You might recall from Physics class that Newton’s 2nd Law of Motion tells us that the net force on an object divided by its mass equals its acceleration, i.e., $a = \frac{F_{net}}{m}$. Let’s look at the case of a person pushing a bobsled and its passengers on the ice.
If the mass of the bobsled and passengers is 300 kg and the person pushes so that there is a net force of 900 Newtons on the bobsled, what is the acceleration of the bobsled (the units are in meters/sec$^{2}$)?
If the net force on the bobsled is tripled, does the acceleration triple?
If, instead, the net force remained at 900 N but the mass of the bobsled and passengers was tripled from part a, what would happen to the acceleration?
What would be the effect on the acceleration of tripling the net force and the mass from part a at the same time?
What would happen to the acceleration if the net force were the same as part a, but the mass of the bobsled and passengers was cut in half?
Finally, what would be the effect on the acceleration if one both doubled the net force from part a and halved the mass at the same time?
As you know, the circumference of a circle and the area of a circle of radius $r$ can be found through the formulas $C = 2\pi r$ and $A = \pi {r^2}$.
If one circle has a radius of 1 meter and another has a radius of 3 meters, how many times bigger is the circumference of the second circle? How many times bigger is its area?
If one circle has a radius of 2.687 meters and another has a radius 3 times as big, how many times bigger is the circumference of the second circle? How many times bigger is its area?
Based on parts a and b, what do you think you can conclude in general? Prove your answer by considering two circles, one with radius $X$ and the other with radius $3X$, and by doing some minor algebra.
The volume of a sphere of radius $r$ can be found by using the formula $V=\frac{4}{3}\pi r^{3}$. If you have two spheres, where the larger sphere has a radius 5 times that of the smaller sphere, how many times bigger must the larger sphere’s volume be? Prove that your answer is correct no matter what the radius of the smaller sphere is.
If one sphere is $\frac{1}{4}$ the radius of another, what fraction of the larger sphere is the volume of the smaller sphere? Again, prove your answer works for any two spheres that have this relationship.
The braking distance of a car is the minimum distance a car going at speed $v$ can stop in (by using the brakes, of course). By using some fundamental physics, one can calculate the braking distance, assuming a car of average weight and a dry road, by the equation $d = .06{v^2}$, where $v$ is in miles per hour and $d$ is in feet.
If Jessup hits the brakes while driving in his jalopy at 20 mph, what is his braking distance? Alternatively, if Tess takes 384 ft. to stop, how fast was she going in her Miata?
Julie says that if she goes 40 mph she will stop in twice the distance than if she was going 20 mph. Is Julie correct? If so, show why. If not, how many times bigger is the 40 mph braking distance than the 20 mph braking distance?
Determine how fast Julie would actually have to go to stop in twice the distance.
How many times faster than 20 mph would she have to go to stop in 16 times as much distance as she would have stopped in at 20 mph?
How many times faster than 20 mph would she have to go to stop in $M$ times as much distance as she would have stopped in at 20 mph?
Given a cube of side length $x$,
What is its volume?
What is its surface area?
If the length of the cube were to change to $5x$, then by what multiple would its volume increase?
If the length of the cube were to change to $5x$, by what multiple would its surface area increase? Why doesn’t the “$6$” in the surface area formula affect your answer?
In the previous problem you examined the volume and surface area of a cube; you found that they were related to the side length $x$ by a cubic $({x^3})$ and a quadratic ($6{x^2}$). Is this true of more complicated shapes? Let’s see.
Say that there is an
irregularly shaped drawing that looks like, say, Big Bird.
Now if you doubled all the dimensions of this drawing (i.e. its height, its width, the distance between Big Bird’s eyes — everything), how would its area change? Why? What if you scaled it up by a factor of 5 instead?
At the Macy’s Day parade last year, there was an enormous balloon of Spongebob Squarepants that was 40 ft high. Toy copies of this balloon were being sold on the parade route and were 3 inches high.
Assuming the other dimensions were proportionally reduced, what is the scale factor of the copy in comparison to the original balloon?
Approximately how many of the toy copies would it take to fill the 40 ft. balloon?
The types of relationships between variables that you have been exploring in the previous problems are special cases of two broad categories: direct proportionality and inverse proportionality.
$y$ is directly proportional to $x$ when $y = kx$, where $x$ and $y$ are variables and $k$ is a constant not equal to 0. Another way of saying this is that $y$ varies directly with $x$. For example, in problem 2, where $C = 2\pi r$, we say that $C$ is directly proportional to $r$, or that $C$ varies directly with $r$.
