Video Transcript
In this video, we will learn how to
recognize direct and inverse variation, involving different roots and powers of
𝑥. We will begin by defining what we
mean by direct and inverse variation or proportion. In direct variation, as one number
increases, so does the other. Direct variation is also called
direct proportion. If two quantities 𝑦 and 𝑥 vary
directly, we can say that 𝑦 is proportional to 𝑥. The 𝛼 symbol shown means is
proportional to. The ratio or proportion of the two
quantities must be equal to some constant 𝑘. This is more commonly written as 𝑦
is equal to 𝑘 multiplied by 𝑥. We have multiplied both sides of
the equation by 𝑥. 𝑘 is known as the constant of
proportionality, and its value will change for each question.
In inverse variation, as one number
increases, the other decreases. This is also known as inverse or
indirect proportion. If we once again consider two
variables 𝑥 and 𝑦, if 𝑦 is inversely proportional to 𝑥, we can write that 𝑦 is
proportional to one over 𝑥. This time, the product of the two
variables is equal to a constant 𝑦 multiplied by 𝑥 is equal to 𝑘. Making 𝑦 the subject, we have 𝑦
is equal to 𝑘 over 𝑥 or 𝑘 divided by 𝑥. As previously mentioned, in this
video, we will look at direct and inverse variation with regards different powers
and roots of 𝑥. This means that 𝑦 will be
proportional to 𝑥 to the 𝑛th power or one over 𝑥 to the 𝑛th power.
This will lead us into solving
equations of the type 𝑦 equals 𝑘 multiplied by 𝑥 to the 𝑛th power or 𝑦 equals
𝑘 divided by 𝑥 to the 𝑛th power. Our first two questions will
involve direct variation.
Given that 𝑦 varies directly as 𝑥
squared, write an equation for 𝑦 in terms of 𝑥 using 𝑘 as a nonzero constant.
If two variables vary directly, we
can say that they are in direct proportion to one another. This means that as one value
increases, so does the other. Our two variables are 𝑦 and 𝑥
squared, which means that 𝑦 is directly proportional to 𝑥 squared. This also means that the ratio or
quotient of the two variables is equal to some constant 𝑘. 𝑦 divided by 𝑥 squared is equal
to 𝑘. We can make 𝑦 the subject of this
equation by multiplying both sides by 𝑥 squared. 𝑦 is therefore equal to 𝑘
multiplied by 𝑥 squared.
When dealing with questions of this
type, we usually go from the proportion expression to our final answer. We replace the proportion symbol
with an equal symbol and a constant 𝑘. If 𝑦 varies directly as 𝑥
squared, then 𝑦 is equal to 𝑘 multiplied by 𝑥 squared.
Given that 𝑥 is proportional to 𝑦
cubed and 𝑥 equals 81 when 𝑦 is equal to three, what is 𝑥 when 𝑦 is equal to
four?
In this question, we’re dealing
with direct proportion or variation. We are told that 𝑥 varies directly
with 𝑦 cubed. This can be rewritten as the
equation 𝑥 is equal to the constant 𝑘 multiplied by 𝑦 cubed. Dividing both sides of this
equation by 𝑦 cubed gives us the constant 𝑘 is equal to 𝑥 divided by 𝑦
cubed. We are also told that when 𝑥 is
equal to 81, 𝑦 is equal to three. This means that we can calculate
the value of 𝑘 by dividing 81 by three cubed. Three cubed is equal to 27, as
three multiplied by three is nine and multiplying this by three gives us 27. This means that 𝑘 is equal to 81
divided by 27. There are three 27s in 81. Therefore, 𝑘 is equal to
three.
If we didn’t spot this, we could
have firstly canceled the fraction by dividing the numerator and denominator by
nine. This would leave us with nine
divided by three, which we know is equal to three. Alternatively, we might have
noticed that 81 is equal to three to the fourth power. And dividing this by three to the
third power or three cubed would give us three to the power of one, which is
three. Substituting this value of 𝑘 back
into our equation gives us 𝑥 is equal to three 𝑦 cubed. We now need to calculate the value
of 𝑥 when 𝑦 is equal to four. This gives us 𝑥 is equal to three
multiplied by four cubed. Four cubed is equal to 64. And multiplying this by three gives
us 192. If 𝑥 is proportional to 𝑦 cubed
and 𝑥 equals 81 when 𝑦 is equal to three, then 𝑥 is equal to 192 when 𝑦 is equal
to four.
We will now look at a couple of
questions involving inverse variation.
Given that 𝑦 varies inversely as
𝑥 squared, write an equation for 𝑦 in terms of 𝑥 using 𝑘 as a nonzero
constant.
We recall that when two variables
vary inversely, we can say that they are inversely proportional. If 𝑦 is inversely proportional to
𝑥 squared, then 𝑦 is proportional to one over 𝑥 squared. We also know that when two
variables are inversely proportional to one another, that product is equal to some
constant 𝑘. 𝑦 multiplied by 𝑥 squared is
equal to 𝑘. Dividing both sides of this
equation by 𝑥 squared gives us the equation 𝑦 is equal to 𝑘 divided by 𝑥
squared.
