6.2 Rearranging Formulas
Marilyn Nielson
Applications
When we use formulas in our fields, sometimes we’ll sketch out an idea on a sheet of paper and plug numbers in to the formula to look at one specific location or condition. More often, we have data we’ve collected in the field, and we enter it into a spreadsheet or database to run calculations efficiently.
For example, chlorine needs to be added to a water treatment plant at a concentration sufficient to kill harmful bacteria. The chlorine will be added in pounds per day and we want to know the concentration over varying volumes of water treated per day (in million gallons per day).
[latex]L = 8.34QC[/latex]
So, if we add 50 lbs of chlorine per day for the month of June, can we arrange this formula to give us the concentration of chlorine over over discharges ranging from 1 to 4 million gallons per day?
Rearranging formulas relies on the rules we learned for algebra. Let’s recap a few of those here.
- We rearrange a formula to “solve for” one of the variables. This means the target variable will be alone on one side of the equation and everything else will be on the other side.
- Whatever you do to one side of an equation needs to be done to the other side.
- It’s often easiest to start by “clearing” any fractions by multiplying both sides of the equation by the denominator.
- We may also need to deal with a square root at this point. Wait until everything is under the square root symbol on one side before squaring each side of the equation to remove the square root (if needed).
- Then, in general, we “undo” addition or subtraction by adding (for subtraction) or subtracting (for addition).
- Next, we separate variables from their coefficients (undo multiplication) by dividing.
- Finally, if our target variable is squared, we’ll get it alone by taking the square root of both sides.
All of the formulas we use are applied – so they are numbers that are a measurement or count of something real. This means we will need to keep track of the units that we input into a formula and what we will get out of the formula with the units we are using.
For some students rearranging formulas is the easiest math we do all quarter and for others it’s the most challenging. If this sort of math gets frustrating for you – and you aren’t dealing with a spreadsheet – you can work a single problem by plugging each of the numbers in for the known variables and working with numbers rather than letters. We’ll look at how this works in the examples below.
Examples
[latex]dib = dbh-2(abt)[/latex]
A) Rearrange to solve for dbh
B) What is dbh if dib = 23″ and abt = 1.5″?
C) Rearrange to solve for abt
D) What is abt if dib = 18″ and dbh = 1.2″?
[latex]Q=VA[/latex]
A) Rearrange to solve for V
B) Rearrange to solve for A
C) What is V if Q is 1200 cubic feet per second and A is 8 square feet?
D) What is A if Q is 1200 cubic feet per second and V is 12 feet per second?
[latex]L=8.34QC[/latex]
A) Rearrange to solve for C
B) Rearrange to solve for Q
C) What is C if L is 2 pounds per day and Q is 1 million gallons per day?
Problem Set 6.2
- [latex]Q=VA[/latex]
-
- Solve for V
- Solve for A
- If Q is 200 cubic feet per second and A is 30 square feet, what is the velocity (V)?
2. [latex]L=8.34QC[/latex]
-
- Solve for C
- If the loading (L) is 500 lbs per day and the discharge (Q) is 2 million gallons per day, what is the concentration in mg/L?
3. [latex]A = LW[/latex]
-
- Solve for W
- I need to fence 12,000 square feet to keep cattle out for a mitigation project. If the length of one side is 800 ft, how wide dos the site need to be to enclose 12,000 square feet?
- I need to fence 5 acres of pasture. If one side of my fence needs to be 1200 feet – how long should the other side be (in feet)?
4. [latex]A = \frac{bh}{2}[/latex]
-
- Solve for b
- If my triangle has an area of 400 square inches, and the height is 15 inches, what is the length of the base (in inches)?
5. [latex]A = \prod_{}^{}r^{2}[/latex]
-
- Solve for r
- If my plot has a radius of 150 feet, what is its area in square feet? What is its area in acres?
- I need to put in 1/4 (one-quarter) acre circular plots. What will their radius be?
6. [latex]V = H\prod_{}r^{2}[/latex]
-
- Solve for H
- Solve for r
- I have a culvert with a radius of 6 inches and a height of 8 feet. How many cubic feet of water can it hold? How many gallons can it hold?
- I have a culvert with a width of 3 feet. How long does it need to be to hold 500 gallons?
7. [latex]V = \frac{4}{3}\prod_{}r^{2}[/latex]
-
- Solve for r
- What radius would a ball need to be to hold 100 gallons of water?
8. If my circular plot needs to encompass one-eighth of an acre, what will its radius be?
9. I add 250 lbs of chlorine to my water treatment system daily. If I treat 100,000 gallons per day, what its the average concentration of chlorine in the system (in mg/L)?
10. A quarter barrel keg has the dimensions shown at left (width 16.125″ x 13.875″ high). How many cubic feet does it hold? How many gallons? If 2 cups are in a pint and 16 cups are in a gallon – does the keg indeed hold 62 pints?
11. A 55-gallon drum is 22.5 inches wide. How tall is it?
12. What is the average bark thickness if the DBH of my log is 24 inches and the diameter inside bark is 21 inches?
13. What is the representative area of a plot if I place 400 plots in a stand that is 500 acres?
14. I have a pile of wood chips that needs to maintain a volume of 350 cubic yards. If the pile naturally forms a cone with a base of 250 feet, how tall must the pile be ?
15. The Spokane River has a cross-sectional area of 2,000 square meters near Post Falls. If the river discharge is 10,000 cubic feet per second during spring runoff. How fast is the water moving in feet per second? How fast in miles per hour?
