Topic 4 Formula Introduction: In many situations in science and business we use formulas. these formula are essentially just an algebraic expression where the variables used have very specific meanings. In order to evaluate a formula we essentially just evaluate the algebraic expression that is the formula using the given values of the variables. Example 1: In electronics the formula V = IR will help you find the voltage V when you know the current I and the resistance R. So if you had a circuit with a current of 5 amps and a resistance of 10 ohms then the Voltage would be found by doing the following. V = IR Write down the formula being used V = (5)(10) Substitute I with 5 and R with 10 V = 50 volts Perform the calculation Example 2: The Volume of a Cuboid V with a Length L, a width W and a height H is given by the formual V = LWH. What is the volume of a cuboid with length 10 cm, width 5 cm and height 8 cm? V = LWH Write down the formula being used V = (10)(5)(8) Substitute L with 5, W with 5 and H with 8 V = 400 cm 3 Perform the calculation Page 1
Example 3: The first formula of motion states that where a is the acceleration, v is the final velocity, u is the initial velocity and t is time. a = What is the acceleration of an object with an initial velocity of 10 ft/sec, a final velocity of 40 ft/sec that does this movement in 4 seconds? a = Write down the formula being used a = Substitute v with 40, u with 10 and t with 4 a = 7.5 ft/sec 2 Perform the calculation Example 4: The second formula of motion states that s = ut + ½at 2 where s is the distance travelled, u is the initial velocity, a is the acceleration and t is time. What is the displacement (distance travelled) for an object with an initial velocity of 5m/sec travelling for 8 seconds and having an acceleration of 10 m/sec 2. s = ut + ½at 2 Write down the formula being used s = (5)(8) + ½(10)(8) 2 Substitute u with 5, a with 10 and t with 8 s = 40 + 320 Use PEMDAS to perform the calculations in s = 360 ft the appropriate order Page 2
Example 5: The simple interest I received when you invest an amount of money called the principal P, with an interest rate r (written as a decimal) for t years is I = Prt. What is the simple interest you would receive if you invested $400 for 3 years at a 2% interest rate? I = Prt Write down the formula being used I = (400)(0.02)(3) Substitute P with 400, r with 0.02 and t with 3 I = $24 Perform the calculation Notice that in this formula we needed to change the interest rate from 2% into the equivalent decimal 0.02 before we could make the appropriate calculation. Example 6: The simple interest I received when you invest an amount of money called the principal P, with an interest rate r (written as a decimal) for t years is I = Prt. What is the simple interest you would receive if you invested $400 for 18 months at a 3% interest rate? I = Prt Write down the formula being used I = (400)(0.03)(1.5) Substitute P with 400, r with 0.03 and t with 1.5 I = $18 Perform the calculation Notice that in this formula we needed to change the interest rate from 3% into the equivalent decimal 0.03 and that 18 months had to be changed into 1.5 years before we could make the appropriate calculation. The units used in much formula are important as they add meaning to the result and in many practical situations you will need to know the appropriate units to use. In the last two examples we needed to be careful to use the correct values in order to get a realistic result. Page 3
Exercise 1A 1. In electronics the formula for calculating the voltage V across a circuit with a current I and a resistance R is V = IR. (a) What is the voltage across a circuit with a current of 12 amps and a resistance of 5 ohms? (b) What is the voltage across a circuit with a current of 4.5 amps and a resistance of 12.2 ohms? (c) What is the voltage across a circuit with a current of ½ amps and a resistance of 4¼ ohms? 2. The Volume of a Cuboid V with a Length L, a width W and a height H is given by the Formula V = LWH. (a) What is the volume of a cuboid with length 4 cm, width 8 cm and height 2 cm? (b) What is the volume of a cuboid with length 15 cm, width 0.2 cm and height 1.5 cm? (c) What is the volume of a cuboid with length 1½ ft, width 2¼ ft and height 8 ft? 3. The Surface area of a cuboid S with a Length L, a width W and a height H is given by the Formula S = 2(LW + LH + WH) (a) What is the Surface Area of a cuboid with length 6 ft, width 12 ft and height 4 ft? (b) What is the Surface Area of a cuboid with length 0.5 m, width 0.2 m and height 0.3 m? (c) What is the Surface Area of a cuboid with length 1½ ft, width 2 ft and height 3½ ft? Page 4
