3.5 Joint Variation

Which is true about a function of two variables?

_____

Which statement is true about a function of two variables?

1. It has one input and two outputs.
2. It has two inputs and one output.
3. It has two inputs and two outputs.
4. None of the above.

The maximum height that the water stream from a fire hose can reach depends on the water pressure and the diameter of the nozzle, and is given by the function

$$H=f(P,n)=26+{\displaystyle \frac{5}{8}}P+5n$$

where $P$ is the nozzle pressure in psi, and $n$ is measured in ${\displaystyle \frac{1}{8}}$-inch increments over the standard nozzle diameter of ${\displaystyle \frac{3}{4}}$ inch.

1. Evaluate $f(40,2)$ and explain what it means.
   $f(40,2)=$_____
   _____
2. What nozzle pressure is needed to reach a height of $91$ feet with a $1{\displaystyle \frac{1}{8}}$-inch nozzle?
   __________

The maximum height that the water stream from a fire hose can reach depends on the water pressure and the diameter of the nozzle, and is given by the function

$$H=f(P,n)=26+{\displaystyle \frac{5}{8}}P+5n$$

where $P$ is the nozzle pressure in psi, and $n$ is measured in ${\displaystyle \frac{1}{8}}$-inch increments over the standard nozzle diameter of ${\displaystyle \frac{3}{4}}$ inch.

1. Evaluate $f(40,2)$ and explain what it means.
2. What nozzle pressure is needed to reach a height of $91$ feet with a $1{\displaystyle \frac{1}{8}}$-inch nozzle?

A retirement plan requires employees to put aside a fixed amount of money each year until retirement. The amount accumulated, $A$, includes 8% annual interest on employee's annual contribution, $c$. $A$ is a function of $c$ and the number of years, $t$, that the employee makes contributions, so $A=f(c,t)$.

1. How much will an employee accumulate if she contributes $500 a year for 40 years? (Do not enter any commas. For example, enter "10000" for 10,000.
   $_____
   Write your answer with function notation:
   $f($_____,_____$)=$_____
2. How much must she contribute each year in order to accumulate $573,770 after 50 years? $_____
   Write your answer with function notation.
   $f($_____,_____$)=$_____
3. Find a value of $t$ that solves the equation $137,286=f(2500,t)$.
   $t=$_____
   What does this equation tell you about the retirement fund?
   _____

A retirement plan requires employees to put aside a fixed amount of money each year until retirement. The amount accumulated, $A$, includes 8% annual interest on employee's annual contribution, $c$. $A$ is a function of $c$ and the number of years, $t$, that the employee makes contributions, so $A=f(c,t)$.

1. How much will an employee accumulate if she contributes $500 a year for 40 years? Write your answer with function notation.
2. How much must she contribute each year in order to accumulate $573,770 after 50 years? Write your answer with function notation.
3. Find a value of $t$ that solves the equation $114,405=f(2500,t)$. What does this equation tell you about the retirement fund?

1. For the table in the previous example, consider the column corresponding to a beam depth of $3$ cm. Graph $L$ as a function of $w$ when the depth is constant at $d=3$.
2. Find a formula for $L$ as a function of $w$ for $d=3$.
   $L=$_____

1. For the table in the previous example, consider the column corresponding to a beam depth of $3$ cm. Graph $L$ as a function of $w$ when the depth is constant at $d=3$.
2. Find a formula for $L$ as a function of $w$ for $d=3$.

Which of the following functions represents joint variation, where $k$ is a constant?

_____

Which of the following functions represents joint variation, where $k$ is a constant?

1. $H=kst$
2. $L=m+kn$
3. $y=k{r}^2$
4. $Z=k(x+y)$

The cost, $C$, of tiling a rectangular floor depends on the dimensions (length and width) of the floor, so $C=f(w,l)$. The table shows the costs in dollars for some dimensions.