In general, when dealing with
questions of this type, we go from our proportional expression straight to the
equation in the answer. We replaced the proportional symbol
with equals 𝑘. We know that 𝑘 multiplied by one
over 𝑥 squared can be written as 𝑘 over 𝑥 squared.
We will now use our knowledge of
inverse variation to match the equation with the correct statement.
If 𝑦 is proportional to one over
the square root of 𝑥, then which of the following is true? Is it (A) 𝑥 is directly
proportional to 𝑦? (B) 𝑥 is directly proportional to
𝑦 squared. (C) 𝑥 is inversely proportional to
𝑦. (D) 𝑥 is inversely proportional to
𝑦 squared. Or (E) 𝑥 is inversely proportional
to 𝑦 cubed.
We know that the expression 𝑦 is
proportional to one over the square root of 𝑥 is the same as 𝑦 is inversely
proportional to the square root of 𝑥. When two variables are inversely
proportional to one another, as one increases, the other decreases. This means we can immediately rule
out options (A) and (B) as in direct proportion, as one variable increases, the
other also increases. In this question, as 𝑦 increases,
the square root of 𝑥 decreases and vice versa. If we consider the expression we
were given, we can begin by multiplying both sides by the square root of 𝑥. This means that the square root of
𝑥 multiplied by 𝑦 is proportional to one.
Dividing both sides of this by 𝑦,
we get that the square root of 𝑥 is proportional to one over 𝑦. We can then square both sides of
this such that 𝑥 is proportional to one over 𝑦 squared. When squaring a fraction, we can
square the numerator and denominator separately. We can therefore conclude that 𝑥
is inversely proportional to 𝑦 squared. The correct answer is option
(D).
In our final question, we will
solve a real-life problem involving variation.
The height of a right circular
cylinder ℎ varies inversely with the square of its radius 𝑟. If ℎ equals 93 centimeters when 𝑟
equals 7.5 centimeters, determine ℎ when 𝑟 is equal to 1.5 centimeters.
We know that if two variables vary
inversely, as one increases, the other decreases. This means that in this question, ℎ
and 𝑟 squared are inversely proportional. This can be written as ℎ is
proportional to one over 𝑟 squared. This can be rewritten as an
equation using the constant of proportionality 𝑘 such that ℎ is equal to 𝑘 divided
by 𝑟 squared. Multiplying both sides of this
equation by 𝑟 squared gives us ℎ multiplied by 𝑟 squared is equal to 𝑘. When dealing with inverse
proportion or variation, our two variables will have a product equal to some
constant 𝑘.
We are told that when the height of
the cylinder is 93 centimeters, the radius is 7.5 centimeters. This means that we can calculate
the value of 𝑘 by multiplying 93 by 7.5 squared. 7.5 squared is equal to 56.25. Multiplying this by 93 gives us a
value of 𝑘 equal to 5231.25. We can substitute this constant
back into our equation such that ℎ is equal to 5231.25 divided by 𝑟 squared. We want to calculate this value of
ℎ when 𝑟 is equal to 1.5. 1.5 squared is equal to 2.25. This means that ℎ is equal to
5231.25 divided by 2.25. Typing this into the calculator
gives us 2325. The height of the cylinder when the
radius is 1.5 centimeters is 2325 centimeters.
There is a slightly quicker way of
calculating the value of ℎ without working out the constant 𝑘. We begin by considering the fact
that the product of the height and the radius squared must be equal to some constant
𝑘 for any height and radius in this cylinder. This means that in our first
scenario, with a height of 93 centimeters and a radius of 7.5 centimeters, we have
93 multiplied by 7.5 squared. In our second scenario, we have ℎ
multiplied by 1.5 squared as the radius is 1.5 centimeters. We can then divide both sides of
this equation by 1.5 squared. Once again, typing this into the
calculator gives us an answer for ℎ equal to 2325. This confirms that this is the
height of the cylinder when the radius is 1.5 centimeters.
We will now summarize the key
points from this video. We can express the direct variation
or direct proportion of two variables 𝑥 and 𝑦, as follows. 𝑦 is proportional to 𝑥 or 𝑦 is
equal to 𝑘 multiplied by 𝑥, where 𝑘 is a nonzero constant. In the same way, if 𝑦 is inversely
proportional to 𝑥, we can write this as 𝑦 is proportional to one over 𝑥 or 𝑦 is
equal to 𝑘 divided by 𝑥. We saw in this video that the two
equations can be rewritten such that the constant of proportionality 𝑘 is equal to
the quotient or product of the two variables.
In this video, we saw direct and
inverse variation with different powers and roots of 𝑥. When dealing with problems of this
type, the relationship holds. If 𝑦 is proportional to 𝑥 to the
𝑛th power, then 𝑦 is equal to 𝑘 multiplied by 𝑥 to the 𝑛th power. In the same way, if 𝑦 is
proportional to one over 𝑥 to the 𝑛th power, then 𝑦 is equal to 𝑘 over 𝑥 to the
𝑛th power. We also saw in this video that by
substituting given values of the variables 𝑥 and 𝑦, we can calculate the value of
𝑘 in abstract and real-life problems.