4. The first formula of motion states that a = where a is the acceleration, v is the final velocity, u is the initial velocity and t is time. (a) What is the acceleration of an object with an initial velocity of 4 ft/sec, a final velocity of 8 ft/sec that does this movement in 2 seconds? (b) What is the acceleration of an object with an initial velocity of 10 m/s, a final velocity of 110 m/s that does this movement in 10 seconds? (c) What is the acceleration of an object with an initial velocity of 2.5 ft/sec, a final velocity of 12.5 ft/sec that does this movement in 2.5 seconds? 5. The second formula of motion states that s = ut + ½at 2 where s is the distance travelled, u is the initial velocity, a is the acceleration and t is time. (a) What is the displacement (distance travelled) for an object with an initial velocity of 4m/sec travelling for 6 seconds and having an acceleration of 8 m/sec 2. (b) What is the displacement (distance travelled) for an object with an initial velocity of 12ft/sec travelling for 4 seconds and having an acceleration of 5 ft/sec 2. (c) What is the displacement (distance travelled) for an object with an initial velocity of 0.5 m/sec travelling for 2 seconds and having an acceleration of 1.2 m/sec 2. 6. The simple interest I received when you invest an amount of money called the principal P, with an interest rate r, written as a decimal for t years is I = Prt. (a) What is the simple interest you would receive if you invested $600 for 2 years at a 6% interest rate? (b) What is the simple interest you would receive if you invested $200 for 6 mont6hs at a 3% interest rate? (c) What is the simple interest you would receive if you invested $600 for 30 months at a 1.5% interest rate? Page 5
Rearranging a Formula Definition: When we write a formula such as V = LWH we say that the subject of the formula is the variable V. It is common to take a formula and to rearrange the formula so that it has a different subject. In order to do this we use similar techniques as those used to solve equations. Example 1: Rearrange the formula V = LWH to find W Solution: V = LWH = W Divide both sides by LH W = Switch the order Example 2: Change the subject of the formula E = ½mv 2 to m. Solution: E = ½mv 2 2E = mv 2 Multiply both sides by 2 = m Divide both sides by v 2 m = Switch the order Example 3: Change the subject of the formula V = to h. Solution: V = 3V = Multiply both sides by 3 = h Divide both sides by h = Switch the order Page 6
Example 4: Change the equation y = in terms of x. Solution: y = y + 7 = Add 7 to both sides = x Divide both sides by x = Switch the order Example 5: Change the equation L = in terms of B. Solution: L = 5L = Multiply both sides by 6 5L + 7C = Add 7C to both sides = B Divide both sides by B = Switch the order Page 7
Exercise 1B: 1. Change the subject of the formula, to b. 2. Change the subject of the formula, to P 3. Change the subject of the formula Ax + By = C to x. 4. Change the subject of the formula Ax + By = C to y. 5. Change the subject of the formula y = mx + c to x. 6. Change the subject of the formula y b = m(x a) to m 7. Change the subject of the formula PV = KT to T. 8. Change the subject of the formula D = rt to r. 9. Rearrange each of these formulae to make d the subject. (a) c = 3d (b) c = 5d (c) c = 2d (d) c = 3 1 d (e) c = 3d + 2 (f) c = 5d + 2 (g) c = 3 1 d 17 (h) c = 4 1 d 5 (I) c = 4 1 d + 1 10. Rearrange each of these formulae to make x the subject. (a) y = 3x + 5 (b) y = 8x + 13 (c) y = 4x 7 (d) y = x 14 (e) y = 2 (x + 3) (f ) y = 5 (x + 7) (g) y = 2 (x 5) (h) y = 17 (x 5) (i) y = (j) y = x 4 3 (k) y = 2x 7 5 (l) y = x 5 9 3x 8 5 Page 8
Solutions Exercise 1A 1.(a) 60 volts 1.(b) 54.9 volts 1.(c) 2.125 or volts 2.(a) 64 cm 3 2.(b) 4.5 volts 2.(c) 27 cm 3 3.(a) 288 ft 2 3.(b) 0.62 m 2 3.(c) 30.5 or m 2 4.(a) 2 ft/s 2 4.(b) 10 ft/s 2 4.(c) 4 ft/s 2 5.(a) 168 m 5.(b) 88 ft 5.(c) 3.4 or m 6.(a) $72 6.(b) $3 6.(c) $22.50 Exercise 1B: 1. B = 2. 3. x = 4. y = 5. x = 6. m = 7. B = 8. r = 9.(a) d = 9.(b) d = 9.(c) d = 9.(d) d = 3c 9.(e) d = 9.(f) d = 9.(g) d = 3(c + 17) 9.(h) d = 4(c + 6) 9.(i) d = 4(c 1) 10.(a) x = 10.(b) x = 10.(c) x = 10.(d) x = y+14 10.(e) x = 10.(f) x = 10.(g) x = 10.(h) x = 10.(i) x = 10.(j) x = 10.(k) x = 10.(l) x = Page 9