1. Consider the row corresponding to $6$ feet in width. Does cost vary directly with length? _____
2. Consider the column corresponding to a length of $10$ feet. Does the cost vary directly with width? _____
3. Given that the cost varies jointly with the length and width of the floor, find a formula for $C=f(w,l)$.
   $C=$_____

The cost, $C$, of tiling a rectangular floor depends on the dimensions (length and width) of the floor, so $C=f(w,l)$. The table shows the costs in dollars for some dimensions.

1. Consider the row corresponding to $6$ feet in width. Does cost vary directly with length?
2. Consider the column corresponding to a length of $10$ feet. Does the cost vary directly with width?
3. Given that the cost varies jointly with the length and width of the floor, find a formula for $C=f(w,l)$.

Explain the difference between the symbols $f(ab)$ and $f(a,b)$.

_____

Explain the difference between the symbols $f(ab)$ and $f(a,b)$.

We can represent a function of two variables by

_____

We can represent a function of two variables by

1. a table of values.
2. a graph in three dimensions.
3. several graphs in two dimensions.
4. all of the above.

The period of a satellite orbiting the Earth varies directly with the radius of the orbit and inversely with the speed of the satellite.

1. Write a formula for the period, $T$, as a function of orbital radius, $r$, and velocity, $v$. Your formula should contain a constant of variation $k$ (whose value we will determine later).
   $T=f(r,v)=$_____
2. GPS satellites orbit at an altitude of $20,200$ kilometers and a speed of 233 kilometers per minute. The period of a GPS satellite is 11 hours and 58 minutes. Find the constant of variation in your formula for $T$. (The radius of the Earth is 6360 km.)
   $k\approx$_____
3. Satellites in polar orbits are used to measure ozone concentrations in the atmosphere. One such satellite orbits at an altitude of 833 km and has a period of 101.2 minutes. What is the speed of this satellite?
   Speed: __________
4. Graph $T$ as a function of $v$ for $r=5000$, $r=10,000$, $r=20,000$, and $r=30,000$.

The period of a satellite orbiting the Earth varies directly with the radius of the orbit and inversely with the speed of the satellite.

1. Write a formula for the period, $T$, as a function of orbital radius, $r$, and velocity, $v$.
2. GPS satellites orbit at an altitude of $20,200$ kilometers and a speed of 232 kilometers per minute. The period of a GPS satellite is 11 hours and 58 minutes. Find the constant of variation in your formula for $T$. (The radius of the Earth is 6360 km.)
3. Satellites in polar orbits are used to measure ozone concentrations in the atmosphere. One such satellite orbits at an altitude of 833 km and has a period of 101.2 minutes. What is the speed of this satellite?
4. Graph $T$ as a function of $v$ for $r=5000$, $r=10,000$, $r=20,000$, and $r=30,000$.

Melody Airlines charges $$129$ for a coach ticket from San Francisco to Seattle, and $$240$ for a first-class ticket.

1. Write a function of two variables for the revenue, $R$, that Melody Airlines will collect from the flight.
2. The airplane has $12$ first-class seats and $24$ coach seats. What is the maximum revenue the airline can collect? Write your answer with function notation.

A manufacturing firm calculates its profit (or loss) by subtracting the cost of production from its revenue. The firm can produce 100 items per week, with fixed costs (overhead) of $$500$.

1. Write a function for the firm's weekly profit, $P$, if they charge a price of $p$ dollars per item and it costs them $c$ dollars to produce each item.
2. If each item costs $$80$ to produce, what price should the firm charge in order to make a profit? Write your answer with function notation.

Archaeologists can calculate the size of a pot from just a fragment, or sherd, of the original. If $L$ and $h$ are the dimensions of an arc of a circle, as shown in the figure, then the radius of the entire circle is given by the function

$$r=f(L,h)=\frac{L^2}{8h}+\frac{h}{2}$$

1. A pottery sherd has dimensions $L=4$ inches and $h={\displaystyle \frac{3}{2}}$ inch. What was the radius of the whole pot? Write your answer with function notation.
2. Does $r$ vary directly with $L^2$? Why or why not?
3. Show that the formula gives the correct value for $r$ when the sherd is actually a semicircle. Hint: What are the values of $L$ and $h$ in this case?

