu inches?

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A piece of cardboard measuring 10 inches by 15 inches is to be made into a box with an open top by cutting equal-sized squares from each corner and folding up the sides. Let x represent the length of a side of each such square in inches.
:
a) Draw a diagram
Draw a rectangle about 10 by 15 cm. Label it 10 by 15 inches. Draw a small square at each corner. Label it's side as x
:
b) What are the restrictions on x?
From the diagram you can see the box dimensions will be (10-2x) by (15-2x)
Therefore you know the value of x has to be less than 5. ie (10-2(5)) = 0
:
c) Determine a function V that gives the volume of the box.
The length = (15-2x)
The width = (10-2x)
The height = x
:
V = (15-2x)*(10-2x) * x
FOIL
V = (150 - 30x - 20x + 4x^2) * x
Which is:
V = x(4x^2 - 50x + 150)
V = 4x^3 - 50x^2 + 150x; is the function of the volume
:
d) Draw a graph the function and find the value of x that produces the maximum
volume.
Plot your graph x = .5 to x = +5, every .5 inches
x | y
-------
.5 | 63
1.0|104
1.5|126
2.0|132
2.5|125
3.0|108
3.5|84
4.0|56
4.5|27
5.0| 0
:
Your graph should look like this:
+graph%28+300%2C+200%2C+-2%2C+5%2C+-20%2C+140%2C+4x%5E3-50x%5E2%2B150x%29+
:
:
What is the maximum volume of the box?
Looking at the graph we can see max vol occurs when x = 2, vol = 132 cu in
:
Substitute 2 for x in the original equation to confirm this:
:
:
e) When will the volume of the box be greater than 80 cu inches?
:
Looking at the graph (and the table) we can say between 1 and 3.5 inches the vol > 80
:
You can confirm this also by substituting these values in the equation
:
Any questions about this?