When it comes to word problems such as these, always begin by drawing an illustration.  Below is a picture of our 12 x 12 area we need to fill with tiles and one 2 x 2 tile that we are going to use to fill this area.

There are two methods that we can use to solve this problem:

Method 1:

First we need to ask, how many tiles are needed to line the width of this 12 x 12 area.  So if we have tiles that are 2 ft each we need to divide 12 by 2 to figure out how many tiles we need to line the width.

And so we need 6 tiles to line the width.

No we need to ask, how many tiles are needed to line the height of this 12 x 12 area.  So if we have tiles that are 2 ft each we need to divide 12 by 2 to figure out how many tiles we need to line the height.

And so we need 6 tiles to line the width.

Now we need to find the total amount of tiles needed to fill the entire 12 x 12 space.  We can solve for this just like we solve for area.  We will multiply the amount of tiles across the width by the amount of tiles lining the height.  This gives us an “area” in terms of the amount of tiles needed to fill the space.

And so we need 36 tiles to fill the space

Method 2:

We begin by finding the area of the entire 12 x 12 space.  We use the fact that

And so our area of the entire space is .  Now we need to find the area of each individual tile.

And so the area of each individual tile is .  To find the total amount of tiles needed we need to find the amount of sections in the total area.  To do this we divide:

Our units of cancel and we are left with 36 tiles.

We must first recognize that this is a volume problem.  There is a cylinder full of water and we must find the total volume of the cylinder in order to find the amount of water in the cylinder.  We will use the formula for finding the volume of a cylinder, .  is our radius and will be 1ft, is our height and will be 3ft.

But we cannot have represent volume.  So we will use the conversion of cubic feet to liters.  For every cubic foot there is 28.3168 liters.

Our units cancel and we are left with liters.

The goal of this problem is to find the surface area of the triangular prism and see if you have enough 1in x 1in stamps to fill the surface area without leaving any open gaps.  To find the surface area of the prism we know that there are two triangular faces, and three rectangular faces.  We will find the area of each of these and then their sum is the total surface area.

Area for each triangular face:

Area for each rectangular face:

So now we will sum 2 triangular faces and three rectangular faces for the total surface area.

Since we want to find if we have enough stamps, we will divide the total surface area by the surface area of an individual stamp to find how many stamps we would need to cover the prism entirely.

We only have 112 stamps, so we do not have enough stamps.

This is an area problem.  For a fenced in, flat area, we would only consider the height and width.  To be a volume problem we would need to have a 3-dimensional space to fill, one with width, length, and height.

The formula to find the volume of a cylinder is .  While we have the volume of the bottle, we do not have the height.  In order to work backwards to find the radius we would also need the height of the cylinder.

We need to understand that this is a volume question.  We are trying to find the total volume of potting soil needed to fill this rectangular prism.  The formula to find the volume of a rectangular prism is .  The length of the prism is 5ft, the width of the prism is 1.5ft, and the height of the prism is 2.5. We will plug these into our formula to solve this problem.

It is important to recognize that the bleacher, the ground, and the support form a right triangle with the right angle formed by the intersection of the bleacher wall and the ground.  We know the bottom of the support should only be 3ft from the bleacher wall on the ground and the bleacher wall is 10ft high.  We will use the Pythagorean Theorem to solve for the length of the support, which is the hypotenuse of this right triangle.  Our base of the triangle is 3 feet and the leg is 10 feet.

And so we need a support of 10.44 feet long.

This is a volume question, asking for the volume of a sphere.  There is a twist; we only want half of the volume of the globe.  We will start by using the volume formula for a sphere, .

But we want half of the total volume

So we know the volume of half of the globe is 7,065cm³.  Since each candy is 2cm³, we divide the volume of the space we need to take up by this amount to find the number of individual candies we need.

We cannot have half of a candy, however, and we do not want to overfill the globe.  So we round down to fill the globe with 3,532 candies.