$y$ is inversely proportional to $x$ when $y = \frac{k}{x}$, where $x$ and $y$ are variables and $k$ is a constant not equal to 0. Another way of saying this is that $y$ varies inversely (or indirectly) with $x$. For example, in problem 1c, where $a = \frac{{900}}{m}$, we say that $a$ is inversely proportional to $m$, or that $a$ varies inversely with $m$.
Answer the following questions by using the definitions of direct and inverse proportionality given above.
If $WZ = 100$, are $W$ and $Z$ directly or inversely proportional, or neither?
If $\frac{C}{T} = 64$, do $C$ and $T$ vary directly or indirectly, or neither?
If $P = Q + 20$, are $P$ and $Q$ directly or inversely proportional, or neither?
Suppose $Z$ is directly proportional to $S$. As $S$ increases, must $Z$ also increase?
Suppose $M$ varies inversely with $R$. As $R$ increases, must $M$ decrease?
Just as it is useful to say that “$A$ varies directly with $B$”, we can apply this language to a wider range of expressions as well. For example, in problem 4, where $A = 6{x^2}$, we can say that $A$ varies directly with ${x^2}$, as $A$ is equal to a constant times ${x^2}$. Alternatively, we could also have said in words that surface area is directly proportional to the square of the side length of the cube.
In the following questions, either write an equation (which may have an unknown constant $k$ in it) based on the sentence given, or write a sentence based on the equation given.
The weight of a human is directly proportional to its volume.
The surface area of a sphere varies directly with the square of its radius.
From problem 3, $d = .06{v^2}$.
The intensity of light varies inversely with the square of the distance from the light source.
$G = \frac{5}{{7{w^3}}}$, where G are “galumphs” and w are “wagdoodles”.
The height of an extra-terrestrial varies directly with the square root of the length of its yellow antennae.
Practice
If a large pizza is 1.5 times the diameter of a small pizza, how much more should it cost, assuming that price is based on area?
The width of a rectangle is 12 ft and the length is 24 ft. By what multiple does the area change if:
The width is doubled and the length is tripled?
The width is halved and the length is quadrupled?
The width is multiplied by some number $A$ and the length is multiplied by some number $B$?
How would your answer to part a be affected if the original width and length of the rectangle were 100 ft and 500 ft?
You have a rectangular fish tank with dimensions 12 ft x 10 ft x 8 ft Which side, if you double it, will give you the biggest volume?
In the study of waves one discovers the relationship between the velocity, wavelength, and frequency of any wave: $v = \lambda f$. For example, if the wavelength is 4 meters and the frequency is 6 cycles/second, then the velocity would be 24 meters/second.
Say that you were looking at a different wave that had twice the wavelength and half the frequency. How many times bigger or smaller would the velocity be than in the example?
Say instead that you were looking at a different wave that had 2.5 times the wavelength and 4 times the frequency. How many times bigger or smaller would the velocity be than in the example?
Finally, say that you were looking at a different wave that had 6 times the velocity and half the wavelength of the initial example. How many times bigger or smaller would the frequency be?
The heat emitted per second (Q) due to radiation from an object is directly proportional to the fourth power of temperature T (in $^\circ K$) of the object. If the temperature of a quantity of metal triples, how much greater will its heat output be per second as a consequence?
Biologists have determined a good approximation of the relationship between the mass of a flying object and its optimum cruising velocity. “Optimum cruising velocity” is defined as the velocity at which an object travels the most distance for a given amount of energy it expends; the easiest analogy is with cars, where the most fuel efficient speed to drive turns out to be 55 mph.
The relationship is $M = \frac{{{V^6}}}{{729000000}}$,
where $V$ is the optimum cruising speed measured in
meters/second and $M$ is the mass of the object measured in
kilograms. This relationship holds for birds, insects, and
even planes!
Source: http://en.wikipedia.org/wiki/File:Allometric_Law_of_Body_Mass_vs_Cruising_Speed_in_Constructal_Theory.JPEG
If one bird’s $V$ is double another’s, how many times more massive is it? What if its $V$ were triple another’s?
If one bird’s $M$ is double another’s, how many times faster is its optimum velocity?
If $R = PS$, are $R$ and $S$ in direct or inverse variation, or neither?
If $R = {P^3}S$, are ${P^3}$ and $S$ in direct or inverse variation, or neither?
If ${R^4} = PS$, are ${R^4}$ and $P$ in direct or inverse variation, or neither?
In each question below, first find the constant of proportionality so that you can then answer the question being asked.
If $y$ is directly proportional to $x$, and $y = 24$ when $x = 6$, what is $y$ when $x = 13$?
If $y$ varies inversely with $x$, and $y = 8$ when $x = 5$, what is $y$ when $x = 80$?
If $y$ varies directly with ${x^2}$, and $y = 63$ when $x = 3$, what is $y$ when $x = 10$?
