General Quadratic Word Problems

The height is defined in terms of the width, so I'll pick a variable for "width", and then create an expression for the height.

Let "w" stand for the width of the picture. The height h is given as being 4/3 of the width, so:

h = (4/3)w

Then the area, being the product of the width and the height, is given by:

I need to solve this "area" equation for the value of the width, and then back-solve to find the value of the height.

Since I can't have a negative width, I can ignore the "w = −12" solution. Then the width must be the other solution, 12, and the height is then:

I can do a quick check of my answer, since this is a "solving" exercise, by plugging my values back into the original exercise. Since 12 × 16 does indeed equal 192, my answer checks. So my hand-in answer is:

Remember that consecutive integers are one unit apart, so my numbers are n and n + 1. Multiplying to get the product, I get:

The solutions are n = −34 and n = 33. I need a negative value (because the exercise statement specified that the numbers are negative), so I'll ignore the n = 33& and take n = −34. Then the other number is:

Then my hand-in answer is:

The first thing I need to do is draw a picture. Since I don't know how wide the path will be, I'll label the width as "x".

Looking at my picture, I see that the total width of the finished garden-plus-path area will be:

...and the total length will be:

Then the new area, being the product of the new width and height, is given by:

This quadratic is messy enough that I won't bother with trying to use factoring to solve; I'll just go straight to the Quadratic Formula:

Obviously the negative value won't work in this context, so I'll ignore it. Checking the original exercise to verify what I'm being asked to find, I notice that I need to have units on my answer:

When dealing with geometric sorts of word problems, it is usually helpful to draw a picture. Since I'll be cutting equal-sized squares out of all of the corners, and since the box will have a square bottom, I know I'll be starting with a square piece of cardboard.

I don't know how big the cardboard will be yet, so I'll label the sides as having length "w".

Since I know I'll be cutting out three-by-three squares to get sides that are three inches high, I can mark that on my drawing.

The dashed (rather than solid) lines show where I'll be scoring the cardboard and folding up the sides.

Since I'll be losing three inches on either end of the cardboard when I fold up the sides, the final width of the bottom will be the original "w" inches, less three on the one side and another three on the other side. That is, the width of the bottom will be:

The volume will be the product of the width, the length (which is the same as the width), and the depth (which was created by cutting out the corners and folding up the sides).

Then the volume of the box, working from the drawing, gives me the following equation:

This is the quadratic I need to solve. I can take the square root of either side, and then add the to the right-hand side:

...or I can multiply out the square and apply the Quadratic Formula:

Either way, I get two solutions which, when expressed in practical decimal terms, tell me that the width of the original cardboard is either about 2.26 inches or else about 9.74 inches.

How do I know which solution value for the width is right? By checking each value in the original word problem.

If the cardboard is only 2.26 inches wide, then how on earth would I be able to fold up three-inch-deep sides? But if the cardboard is 9.74 inches, then I can fold up three inches of cardboard on either side, and still be left with 3.74 inches in the middle. Checking:

This isn't exactly 42, but, taking round-off error into account, it's close enough that I can trust that I have the correct value. So my hand-in answer, complete with units, is: