VARIATIONS AND FORMULAS
DIRECT VARIATION
1. The graph below shows the relationship between the cost, C, in dollars of trail mix and
the mass, m, measured in kg’s to be purchased.
a) Identify the dependent and independent variables.
b) Determine the slope of the line.
c) State a formula for the direct variation.
2. The time needed to manufacture CD’s is shown in the following table:
time (min), T
number of CD's,
N
0
10
20
50
100
0
80
160
400
800
a) Graph this variation
b) Identify the dependent and independent variables
c) Determine the slope of the line
d) Write a formula for the direct variation
e) What does the constant of variation represent?
3. The estimated cost C, of building a house in Flin Flon varies directly as the area in
square metres of floor space, a. It costs $30,000 for 100 sq.metres.
a) Determine the constant of variation.
b) State the direct variation formula.
c) Find the cost of building a 300 sq.metre house in Flin Flon.
4. The distance D, of a vehicle traveling at constant speed is directly proportional to the
time, t, it travels. A vehicle traveling at a constant speed travels 255 km in 3 hours.
a) Determine the constant of variation.
b) State the direct variation formula.
c) Find the distance traveled in 5 hours.
d) Find how long it would take to travel 800 km.
e) What does the constant of variation represent?
f) Identify the dependent and independent variables.
5. The cost C, of long distance calls varies directly as the length of time t, of the call. A
60 minute call costs $4.80.
a) Determine the constant of variation.
b) State the direct variation formula.
c) Determine the cost of a 90 minute call.
d) Determine how many minutes you would have for a call that costs $0.80.
e) What does the constant of variation represent?
f) Identify the dependent and independent variables.
6. The amount of gas G, in litres consumed by a vehicle with steady highway driving is
directly proportional to the distance, d, traveled in km. The vehicle consumes 7.5 litres
per 100 km.
a) Determine the constant of variation.
b) State the direct variation formula.
c) Determine the amount of gasoline consumed on a highway trip of 45 km in this
vehicle.
d) Determine how far you could travel on 30 litres of gas.
e) What does the constant of variation represent?
f) Identify the dependent and independent variables.
7. The voltage V, in an electrical circuit is directly proportional to the current, A. The
voltage is 120V with a current of 15 amps.
a) Determine the constant of variation.
b) State the direct variation formula.
c) Determine the voltage when the current is 12 amps.
8. The commission C, a salesperson receives is directly proportional to her sales, s. If
the salessperson earns a commission of $1,575 on $22,500 worth of sales, find the
commission she would earn on $40,000 worth of sales.
9. The depth of water formed when a given area of snow melts varies directly with the
depth of snow. If a depth of 250cm of snow in a certain area melts to give 28cm of
water, calculate the depth of water formed when a depth of 75cm of snow melts.
10. The amount of grass G, cut is directly proportional to the length of time t, spent
cutting the grass. It takes a student 30 minutes to cut 174 sq. metres of grass.
a) Find the amount of grass the student would cut in five hours.
b) How long would it take the student to cut 1000 sq. metres of grass?
11. The highway department calculates that it costs $3.5 million to repave 20km of
highway. They have a budget of $12,000,000 for a highway repaving project. How
many km can they pave? Graph this variation. How much would it cost to pave 1200
km?
INVERSE VARIATIONS
1. The temperature T of the air in degrees Celsius varies inversely with the height h in
metres above sea level. The constant of variation is 4800.
a) Give a formula for this variation.
b) Find the temperature at 200 m above sea level.
c) Find the temperature at 1500 m above sea level.
d) Why do you think that there is often snow on high mountains?
2. The time, T, in months it takes you to earn enough money for a vacation varies
inversely with the amount you save each month s. It takes eight month when you save
$300 a month.
a) Find the constant of variation.
b) Give the formula for this variation.
c) How long will it take you to earn enough money for the vacation if you save $400 a
month?
3. The frequency F of a note produced by a guitar string with constant tension varies
inversely as the length L of the string. The frequency of a note produced by a guitar
string with a length of 60 cm is 640 hertz.
a) Find the constant of variation.
b) Give the formula for this variation.
c) Find the frequency of a guitar string with a length of 75 cm.
4. The volume V of a fixed mass of gas at constant temperature varies inversely as the
pressure P. The volume of a fixed mass of gas is 12 litres when the pressure is 2
atmospheres.
a) Find the formulas for this variation.
b) Find the volume when the pressure of the same mass of gas is 6 atmospheres.
5. In electromagnetic transmission the signal frequency F in kilohertz is inversely
proportional to the wavelength w, in metres. An AM radio station has a signal frequency
of 608 kHz with a wavelength of 500 m. Find the signal frequency of an electromagnetic
transmission with a wavelength of 650 m.
6. The strength S of a radio signal is inversely proportional to the square of its distance d
from the transmitter. A signal received from a transmitter 75 km away is 80% of the
standard signal.
a) Determine the constant of variation.
b) Determine the formula for this variation.
c) What would the strength of the same signal be at a distance of 100 km?
d) At what distance would the transmitter need to be for a signal of 90% strength?
PARTIAL VARIATIONS
1. To rent a mid sized car a rental company charges $30 a day plus thirty five cents per
km.
a) Express the variation as a formula.
b) Graph this equation.
c) Identify the dependent variable, the independent variable, the constant of variation, and
the fixed value.
2. The cost of renting a community centre for an event is $100 plus a $5 hospitality fee
for each person attending the event.
a) Give the formula for this variation.
b) Graph this variation.
3. A plumber charges its customers a base service charge of $25 plus an hourly rate of
$50.
a) Express this variation as a formula.
b) Determine the total charge for a repair that takes 2 hours.
c) Determine the total charge for a repair that takes 30 minutes.
4. A salesperson receives a base monthly salary of $500 plus a commission of 6% on his
sales.
a) Give a formula for this variation.
b) Determine the monthly salary if he sells $30,000 worth of goods.
5. Doe Doe’s Pizza charges $8.40 for its basic medium pizza. Each additional topping
costs $0.90.
a) Write a formula for this variation.
b) What would the cost be for a pizza with six additional toppings?
6. The cost of making uniforms for a softball team involves a fixed cost and a cost per
player. The cost for 15 players is $695 and the cost for 20 players is $920.
a) Determine a formula for this variation.
b) What would the cost for uniforms be for 25 players?
7. A taxi company charges its customers a fixed amount plus an amount per kilometer
driven. The cost for a ride of 10 km is $6 while the cost for 25 km is $11.25.
a) Express this variation as a formula.
b) Find the cost for a ride of 36 km.
8. A garage charges a fixed rate as well as a fee per hour. Repair work which takes half
an hour costs $45. Repair work that takes three hours costs $145.
a) Express this variation as a formula.
b) Find the cost for a repair that takes four hours and 45 minutes.
9. You have two job offers. One has a starting salary of $15,000 for the first year with
raises of $500 per year. The other has a starting salary of $12,000 for the first year with
raises of $800 per year. After how many years will these salaries be the same?
DIRECT SQUARED
1. The kinetic energy in joules, J, of a moving mass varies as the square of its speed, s, in
m/s. The constant of variation is 0.5.
a) Express this variation as a formula.
b) Find the kinetic energy of a moving mass with a speed of 2.5 m/s.
c) Graph this variation for values of s from 0 to 10.
2. The power P in watts in a given circuit varies directly as the square of the current I in
amps. The power is 2000 watts when the current is 40 amps.
a) Determine the value of the constant k.
b) State the formula for this variation.
c) Determine the power when the current is 30 amps.
3. The number N of bacteria in a culture is directly proportional to the square of the time
t in minutes it has been growing. A culture that has been growing for 20 minutes has
1000 bacteria.
a) Determine the formula for this variation.
b) Find the number of bacteria after 30 minutes.
4. The distance D in metres required to stop a vehicle by braking varies directly as the
square of the speed s in km/h. The constant of variation is 0.006.
a) Determine the formula for this variation.
b) Compare the stopping distances at 50 km/h and at 100 km/h.
c) Is it true that if you double your speed, you will double the stopping distance?
5. The distance D a hockey puck travels on ice varies directly as the square of the speed s
of the puck. A puck moving at a speed of 4 m/s travels 220 m before stopping.
a) Find the constant of variation.
b) Give a formula for this variation.
c) Determine how far a hockey puck would travel if it is hit with a speed of 3 m/s.
6. The power P required to run a boat varies as the cube of its speed v. A boat motor
with a power of 500kW will run a boat at 4 m/s. Write a formula representing this
variation. Determine the power needed to run the same boat at 5 m/s.
FORMULA USE
a) Find the area of a circle whose radius is 20 cm.
b) Find the simple interest earned after six years on a $1,000 investment at an interest rate
of 5.5% per year.
c) Find the Fahrenheit temperature if the temperature is 200º C.
d) Find the surface area of a sphere with a radius of 18 cm.
e) Find the hypotenuse of a right angled triangle if the other two sides of the triangle are 5
and 12 cm.
f) Find the final amount of a compound interest investment of $1,000 compounded semiannually (twice a year) at 6% over three years. Answer to the nearest one cent.
g) Find the width of a rectangle with an area of 34 cm2 and a length of 9 cm.
h) Find the length of a rectangle with a perimeter of 186 m and a width of 32 m. The
formula is P = 2l + 2w.
i) Find the height of a cone with a radius of 24 cm and a volume of 9047 cm3.
Eff
where hp represents horsepower, I
746
represent current, V represents voltage and Eff represents efficiency. Determine the
horsepower of an electric motor with voltage 240 V, current 20 amps, and efficiency
85%.
j) Horsepower is given by the formula hp  IV
k) A solid sphere just fits into a cylinder. Calculate the volume of empty space in the
cylinder if the height and diameter of the cylinder are 10 centimeters.