Find Min Area of Box for 5in x 7in Book: Explanations

[link2110]

Homework Statement

A book with cover dimensions 5in x 7in is to be placed symmetrically in a diamond shaped gift box. What is the smallest possible area of the box? Explain how you know your answer is minimal.

Homework Equations

n/a

The Attempt at a Solution

i don't know how to start this question, any hints?

 

[Mark44]

Start by drawing a sketch of the box, with the book in it. Label the book's given dimensions, and label the dimensions of the diamond-shaped box with a variable.

See if you can determine some relationships between the book's dimensions and the box's dimensions.

 

[link2110]

If the book's edges are always touching the diamond's walls, then does the diamond always have the same area, no matter if it is stretched? I tried considering the centre of the box as the origin, and only looked at the first quadrant...basically a rectangle of dimensions 3.5 X 2.5 inside a triangle of unknown size. but I don't know if I am on the right track

 

[link2110]

my initial guess is 2X the size of the book...but I don't have a mathematical proof

 

[Mark44]

If the book's edges are always touching the diamond's walls, then does the diamond always have the same area, no matter if it is stretched? I tried considering the centre of the box as the origin, and only looked at the first quadrant...basically a rectangle of dimensions 3.5 X 2.5 inside a triangle of unknown size. but I don't know if I am on the right track

The diamond won't always have the same area, but otherwise, I think you are on the right track.

In my sketch I have a triangle - the first quadrant portion of the diamond, and a 2.5" X 3.5" rectangle within the triangle, with the 3.5" side vertical.

In my drawing, y is the remainder of the length of the height of the triangle (i.e., y + 3.5 is the length of the vertical side of the triangle. The horizontal leg of the triangle is of length 2.5 + x.

What you want to do is minimize the area of the diamond, which is equivalent to minimizing the area of the triangle, A = 1/2*(y + 3.5)(x + 2.5).

The rectangle within the triangle defines two other similar triangles. From these we get the relationship that y/2.5 = 3.5/x. Solve for y and use it in the area formula to get area as a function of one variable.

 

[link2110]

Thank you! I understand the first part and the overall concept, but I am confused as to how to find y/2.5=3.5/x...

 

[Mark44]

A drawing would have made it more obvious what I was doing. My drawing has the large triangle, with legs of length y + 3.5 and x + 2.5. Inside the triangle is the rectangle of width 2.5 and height 3.5. Inside the large triangle are two other right triangles: one above the rectangle, and one to the right of the rectangle. All three triangles are similar, meaning all the corresponding angles are equal, which makes the corresponding sides proportionate.

For the triangle above the rectangle, its height to base ratio is the same as the height to base ratio of the triangle to the right of the rectangle.

IOW, y/2.5 = 3.5/x. This equation is the key to being able to write the area of the large triangle as a function of one variable.

 

[link2110]

Thank You :D

 

[Phrak]

In my drawing, y is the remainder of the length of the height of the triangle (i.e., y + 3.5 is the length of the vertical side of the triangle. The horizontal leg of the triangle is of length 2.5 + x.

I don't see anything in the problem statement that says the axies of the book and the diamond are aligned in the same direction. What do you think?

 

[Mark44]

The first post has this: "A book with cover dimensions 5in x 7in is to be placed symmetrically in a diamond shaped gift box."
That let's us align the axes of the book and the box.