Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes.
(b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
(c) Write an expression for the volume V in terms of both x and y.
(d) Use the given information to write an equation that relates the variables x and y.
(e) Use part (d) to write the volume as a function of only x.
(f) Finish solving the problem by finding the largest volume that such a box can have.
Also:
A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
A box with a square base and open top must have a volume of 13,500 cm^3. Find the dimensions of the box that minimize the amount of material used.
Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
(a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes.
(b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base.
(c) Write an expression for the volume V in terms of both x and y.
(d) Use the given information to write an equation that relates the variables x and y.
(e) Use part (d) to write the volume as a function of only x.
(f) Finish solving the problem by finding the largest volume that such a box can have.
Also:
A farmer wants to fence an area of 24 million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?
A box with a square base and open top must have a volume of 13,500 cm^3. Find the dimensions of the box that minimize the amount of material used.
