Algebra II : Variable Relationships

Study concepts, example questions & explanations for Algebra II

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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 ?

Possible Answers:

None of these answers are correct.

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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  ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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 ?

Possible Answers:

Correct answer:

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

 

Possible Answers:

Correct answer:

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.

Linear_interp

Possible Answers:

Correct answer:

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 .

Possible Answers:

Correct answer:

Explanation:

Use the formula for interpolation to determine the value of y:

We will use (30, 51) as our x2 and y2 and (20, 36) as our x1 and y1 and we will solve for y using 26.5 for x.

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