The surface area of a cylinder is a function of its diameter and height,

$$S=f(d,h)=\pi dh+\frac{\pi }{2}{d}^2$$

1. What is the surface area of a cylindrical oatmeal container with diameter $4$ inches and height $7$ inches? Write your answer with function notation.
2. Write a formula in terms of $d$ for the surface area of a cylinder whose height is equal to its diameter.
3. How does the surface area of the cylinder in part (b) compare with the surface area of a sphere of the same diameter? Sketch both surfaces with the same center.

The Dubois formula is used to estimate the surface area, $S$, of a person in terms of his or her weight and height. A good estimate of surface area is critical to some forms of cancer treatment. In square centimeters, $S$ is given by

$$S=f(w,h)=71.84w^{0.425}{h}^{0.725}$$

where $w$ is in kilograms and $h$ is in centimeters.

1. Use the Dubois formula to estimate the surface area of a person who weighs $60$ kg and is $160$ cm tall.
2. Does surface area increase more rapidly with weight or with height?
3. What percent increase in surface area does a $10\mathrm{\%}$ increase in weight produce?

In the 1970s, McNeill Alexander proposed a relationship between an animal's running speed, $v$, its hip height, $h$, and its stride length, $s$. If stride and hip height are measured in meters, the running speed is given in meters per second by

$$v=f(h,s)=0.78s^{1.67}{h}^{-1.17}$$

1. What is the speed of a racehorse whose hip height is $1.6$ meters and stride length is $7$ meters?
2. A cheetah can run at $33$ meters per second. If its hip height is $0.8$ meters, what is its stride length?
3. A pronghorn antelope has the same stride length as a cheetah and its hip height is $12\mathrm{\%}$ greater than the cheetah's. How does its running speed compare to a cheetah's?

If you walk for exercise, the number of calories, $C$, you burn per mile depends on your walking speed, $s$, and your weight, $w$.

1. Write this fact in function notation.
2. Use the table to evaluate $f(4.5,160)$ and explain its meaning.
3. Solve the inequality $f(s,160)>110$ and explain its meaning.
4. If you weigh $140$ pounds, how fast should you walk to burn the most calories per mile?
5. How can you use the table to calculate how many calories you burn per hour while walking?

The BMI (body mass index) is used to determine whether a person is a healthy weight, overweight, or obese. $B$ is a function of height, $h$, and weight, $w$.

1. Write this fact in function notation.
2. Use the table to evaluate $f(68,150),$ and explain its meaning.
3. A person is deemed overweight if his or her BMI is at least $25$ but less than $30$. Write this fact in function notation. (A person is obese if the BMI is over $30$.)
4. Solve the inequality $f(66,w)<25$ and explain its meaning.
5. Solve the inequality $f(h,200)>30$ and explain its meaning.

An amortization table shows the monthly payments for a loan or mortgage. The table below gives monthly payments for a loan of $$100,000$. The monthly payment, $P$, is a function of the annual interest rate, $r$, and the length of the loan, $t$, in years.

1. You would like to borrow $$100,000$ for $20$ years. What interest rate, to the nearest percent, can you accept if your monthly payments must be no more than $$800$? What interest rate can you accept if the loan is for $30$ years? If $P=f(r,t)$, use function notation to write both of these questions as inequalities.
2. Suppose you borrow $$100,000$ for $10$ years at $10\mathrm{\%}$ interest. Which would cause a greater reduction in your monthly payment: Reducing the interest rate by $5\mathrm{\%}$ or increasing the length of the loan by $5$ years?
3. At a fixed interest rate of $8\mathrm{\%}$, is the monthly payment a linear function of the length of the loan?
4. For a fixed loan period of $25$ years, is the monthly payment a linear function of interest rate?
5. Does a $1\mathrm{\%}$ increase in the interest rate have a greater affect on the monthly payment for a $15$-year loan or a $30$-year loan?

