Joint Variation and Combined Variation Joint variation is just like direct variation, but involves more than one other variable. All the variables are directly proportional, taken one at a time. Let’s set this up like we did with direct variation, find the k, and then solve for y; we need to use the Formula Method: Joint Variation Problem Formula Method Suppose x varies jointly with y and the square root of z. When x=−18 and y=2, then z=9. Find y when x=10 and z=4. x−18−18k=kyz√=k(2)9–√=6k=−3 xx1010y=kyz√=−3yz√=−3y4–√=−3y(2)=10−6=−53 Again, we can set it up almost word for word from the word problem. For the words “varies jointly�, just basically use the “=� sign, and everything else will fall in place. Solve for k first by plugging in variables we are given at first; we get k=−3. Now we can plug in the new values of x and z to get the new y. We see that y=−53. Really not that bad! Joint Variation Word Problem: We know the equation for the area of a triangle is A=12bh (b= base and h= height), so we can think of the area having a joint variation with b and h, with k=12. Let’s do an area problem, where we wouldn’t even have to know the value for k: Joint Variation Problem Math and Notes The area of a triangle is jointly related to the height and the base. If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area? A=kbh(original)A=k(1.4b)(.9h)(new)A=k(1.4)(.9)bhA=k(1.26)bh Remember that when we increase a number by 40%, we are actually multiplying it by 1.4, since we have to add 40% to the original amount. Similarly, when we decrease a number by 10%, we are multiplying it by .9, since we are decreasing the original amount by 10%. Reduce the original values by the new values, and find the new “multiplier�; we see that there will be a 26% increase in the area (A would be multiplied by 1.26, or be 26% greater.) You can put real numbers to verify this, using the formula A=12bh. Joint Variation Word Problem: Here’s another: Joint Variation Problem Math and Notes The volume of wood in a tree (V) varies directly as the height (h) and the square of the girth (g). If the volume of a tree is 144 cubic meters (m3) when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters? V=k(height)(girth)2V=khg2144=k(20)(1.5)2=45k144=45k;k=3.2V=khg2;1000=3.2h⋅22h=78.125 We can set it up almost word for word from the word problem. For the words “varies directly�, just basically use the “=� sign, and everything else will fall in place. Solve for k first; we get k=3.2. Now we can plug in the new values to get the new height. The new height is 78.125 meters. Question: 1. If quantity y is jointly related to quantity x and z, what happens to y as x and z increases? 2. If quantity y is jointly related to quantity x and z, what happens to y as x and z decreases? 3. What mathematical formulas model joint variation? 4. Complete the statement: If y=kxz then the quantity y varies _________jointly________________. K is called the __________constant________of variation. Joint Variation and Combined Variation
Joint variation is just like direct variation, but involves more than one other variable.  All the variables are directly proportional, taken one at a time.
Let’s set this up like we did with direct variation, find the k, and then solve for y; we need to use the Formula Method:
| Joint Variation Problem | Formula Method |
| Suppose x varies jointly with y and the square root of z.  When x=−18 and y=2, then z=9.  Find y when x=10 and z=4. | x−18−18k=kyz√=k(2)9–√=6k=−3          xx1010y=kyz√=−3yz√=−3y4–√=−3y(2)=10−6=−53 Again, we can set it up almost word for word from the word problem. For the words “varies jointly�, just basically use the “=� sign, and everything else will fall in place. Solve for k first by plugging in variables we are given at first; we get k=−3. Now we can plug in the new values of x and z to get the new y. We see that y=−53. Really not that bad! |
Joint Variation Word Problem:
We know the equation for the area of a triangle is A=12bh (b= base and h= height), so we can think of the area having a joint variation with b and h, with k=12. Let’s do an area problem, where we wouldn’t even have to know the value for k:
