Lesson Solving and Transformation of Formulas

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SOLVING FORMULAS.

A. GENERALITIES.
In the study of math and sciences students often meet formulas. A formula is usually an equation which expresses an inter-relationship between a few variables. One of the main uses of formulas is to find the numeric value of a variable when the other variables are known.

Examples of formulas:
Math formula: a%5E2+=+b%5E2+%2B+c%5E2 ; A = 1/2b.h ; a/sin A = b/sin B = c/sin C ; C = 2.Pi.R ;
Science formulas: D = Vt ; C = 5/9(F - 32) ; F = m.a.

B. SOLVING FORMULAS.

Solving a formula means finding the numeric value of one variable in the formula when the numeric values of other variables are known. Solving a formula proceeds exactly like solving a linear equation using shortcut.

C. EXAMPLES OF SOLVING FORMULAS.

Example 1. Find the velocity V in the formula D = V.t, in which the distance D = 120 km and the time is t = 3 sec.
Solution. Substitute the values of D and t into the formula:
120 = 3.V ; ---> V = 120/3 = 40 km/hr.

Example 2. Find the F temperature knowing the C temperature is 38 deg.
Solution. The conversion temperature formula: C = 5/9(F - 32). Replace C = 38 into the formula:
9.(38)/5 = F - 32 ; ---> F = 32 + 68.4 = 100. 4 deg.

Example 3. Find the time t from the free fall formula: d+=+gt%5E2%2F2 in which the distance is d = 50 m, and the gravity acceleration is g = 9.81 m/s/s.
Solution. Substitute the given values into the formula:
2%2850%29+=+9.81%2At%5E2 ; ---> t%5E2+=+100%2F9.81+=+10.20 ; ---> t = 3.19 sec.

D. EXERCISES ON SOLVING FORMULAS.

1. Formula: C = 2.Pi.R ; Find R, knowing C = 15.7, and Pi = 3.14.
2. Formula: F = m.a Find F; given m = 230 kg, and a =7 m%2Fs%5E2.
3. Formula: a%5E2+=+b%5E2+%2B+c%5E2 ; Find a ; given b = 16 and c = 20.
4. Formula: A+=+Pi%2AR%5E2 ; given A = 78.5, and Pi = 3.14.
5. Formula A = b.h/2 ; given A = 35 and h = 10.

E. TRANSFORMATION OF FORMULAS.

To solve for a variable in a formula, we can transform the formula into another one in which the
selected variable is expressed in terms of other variables, with no numeric values involved. Transformation of a formula proceeds exactly like solving a linear equation using shortcut.

F. EXAMPLES ON TRANSFORMATION OF FORMULAS.

Examples 1. Formula D = Vt ; Transform it to find V or find t.
Solution. To find V, transform the formula into: V = D/t.
To find t, transform to: t = D/V.


Example 2. Transform the temperature conversion formula to find F.
Solution. Formula C = 5/9(F - 32). Solving this linear equation for F using shortcut:
F - 32 = 9C/5 ; ---> F = 32 + 9C/5.

Example 3. Transform the formula d+=+g%2At%5E2%2F2 to find g.
Solution. 2d+=+g%2At%5E2 ; ---> g+=+2d%2Ft%5E2.

Example 4. Formula; 1/f = 1/d1 + 1/d2. Express d1 in terms of other variables.
Solution. 1/d1 = 1/f - 1/d2 = (d2 - f)/(f.d2) ; ---> d1 = f.d2/(d2 - f).

Example 5. Formula: 1/R = 1/r1 + 1/r2 + 1/r3. Find R in terms of r1, r2, and r3.
Solution. 1/R = 1/r1 + 1/r2 +1/r3 = (r2.r3 + r3.r1 + r1.r2)/r1.r2.r3.
R = (r1.r2.r3)/(r2.r3 + r1.r3 + r1.r2).

Example 6. Formula: 1/R = 1/r1 + 1/r2 + 1/r3. Find r1 in terms of other variables.
Solution. Solve for r1 using solving linear equation using shortcut:
1/r1 = 1/R - 1/r2 - 1/r3 = (r2.r3 - R.r3 - R.r2)/(R.r2.r3).
r1 = R.r2.r3/(r2.r3 - R.r3 - R.r2).
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