Joint variation

1. JOINT VARIATION

2. Definition Of Joint Variation Joint variation is the same as direct variation with two or more quantities. Direct variation occurs when two quantities change in the same manner. That is: Increase in one quantity causes an increase in the other quantity Decrease in one quantity causes a decrease in the other quantity

3. Joint Variation : Joint variation is a variable which is proportional to the product of two or more other varia bles Example : Y = K X Z Z = 11a X 15b

4. Introduction The meaning of the phrase "Joint Variation" can be gleaned from the meaning of the two words "Joint" and "Variation".  In other words, it refers to a case where one variable or quantity  varies jointly  with several other variables or quantities. Symbolically, when we say that the variable Z varies jointly with the variables X and Y, we mean that if either X, or Y or both X and Y are varied, then Z will vary accordingly.  This is an example of a variable varying jointly with two other variables. In general. a variable may vary jointly with many variables (two or more).

5. For Example: The cost of a pencil and the number of pencils you buy. Buy more pay more.....Buy less pay less. Direct variation between variables x and y can be expressed as: y = kx , where 'k' is the constant of variation and  k ≠ 0 y = kxz  represents joint variation. Here, y varies jointly as  x  and  z .

6. More Examples on Joint Variation y = 7xz, here y varies jointly as x and z y = 7x 2 z 3 , here y varies jointly as x 2  and z 3 Area of a triangle = is an example of joint variation. Here the constant is 1. Area of a triangle varies jointly with base 'b' and height 'h' Area of a rectangle = L x M represents joint variation. Here the constant is 1. Area of a rectangle varies jointly with length 'l' and width 'w'.

7. Solved Example on Joint Variation Ques :  Assume a varies jointly with b and c. If b = 2 and c = 3, find the value of a. Given that a = 12 when b = 1 and c = 6. Solution: Step 1: First set up the equation. a varies jointly with b and c a = kbcStep 2: Find the value of the constant, k. Given that a = 12 when b = 1 and c = 6 a = kbc 12 = k x 1 x 6 ⇒ k = 2Step 3: Rewrite the equation using the value of the constant 'k' a =  2bc

8. Step 4: Using the new equation, find the missing value. If b = 2 and c = 3, then a = 2 x 2 x 3 = 12Step 5: So, when a varies jointly with b and c and If b = 2 and c = 3, then the value of a is 12 . Real-world Connections for Joint Variation Force = mass × acceleration. The force exerted on an object varies jointly as the mass of the object and the acceleration produced.

9. exercise If y varies jointly as x and z, and y = 33 when x = 9 and z = 12, find y when x = 16 and z = 22 . If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f when g = 3 and h = 6 . Wind resistance varies jointly as an object’s surface area and velocity. If an object traveling at 40 mile per hour with a surface area of 25 square feet experiences a wind resistance of 225 Newtons , how fast must a car with 40 square feet of surface area travel in order to experience a wind resistance of 270 Newtons ?

10. For a given interest rate, simple interest varies jointly as principal and time. If $2000 left in an account for 4 years earns interest of $320, how much interest would be earned in if you deposit $5000 for 7 years ? If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find a when b = 9 and c = 225 . The volume of a pyramid varies jointly as its height and the area of its base. A pyramid with a height of 12 feet and a base with area of 23 square feet has a volume of 92 cubic feet. Find the volume of a pyramid with a height of 17 feet and a base with an area of 27 square feet.

11. answer 868 1.y= — 9 2.f=405 3.velocity=30 miles per hour 4. interest= $1400 5.a=126 6.volume=153 cubic feet

12. Importance of Joint Variation in Real Life We have seen several real world examples of Joint Variation. This concept is widely used in what-if analysis. We will illustrate what-if analysis using an example. If we look at the investment example, we saw that the interest earned varies jointly with the amount deposited, the rate of interest and the period of investment. A what-if analysis would involve varying each of the variables, in turn, and finding out the effect on the rest of the variables. For example we could have several investment plans where the rate of interest would be different, the period of investment would be different and so on. The what-if analysis would find the best option for the investor, depending on his/her requirements, by varying the variables in turn and finding the effect.

13. Thankyou for understanding By; EUNICE LIBAO G9- ILANG-ILANG