1. Word Problems Involving Radicals: Direct and Inverse Variation
Section 8.7
Word Problems Involving Radicals: Direct and Inverse Variation

2. Variation
A variation is an equation that relates one variable to another variable.
If a variable y varies directly as x, then y = kx, where k is the constant of variation.

3. Solving a Direct Variation Problem
Write the direct variation equation.
Solve for the constant k by substituting in known values.
Replace k in the direct variation equation by the value obtained in step 2.
Solve for the desired value.

4. Example
If y varies directly as x, and y = 16 when x = 7, find the value of y when x = 30. Step 1: y = kx Step 2: To find k, substitute y = 16 and x = 7. Step 3: We now write the variation equation with k replaced by 16/7.

5. Example (cont)
If y varies directly as x, and y = 16 when x = 7, find the value of y when x = 30. Step 4: Replace x by 30 and find the value of y.
Therefore, y is approximately 68.6 when x = 30.

6. Sample Direct Variation Situations
Verbal Description Variation Equation
y varies directly as x y = kx
b varies directly as the square of c b = kc2
l varies directly as the cube of m l = km3
V varies directly as the square root of h

7. Example
There are 14 blonde-haired students in a statistics class with 45 students. The number of total students in the class varies directly with the number of blonde-haired students. Approximately how many students will there be in a statistics class if there are 25 blonde-haired students? Step 1: Let S = the total number of students. Let b = the number of blonde-haired students. S = kb

8. Example (cont) Step 2: To find k, substitute S = 45 and b = 14.
There are 14 blonde-haired students in a statistics class with 45 students. The number of total students in the class varies directly with the number of blonde-haired students. Approximately how many students will there be in a statistics class if there are 25 blonde-haired students?
Step 2: To find k, substitute S = 45 and b = 14.
Step 3: Write the variation equation.

9. Example (cont) Step 4: Replace b by 25 and find the value of S.
There are 14 blonde-haired students in a statistics class with 45 students. The number of total students in the class varies directly with the number of blonde-haired students. Approximately how many students will there be in a statistics class if there are 25 blonde-haired students?
Step 4: Replace b by 25 and find the value of S.
There will be approximately 80 students in the class.

10. Solving an Inverse Variation Problem
Write the inverse variation equation.
Solve for the constant k by substituting in known values.
Replace k in the inverse variation equation by the value obtained in step 2.
Solve for the desired value.
If a variable y varies inversely as x, then where k is the constant of variation.

11. Example
If y varies inversely as x, and y = 24 when x = 10, find the value of y when x = 36. Step 1: Step 2: To find k, substitute y = 24 and x = 10. Step 3: Write the variation equation.

12. Example (cont)
If y varies inversely as x, and y = 24 when x = 10, find the value of y when x = 36. Step 4: Replace x by 36 and find the value of y. Therefore, y is approximately 6.7 when x = 36.

13. Sample Inverse Variation Situations
Verbal Description Variation Equation
y varies inversely as x
b varies inversely as the square of c
l varies inversely as the cube of m
d varies inversely as the square root of t

14. Example
Sally spends 1.5 hours watching television and 8 hours studying each week. If the amount of time spent watching TV varies inversely with the amount of time spent studying, find the amount of time Sally will spend watching TV if she studies 14 hours a week. Step 1: Let T = the number of hours spent watching television. Let s = the number of hours spent studying.

15. Example (cont) Step 2: To find k, substitute T = 1.5 and s = 8.
Sally spends 1.5 hours watching television and 8 hours studying each week. If the amount of time spent watching TV varies inversely with the amount of time spent studying, find the amount of time Sally will spend watching TV if she studies 14 hours a week.
Step 2: To find k, substitute T = 1.5 and s = 8.
Step 3: Write the variation equation.
Step 4: Replace s by 14 and find the value of T.
Sally will spend approximately 0.86 hours (or 52 minutes) watching TV.

16. Graphs of Variation Equations
Variation Statement
Equation
Graph
y varies directly as x
y varies directly as x2
y varies directly as x3
y varies directly as the square root of x
y = kx x y x y
y = kx2
y = kx3 x y x y

17. Graphs of Variation Equations
Variation Statement
Equation
Graph
y varies inversely as x
y varies inversely as x2 x y x y