| Joint Variation Problem | Math and Notes |
| The area of a triangle is jointly related to the height and the base.  If the base is increased by 40% and the height is decreased by 10%, what will be the percentage change of the area? | A=kbh(original)A=k(1.4b)(.9h)(new)A=k(1.4)(.9)bhA=k(1.26)bh Remember that when we increase a number by 40%, we are actually multiplying it by 1.4, since we have to add 40% to the original amount. Similarly, when we decrease a number by 10%, we are multiplying it by .9, since we are decreasing the original amount by 10%. Reduce the original values by the new values, and find the new “multiplier�; we see that there will be a 26% increase in the area (A would be multiplied by 1.26, or be 26% greater.) You can put real numbers to verify this, using the formula A=12bh. |
Joint Variation Word Problem:
Here’s another:ÂÂ
| Joint Variation Problem | Math and Notes |
| The volume of wood in a tree (V) varies directly as the height (h) and the square of the girth (g).  If the volume of a tree is 144 cubic meters (m3) when the height is 20 meters and the girth is 1.5 meters, what is the height of a tree with a volume of 1000 and girth of 2 meters? |   V=k(height)(girth)2V=khg2144=k(20)(1.5)2=45k144=45k;k=3.2V=khg2;1000=3.2h⋅22h=78.125 We can set it up almost word for word from the word problem. For the words “varies directly�, just basically use the “=� sign, and everything else will fall in place. Solve for k first; we get k=3.2. Now we can plug in the new values to get the new height. The new height is 78.125 meters. |
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Question:ÂÂ
1. If quantity y is jointly related to quantity x and z, what happens to y as x and z increases?
2. If quantity y is jointly related to quantity x and z, what happens to y as x and z decreases?
3. What mathematical formulas model joint variation?
4. Complete the statement:
If y=kxz then the quantity y varies _________jointly________________. K is called the __________constant________of variation.
The Correct Answer and Explanation is :
Joint variation occurs when a variable depends on the product or quotient of two or more other variables. In mathematical terms, if ( y ) varies jointly with ( x ) and ( z ), we express this relationship as:
[ y = k \cdot x \cdot z ]
where ( k ) is the constant of variation.
1. Effect of Increasing ( x ) and ( z ) on ( y ):
If both ( x ) and ( z ) increase, ( y ) will also increase. This is because ( y ) is directly proportional to both ( x ) and ( z ). For example, if ( x ) and ( z ) each double, ( y ) will increase by a factor of four.
2. Effect of Decreasing ( x ) and ( z ) on ( y ):
Conversely, if both ( x ) and ( z ) decrease, ( y ) will decrease. Since ( y ) is directly proportional to both variables, any reduction in ( x ) and ( z ) leads to a proportional decrease in ( y ).
3. Mathematical Formulas Modeling Joint Variation:
The general formula for joint variation is:
[ y = k \cdot x \cdot z ]
This equation indicates that ( y ) varies jointly with ( x ) and ( z ). If ( y ) varies jointly with ( x ) and the square of ( z ), the formula becomes:
[ y = k \cdot x \cdot z^2 ]
These formulas can be adapted to include more variables or different powers, depending on the specific relationship.
4. Completing the Statement:
If ( y = k \cdot x \cdot z ), then the quantity ( y ) varies jointly. ( k ) is called the constant of variation.
Example Problem:
Suppose ( x ) varies jointly with ( y ) and the square root of ( z ). When ( x = -18 ), ( y = 2 ), and ( z = 9 ), find ( y ) when ( x = 10 ) and ( z = 4 ).
Solution:
- Determine the constant ( k ): Using the initial conditions: [ x = k \cdot y \cdot \sqrt{z} ] Substitute the known values: [ -18 = k \cdot 2 \cdot \sqrt{9} ] [ -18 = k \cdot 2 \cdot 3 ] [ -18 = 6k ] [ k = -3 ]
- Find ( y ) when ( x = 10 ) and ( z = 4 ): Use the formula with the new values: [ 10 = (-3) \cdot y \cdot \sqrt{4} ] [ 10 = (-3) \cdot y \cdot 2 ] [ 10 = -6y ] [ y = -\frac{10}{6} ] [ y = -\frac{5}{3} ]
Therefore, when ( x = 10 ) and ( z = 4 ), ( y = -\frac{5}{3} ).
For a visual explanation and additional examples on joint variation, you might find the following video helpful: