SOLUTION: you decide to make a box. You have one piece of plastic with dimensions 15 in x 10 in, with squares cut from the corners to fold up the sides. a. Write an equation that represent

Algebra ->  Trigonometry-basics -> SOLUTION: you decide to make a box. You have one piece of plastic with dimensions 15 in x 10 in, with squares cut from the corners to fold up the sides. a. Write an equation that represent      Log On


   

Question 388388: you decide to make a box. You have one piece of plastic with dimensions 15 in x 10 in, with squares cut from the corners to fold up the sides.
a. Write an equation that represents the volume of the box without a cover.
b. Graph your equation on graphing calculator
c. Use a graph to find the dimensions of the box, to the nearest whole numbers, with the maximum volume.
THANK YOU SO VERY VERY VERY MUCH!!
Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
The piece of plastic is 15" by 10".
The squares cut from each corner are x" by x".
The volume of the box made from folding up the sides after the square corners have been removed can be expressed by V+=+x%28B%29 where B is the area of the bottom of the box and x is the height of the box sides:
The area of the box bottom can be expressed by:
B+=+%2810-2x%29%2815-2x%29, so the volume equation becomes:
V+=+x%2810-2x%29%2815-2x%29
V+=+x%28150-50x%2B4x%5E2%29 or...
V+=+4x%5E3-50x%5E2%2B150x This is the equation for the volume of the box.
The graph of this cubic equation is:
graph%28400%2C400%2C-5%2C10%2C-50%2C140%2C4x%5E3-50x%5E2%2B150x%29
From the graph, you can see that the relative maximum (of y, the volume) occurs at about x = 2 (it's really 1.96187390745...), so the dimensions of the square cut-outs would be 2" by 2" to obtain the maximum volume.