3.03 Direct and inverse variation
If \left(4, 512\right) is on the curve, P = k Q^{4}, solve for k.
A variable M is directly proportional to the square of N.
Using k as the constant of proportionality, form an equation for M in terms of N.
Given that \left(4, 64\right) satisfies the equation, solve for k.
Consider the relationship where y is directly proportional to the cube of x.
Using k as a constant of proportionality, state the equation relating x and y.
The following table of values shows the relationship between x and y:
| x | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| y | 3 | 24 | 81 | 192 | 375 |
Solve for k, the constant of proportionality.
y is a power function which is directly proportional to x^{n}.
Using k as the proportionality constant, form an equation for y in terms of x.
The table below shows values that satisfy the relationship y = k x^{n}, for the case when n = 2.
| x | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| y | 32 | 128 | 288 | 512 | 800 |
Solve for the constant of proportionality k. Round your answer to the nearest integer.
y is a power function which is directly proportional to x^{n}.
Using k as the proportionality constant, form an equation for y in terms of x.
The table below shows values that satisfy the relationship y = k x^{n}, for the case when n = 3.
| x | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| y | -48 | -384 | -1296 | -3072 | -6000 |
Solve for the constant of proportionality k. Round your answer to the nearest integer.
For the quadratic function pictured here, determine the constant of proportionality k:
For the quadratic shown, determine the constant of proportionality k:
The following graph shows the intersection of the two functions y = m x^{\frac{4}{3}} and y = k x^{\frac{1}{3}}:
Find the value of m.
Find the value of k.
If y varies inversely with x, write an equation that uses k as the constant of variation.
If \left(10, 15\right) is on the curve P = \dfrac{k}{Q}, solve for k.
Determine whether each of the following is an example of a direct variation or an inverse variation:
The variation relating the distance between two locations on a map and the actual distance between the two locations.
The variation relating the number of workers hired to build a house and the time required to build the house.
If r varies inversely with a, write an equation that uses k as the constant of variation.
Determine whether the following equations represent an inverse relationship between x and y:
x = 1 + y^{3}
x = \dfrac {8}{y^{2}}
y = 6 x + 8
x y = - 7
x = \dfrac {2}{y}
x y = 5 x
Consider the equation s = \dfrac {375}{t}.
State the constant of proportionality.
Find the exact value of s when t = 6.
Find the exact value of s when t = 12.
State whether the following tables represent an inversely proportional relationship between x and y.
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 3 | 1.5 | 1 | 0.75 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 36 | 18 | 12 | 9 |
| x | 1 | 5 | 6 | 10 |
|---|---|---|---|---|
| y | 3 | 75 | 108 | 300 |
| x | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| y | 4 | 5 | 6 | 7 |
m is proportional to \dfrac {1}{p}. Consider the values in the table which represents the relationship.
| p | 4 | 6 | 7 | x |
|---|---|---|---|---|
| m | 63 | y | 36 | 28 |
Determine the constant of proportionality, k.
Find the values of x and y.
Find the equation relating t and s for each of the following tables of values:
| s | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| t | 48 | 24 | 16 | 12 |
| s | 3 | 6 | 9 | 12 |
|---|---|---|---|---|
| t | \dfrac{2}{9} | \dfrac{1}{18} | \dfrac{2}{81} | \dfrac{1}{72} |
Use technology to graph the inverse relationship y = \dfrac {6}{x}.
How many x-intercepts does the graph have?
How many y-intercepts does the graph have?
In exchange rate calculations the constant of proportionality is the exchange rate itself. Determine the exchange rate, k, if someone can purchase:
889.00 AUD with 700.00 USD.
553.00 USD with 700.00 AUD.
The heart mass H of a mammal is proportional to its body mass M.
Using k as the constant of proportionality, form an equation relating H and M.
Given that a human with a body mass of 77kg has a heart mass of 0.462kg, determine k, the constant of proportionality.
If a Chimpanzee has a body mass of 57kg, what is the heart mass H?
If a Macaque has a heart mass of 0.084kg, solve for its body mass M.
The blood circulation time C of a mammal is proportional to the fourth root of its body mass B.
Using k as the constant of proportionality, form an equation relating C and B.
Given that an elephant with a body mass of 5290\text{ kg} has a blood circulation time of 148 seconds, determine k, the constant of proportionality. Round your answer to one decimal place.
If a Orangutang has a body mass of 40\text{ kg}, find C, its circulation time. Round your answer to one decimal place.
If a Squirrel has a circulation time of 22 seconds, what is the body mass? Round your answer to one decimal place.