Warmer air can hold more moisture than cooler air. Relative humidity is the amount of moisture in the air, as a fraction of the saturation level at the current temperature. A common measure of humidity is the dewpoint: the temperature at which the current humidity would saturate the air, so that dew forms. The table gives dewpoints, $D$, in ${}^{\circ }$F, as a function of temperature, $T$, and relative humidity, $H$, $D=f(T,H)$.

1. Estimate the relative humidity if the temperature is ${70}^{\circ }$F and the dewpoint is ${40}^{\circ }$F. Write your answer in function notation.
2. Does the dewpoint rise or fall with temperature? (Hint: Consider any column in the table, and notice how dewpoint changes with increasing temperature.)
3. Does the dewpoint rise or fall with humidity? (Hint: Consider any row in the table, and notice how dewpoint changes with increasing humidity.)
4. Suppose that the temperature is ${70}^{\circ }$F and the relative humidity is $70\mathrm{\%}$. Which would cause a larger change in dewpoint: a rise in temperature to ${80}^{\circ }$F or an increase in humidity to $80\mathrm{\%}$?
5. Does dewpoint change more rapidly with temperature when the humidity is low or when the humidity is high?

The table shows automobile fuel efficiency, $E$, as a function of gasoline used, $g$, and miles driven, $m$. Values of $E$ are rounded to tenths where necessary.

1. Choose one row of the table and decide if $E$ varies directly with $m$ or inversely with $m$. Explain your method.
2. Choose one column of the table and decide if $E$ varies directly with $g$ or inversely with $g$. Explain your method.
3. Find the constant of variation and write $E$ as a function of $g$ and $m$. What are the units of $E$?
4. Sketch a graph of $E$ as a function of $m$ for $g=8$, $10$, $12$, and $14$.
5. Sketch a graph of $E$ as a function of $g$ for $m=100$, $200$, $300$, and $400$.

The table shows the productivity, $P$, of a manufacturing plant as a function of the number of items produced, $I$, and the hours of labor used, $w$. Values of $P$ are rounded to tenths where necessary.

1. Choose one row of the table and decide if $P$ varies directly with $w$ or inversely with $w$. Explain your method.
2. Choose one column of the table and decide if $P$ varies directly with $I$ or inversely with $I$. Explain your method.
3. Find the constant of variation and write $P$ as a function of $I$ and $w$. What are the units of $P$?
4. Sketch a graph of $P$ as a function of $w$ for $I=500$, $600$, $800$, and $1000$.
5. Sketch a graph of $P$ as a function of $I$ for $w=100$, $200$, $300$, and $400$.

The water gushing out of a fire hose exerts a backward force that the firefighter must control. This force, called the nozzle reaction, $R$, is a function of the diameter, $d$, of the nozzle and the water pressure, $P$, at the nozzle.

1. Show that $R$ varies directly with $P$.
2. Show that $R$ varies directly with a power of $d$. What is the power?
3. Find the constant of variation and write a formula for $R$ as a function of $d$ and $P$.
4. A typical fire hose has nozzle diameter $2\frac{1}{2}$ inches and nozzle pressure $60$ psi. Use your formula to calculate the nozzle reaction.

The resistance, $R$, of a wire depends on its length, $L$, and diameter, $d$. The table shows the resistance of copper wires of gauges from $10$ to $20$. The diameters of the wires are given in mils, where $1\text{ mil}=0.001\text{ inch}$.

1. Show that $R$ varies directly with $L$.
2. Show that $R$ varies inversely with a power of $d$. What is the power?
3. Find the constant of variation and write a formula for $R$ as a function of $L$ and $d$.
4. Household current uses 12-gauge wire, with a diameter of 0.081 inches. Use your formula to calculate the resistance of a 100-foot length of 12-gauge wire, and verify with the table.

