# Algebra II : Variable Relationships

## Example Questions

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### Example Question #1 : Variable Relationships

varies directly with the square root of . If , then  . What is the value of  if ?

None of these answers are correct.

Explanation:

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 .

### Example Question #2 : Variable Relationships

If  varies directly with  and when  due to the effect of a constant, what is the value of  when ?

Explanation:

Since  varies directly with  where  is a constant.

1. Solve for  when  and .

2. Use your equation to solve for  when .

### Example Question #3 : Variable Relationships

If  varies indirectly with  and when  due to the effect of a constant, what is the value of  when ?

Explanation:

Since  varies indirectly with

1. Solve for  when  and .

2. Use the equation you found to solve for  when .

### Example Question #4 : Variable Relationships

varies directly with . If , what is  if

Explanation:

1. Since  varies directly with

with K being some constant.

2. Solve for K using the x and y values given:

3. Use the equation you solved for to find the value of y:

### Example Question #5 : Variable Relationships

varies inversely with . If , then what is  equal to when  ?

Explanation:

1. Since  varies indirectly with :

2. Use the given x and y values to determine the value of K:

3. Using the equation along with the value of K, find the value of y when x=5:

### Example Question #6 : Variable Relationships

varies directly with  and when . What is  when ?

Explanation:

1. Since  varies directly with :

2. Use the values given for x and y to solve for K:

3. Use your new equation with the K you solved for to solve for y when x=27:

### Example Question #7 : Variable Relationships

varies inversely with . When . What is the value of  when ?

Explanation:

1. Since y varies indirectly with :

2. Solve for K using the x and y values given:

3. Using the equation you created by solving for K, find y when x=100:

### Example Question #8 : Variable Relationships

Given the two following points, use interpolation to determine the best estimate for the value

Explanation:

Using our two known points, we can use interpolation to determine the value at any point between them with the following formula:

Where is our first given point, is our second given point, and is the point we want to find. We know our two given points, as well as the x value of our unknown point, so now all we must do is plug in all of our known values and solve for y, our only unknown:

### Example Question #9 : Variable Relationships

The output of a factory in units per day versus the number of employees working is plotted on the graph below, with the following data points collected:

(Workers, Units of output per day):

Assuming a linear relationship, interpolate to find how many units will be made per day if  workers are present.

Explanation:

We want to do a linear interpolation since the relationship between workers and units can be assumed to be linear. This means there is a constant slope between the points, so the slope between two known points will be equal to the slope between the point we are trying to find and some known point. This is expressed in the relation:

,

where  and  are the points we want to find and  and  are known. We choose the known points to be those that are just to the left and right of the point we are trying to find,

and .

Plugging these into our interpolation formula and knowing , we can find , the units output per day.

.

Simplifying and rearranging to solve for :

.

So there are  units produced when the number of workers is .

### Example Question #10 : Variable Relationships

Given the points and , use linear interpolation to find the value of  when .