Applications

You are managing a hazardous spill. It’s soaked an area of soil 8 inches deep, 40 feet long and 30 feet wide. If a dump truck typically hauls 10 cubic yards of soil, how many loads will need to be hauled to remove the contaminated earth?

This is the type of problem we want to get good at solving! We’ll learn how to approach this multifaceted problem in an organized way to reach a solution we are confident in.

Let’s look at the problem from the Applications section above to come up with a general approach that will help us successfully solve applied problems that include a geometry component.

Step 1: Sketch a picture

On the job, sometimes you’ll be given drawings for a project. Other times, you’ll have a vision that you’re in charge of making real. You might sketch this out on a sheet of paper, or using CAD (computer aided design) or GIS (geographic information systems). A sketch, map, or drawing translates the instructions or idea you’ve been given into tangible shapes. This lets you find the formulas you’ll work with, check for feasibility, and share the plan with others for review.

Step 2: What shapes are involved & what formulas do you have for these shapes?

Identify rectangles, circles, triangles and three-dimensional forms like cylinders and cubes. Find the formulas for perimeter, area, and volume that you’ll use.

Step 3: Are the units you are given different than the units the answer will be in?

The units used for measuring length will need to be the same before inputting them into formulas for area or volume. When calculating volume, it’s okay to use units of feet and square feet, meters and square meters, or inches and square inches. For example, square feet multiplied by feet gives us cubic feet. We can’t multiply inches by square feet or acres by centimeters and arrive at a useful answer.

If you’re looking for an answer in acres, easiest conversions will come from feet. Similarly, the conversion to gallons is easiest when dealing in cubic feet.

Use your dimensional analysis skills!

Step 4: Use the formulas

Use the formulas you identified and the units you arrived at to calculate your answer. Showing your work in an organized way will let others check it and prevent costly mistakes when ordering materials.

Step 5: Does you answer need to be in different units?

When needed, you can convert an answer from square feet to acres or square miles. You can convert from cubic feet to gallons or acre-feet. Double check to make sure your answer is in units you can use.

Step 6: What else does the situation require?

Your situation may include components that aren’t just geometry. You may need to calculate the number of rolls of fencing to use, the cost of concrete for a given volume, or the amount of chlorine to add to a volume of water for disinfection. Often, a dimensional analysis approach will work well here. Any component that uses the term “per” can make an easy conversion factor.

Again, show your calculations so that you can double check them and others on the job can review your work.

Step 7: Does my answer make sense?

Look back at your drawing. Do your final numbers feel about right? For example, if you are working on the scale of a building, expect lengths 10 to 100 feet, areas from 200 to 3000 square feet, volumes from 2,400 to 120,000 cubic feet. If you find your self an order of magnitude off – it’s likely that something went wrong in your calculations.

If your answer feels wrong – like $20,000 for fencing a small paddock – it likely is. Check each step of your work again. If you don’t find an error, have a colleague review your calculations.