In 2005, Popular Mechanics tested the acceleration from rest for eight sports cars.

1. Acceleration, $a$, varies directly with distance, $d$, and inversely with a power of time, $t$. Use the information in the table to write $a$ as a function of $d$ and $t$.
2. Use your formula to compute the acceleration of the other five cars.
3. Use your formula to plot $a$ against $d$ for $t=1$, $2$, $3$, and $4$ seconds.
4. Use your formula to plot $a$ against $t$ for $d=100$, $200$, $300$, and $400$ feet.

The rate of water flow, $F$, through a fire hose is a function of the diameter, $d$, of the nozzle and the nozzle pressure, $p$.

1. Plot $F$ as a function of $d$ for $p=30$, $40$, $50$, and $60$. Which basic function do your graphs resemble?
2. Plot $F$ as a function of $p$ for $d=1$, $1.25$, $1.5$, and $1.75$. Which basic function do your graphs resemble?
3. Write a formula for $F$ as a function of $d$ and $p$, and find the constant of variation. Check your formula against the table.

Railroad engineers use a transition curve between straight sections of track and bends, which are designed as arcs of circles. The length, $L$, of the transition curve varies directly with the cube of the train's speed, $v$, and inversely with the radius, $R$, of the circular arc.

1. On a section of track where the speed limit is $50$ mph, a circular bend has a radius of $2000$ feet, and the transition curve is $200$ feet long. Find the constant of variation and write a formula for $L$ as a function of $v$ and $R$.
2. If the speed limit is increased by $20\mathrm{\%}$, how is the length of the transition curve affected?
3. If the radius of the bend is increased by $20\mathrm{\%}$, how is the length of the transition curve affected?

The density, $D$, of a planet varies directly with its mass, $M$, and inversely with the cube of its radius, $r$.

1. Use the data for Earth to find the constant of variation, then write a formula for $D$ as a function of $M$ and $r$.
2. Calculate the densities of the other planets.
3. The planets are composed of three broad categories of materials: rocky materials, icy materials (including water), and the materials that dominate the sun, namely hydrogen and helium. The density of rock varies from $3000$ to $8000\text{ kg}/{\text{m}}^3$. Which of the planets could be composed mainly of rock?

Ammonia has many uses in industry and agriculture, including the production of fertilizers. It is produced in the laboratory from nitrogen and hydrogen, but the process requires high pressure and temperature for significant yield. The graph illustrates the relationship. (Source: Hunt and Sykes, 1984)

1. Complete the table showing the yield of ammonia, as a percent of the gas mixture leaving the reactor, at various pressures and temperatures.

   Percent Ammonia
   	Pressure (atmospheres)
   Temperature (${}^{\circ }$C)	$50$	$100$	$150$	$200$	$250$	$300$	$350$	$400$
   $350$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$
   $400$
   $450$
   $500$
   $550$

2. What happens to the yield of ammonia if the pressure is held constant but the temperature is increased beyond ${350}^{\circ }$C?
3. Sketch a graph of the yield of ammonia as a function of temperature when the pressure is $300$ atmospheres.

The graph shows the heat index, which combines air temperature and relative humidity to determine an apparent temperature, or what the temperature actually feels like. (Source: Ahrens, 1998)

1. Complete the table showing the heat index for various combinations of air temperature and relative humidity.

   Heat Index
   	Relative humidity ($\mathrm{\%}$)
   Air temperature (${}^{\circ }$F)	$0$	$20$	$40$	$60$	$80$	$100$
   $80$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$
   $90$
   $100$
   $110$
   $120$

2. Complete the table showing the relative humidity at which the heat index is equal to the actual air temperature.

   Air Temperature (${}^{\circ }$F)	$80$	$90$	$100$	$110$	$120$
   Relative humidity ($\mathrm{\%}$)	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$	$\phantom{0000}$

3. Sketch a graph of the heat index as a function of air temperature if the relative humidity is $70\mathrm{\%}$.