Download Topic 4 Formula - Olympic College

Olympic College - Topic 4 Formula
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 cm3
Perform the calculation
Page | 1
Olympic College - Topic 4 Formula
Example 3:
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.
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/sec2
Perform the calculation
Example 4:
The second formula of motion states that
s = ut + ½at2
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/sec2.
s
=
ut + ½at2
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
Olympic College - Topic 4 Formula
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
Olympic College - Topic 4 Formula
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
Olympic College - Topic 4 Formula
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 + ½at2 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/sec2.
(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/sec2.
(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/sec2.
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
Olympic College - Topic 4 Formula
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
W =
Divide both sides by LH
Switch the order
Example 2: Change the subject of the formula E = ½mv2
Solution:
E =
½mv2
2E =
mv2
Multiply both sides by 2
m
Divide both sides by v2
=
m
=
Switch the order
Example 3: Change the subject of the formula V =
Solution:
to m.
to h.
V =
3V =
=
h
=
Multiply both sides by 3
h
Divide both sides by
Switch the order
Page | 6
Olympic College - Topic 4 Formula
Example 4: Change the equation y =
Solution:
y
y+7
=
=
=
x
Add 7 to both sides
x
=
Divide both sides by
Switch the order
Example 5: Change the equation L =
Solution:
in terms of x.
in terms of B.
L =
5L =
Multiply both sides by 6
5L + 7C =
=
B
=
Add 7C to both sides
B
Divide both sides by
Switch the order
Page | 7
Olympic College - Topic 4 Formula
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
1
d
3
1
(g) c = d – 17
3
(d) c =
(b) c = 5d
(c) c = 2d
(e) c = 3d + 2
(f) c = 5d + 2
(h) c =
1
d–5
4
(I) c =
1
d+1
4
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)
(j) y =
x4
3
(k) y =
2x  7
5
x5
9
3x  8
(l) y =
5
y=
Page | 8
Olympic College - Topic 4 Formula
Solutions
Exercise 1A
Exercise 1B:
1.(a) 60 volts
1.(b) 54.9 volts
1.(c) 2.125 or
2.(a) 64 cm3
2.(b) 4.5 volts
2.(c) 27 cm3
3.(a) 288 ft2
3.(b) 0.62 m2
3.(c) 30.5 or
4.(a) 2 ft/s2
4.(b) 10 ft/s2
4.(c) 4 ft/s2
5.(a) 168 m
5.(b) 88 ft
5.(c) 3.4 or
6.(a) $72
6.(b) $3
6.(c) $22.50
volts
m2
m
1. B =
2.
3. x =
4. y =
5.
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 =
x=
6. m =
Page | 9