How to find direct variation - Algebra 1
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The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying $420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly?
The monthly cost to insure your cars varies directly with the number of cars you own. Right now, you are paying $420 per month to insure 3 cars, but you plan to get 2 more cars, so that you will own 5 cars. How much does it cost to insure 5 cars monthly?
The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as 
. M is the monthly cost, C is the number of cars owned, and k is the constant of variation.
Given that it costs $420 a month to insure 3 cars, we can find the k-value.
Divide both sides by 3.
Now, we have the mathematical relationship.
Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.
The statement, 'The monthly costly to insure your cars varies directly with the number of cars you own' can be mathematically expressed as
Given that it costs $420 a month to insure 3 cars, we can find the k-value.
Divide both sides by 3.
Now, we have the mathematical relationship.
Finding how much it costs to insure 5 cars can be found by substituting 5 for C and solving for M.
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varies directly with the square root of 
. If 
, then 
. What is the value of 
if 
?
If 
varies directly with the square root of 
, then for some constant of variation 
,
If 
, then 
; therefore, the equation becomes
,
or
.
Divide by 5 to get 
, making the equation
.
If 
, then 
.
If
If
or
Divide by 5 to get
If
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If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?
If an object is hung on a spring, the elongation of the spring varies directly as the mass of the object. A 20 kg object increases the length of a spring by exactly 7.2 cm. To the nearest tenth of a centimeter, by how much does a 32 kg object increase the length of the same spring?
Let 
be the mass of the weight and the elongation of the spring. Then for some constant of variation 
,
We can find 
by setting 
from the first situation:
so 
In the second situation, we set 
and solve for 
:
which rounds to 11.5 centimeters.
Let
We can find
so
In the second situation, we set
which rounds to 11.5 centimeters.
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If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?
If an object is hung on a spring, the elongation of the spring varies directly with the mass of the object. A 33 kilogram object increases the length of a spring by exactly 6.6 centimeters. To the nearest tenth of a kilogram, how much mass must an object posess to increase the length of that same spring by exactly 10 centimeters?
Let 
be the mass of the weight and the elongation of the spring, respectively. Then for some constant of variation 
,
.
We can find 
by setting 
:
Therefore 
.
Set 
and solve for 
:
kilograms
Let
We can find
Therefore
Set
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The volume of a fixed mass of gas varies inversely as the atmospheric pressure, as measured in millibars, acting on it, and directly as the temperature, as measured in kelvins, acting on it.
A balloon is filled to a capacity of exactly 100 cubic meters at a time at which the temperature is 310 kelvins and the atmospheric pressure is 1,020 millibars. The balloon is released, and an hour later, the balloon is subject to a pressure of 900 millibars and a temperature of 290 kelvins. To the nearest cubic meter, what is the new volume of the balloon?
The volume of a fixed mass of gas varies inversely as the atmospheric pressure, as measured in millibars, acting on it, and directly as the temperature, as measured in kelvins, acting on it.
A balloon is filled to a capacity of exactly 100 cubic meters at a time at which the temperature is 310 kelvins and the atmospheric pressure is 1,020 millibars. The balloon is released, and an hour later, the balloon is subject to a pressure of 900 millibars and a temperature of 290 kelvins. To the nearest cubic meter, what is the new volume of the balloon?
If 
are the volume, pressure, and temperature, then the variation equation will be, for some constant of variation 
,
To calculate 
, substitute 
:
The variation equation is
so substitute 
and solve for 
.
If
To calculate
The variation equation is
so substitute
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If 
is directly proportional to 
and when 
at 
, what is the value of the constant of proportionality?
If
The general formula for direct proportionality is
where 
is the proportionality constant. To find the value of this 
, we plug in 
and 
Solve for 
by dividing both sides by 12
So 
.
The general formula for direct proportionality is
where
Solve for
So
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The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn $48.
The amount of money you earn is directly proportional to the nunber of hours you worked. On the first day, you earned $32 by working 4 hours. On the second day, how many hours do you need to work to earn $48.
The general formula for direct proportionality is
where 
is how much money you earned, 
is the proportionality constant, and 
is the number of hours worked.
Before we can figure out how many hours you need to work to earn $48, we need to find the value of 
. It is given that you earned $32 by working 4 hours. Plug these values into the formula
Solve for 
by dividing both sides by 4.
So 
. We can use this to find out the hours you need to work to earn $48. With 
, we have
Plug in $48.
Divide both sides by 8
So you will need to work 6 hours to earn $48.
The general formula for direct proportionality is
where
Before we can figure out how many hours you need to work to earn $48, we need to find the value of
Solve for
So
Plug in $48.
Divide both sides by 8
So you will need to work 6 hours to earn $48.
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Does the equation below represent a direct variation? If it does, find the constant of variation.
Does the equation below represent a direct variation? If it does, find the constant of variation.
Direct Variation is a relationship that can be represented by a function in the form
, where 
is the constant of variation for a direct variation. 
is the coefficient of 
.
The equation is in the form 
, so the equation is a direct variation.
The constant of variation or 
is 
Therefore, the answer is,
Yes it is a direct variation, 
with a direct variation of 
Direct Variation is a relationship that can be represented by a function in the form
The equation is in the form
The constant of variation or
Therefore, the answer is,
Yes it is a direct variation,
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Suppose 
and 
, and that 
is in direct proportion with 
. What is the value of proportionality?
Suppose
The general formula for direct proportionality is
where 
is our constant of proportionality. From here we can plug in the relevant values for and to get
Solving for 
requires that we divide both sides of the equation by 
, yielding
The general formula for direct proportionality is
where
Solving for
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The cost of a catering company varies directly with the number of people attending. If the cost is $100 when 20 people attend the party, find the constant of variation.
The cost of a catering company varies directly with the number of people attending. If the cost is $100 when 20 people attend the party, find the constant of variation.
Because the cost varies directly with the number of people attending, we have the equation
Where 
is the cost and 
is the number of people attending.
We solve for 
, the constant of variation, by plugging in 
and 
.
And by dividing by 20 on both sides
Yields
Because the cost varies directly with the number of people attending, we have the equation
Where
We solve for
And by dividing by 20 on both sides
Yields
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The amount of money Billy earns is directly proportional to his hours worked. Suppose he earns 
every eight hours of work. What is the minimum hours Billy must work in order to exceed 
? Round to the nearest integer.
The amount of money Billy earns is directly proportional to his hours worked. Suppose he earns
Write the formula for direct proportionality.
Let:
Substitute twelve dollars and eight hours into this equation to solve for 
.
Divide by eight on both sides.
Substitute 
back into the formula.
To find out the minimum number of hours Billy must work to make 
, substitute 
into 
and solve for 
.
Multiply by two thirds on both sides.
Simplify both sides.
Billy must work at least 
hours to earn as much required.
Write the formula for direct proportionality.
Let:
Substitute twelve dollars and eight hours into this equation to solve for
Divide by eight on both sides.
Substitute
To find out the minimum number of hours Billy must work to make
Multiply by two thirds on both sides.
Simplify both sides.
Billy must work at least
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