If $y$ is inversely proportional to ${x^3}$, and $y = 12$ when $x = 2$, what is $y$ when $x = 6$?
A cylinder has a base radius of 4 cm and a height of 10 cm. How will the volume of the cylinder be affected if you scale up the cylinder in all dimensions by a factor of 10? How will the surface area be affected?
A cylinder has a base radius of $R$ cm and a height of $H$ cm. How will the volume of the cylinder be affected if you scale up the cylinder in all dimensions by a factor of 10? How will the surface area be affected?
The area of a standard ghost in Mac-Pan is about 6.5 square units. What is the area of a ghost that has been scaled up by a factor of 2?
A beach ball has 8 times the volume of another beachball. How many times bigger is its circumference?
A beach ball has $K$ times the volume of another beach ball. How many times bigger is its circumference?
Daddy bear, Mommy bear, and Preteen bear are all perfect copies of each other, except that each is .9 times the scale of the previous bear. How many more times than Preteen bear does Daddy bear weigh?
In the ideal gas equation from Chemistry, $PV = nRT$. $R$ is an unchanging constant, but the other 4 quantities can change. Which of these pairs of quantities are in direct variation? Which pairs are in inverse variation?
Write an equation that predicts the number of rotations a wheel makes in a mile, given that you know the diameter of that wheel in inches (and note that 5280 feet = 1 mile).
You have an enormous vat of punch, 10 gallons (160 cups), which you have made for the Kiwanis-Elks Club-Rotary-March of Dimes annual dinner. You’re expecting a big turnout. Write an equation relating the number of people who show up and the number of cups of punch each person can have.
Problems
You have two cans of soup. The large can is 3 times the scale of the small can. For each feature of the cans listed below, write how many times bigger it would get for the large can.
|
Feature |
How many times bigger? |
|
Example: Height of can |
3 times |
|
Amount of soup in can Amount of metal used to make the can How many people you can feed with the soup Number of calories if you eat all the soup Amount of ink needed to print label How many cans it takes to stack up to the ceiling Time it takes to open it with a can opener Number of peas in the soup How much heat rises off the can if it’s hot |
For the following, make an educated guess as to whether the relationship between the given variables is approximately direct, inverse, or neither.
Average income in a town vs. average education level
Time it takes to read a book vs. how interesting the book is to you
Monthly sales of a video game vs. time the video game has been on the shelves
Time it takes to read a book vs. number of pages in the book
The number of hours of daylight vs. day of the year
Birth rate in a country vs. average income per family in that country
Pressure on a tire vs. the amount of air in a tire
Price of a new electronic device vs. number of people who want to buy one but are unable to due to limited availability
Number of diagonals in a polygon vs. number of vertices of the polygon.
It turns out that, to a very good degree of accuracy, the mass of an animal is inversely proportional to the fourth power of its resting heart rate. Given that a typical adult human has a resting heart rate of about 72 beats/minute and a mass of 160 pounds, what would you estimate the heart rate of a mouse (.055 pounds) and dinosaur (70000 pounds) to be?
You have a set of six Russian dolls. Each is a perfect copy of the others, except that each is only .8 times the scale of the previous one. If it took 3 oz of paint to paint the largest doll, how much paint was needed to paint the second-largest doll? How much paint was needed to paint the smallest doll?
The force on a car that is moving in a circle is described by the centripetal force equation: $F = \frac{{m{v^2}}}{r}$, where $m$ is the mass of the car in kilograms, $v$ is the velocity of the car in meters/second, $r$ is the radius of the circle the car is moving in, and $F$ is the force in Newtons. Suppose the force on a car is 12000 Newtons as it travels in a circle.
If the radius of the circle the car is traveling in doubles, what will the new force on the car be?
If the speed of the car doubles, what will the new force on the car be?
If the speed of the car doubles at the same time that the radius of the circle it is moving in doubles, what will the new force on the car be?
If the force on another car is 27000 Newtons as it travels in a circle, and the radius of the circle is suddenly cut to a ninth of the original radius, how much slower would the car have to travel for the force to stay unchanged?
The gravitational attraction of two objects is described by the equation $F = \frac{k}{{{d^2}}}$ (where $d$ is the distance in meters between the centers of the objects, $k$ is a constant based on the masses of the objects, and $F$ is the force of attraction between them, measured in Newtons). For brevity, one often says that Gravity is an “inverse square” force, which just means that the force is inversely proportional to the distance squared.
If initially the force between a person and a planet is ${Q_0}$ and the distance between their centers is ${d_0}$, and then the person moves so that the distance between their centers is doubled, what is the new Force ${Q_1}$ between the objects, in terms of ${Q_0}$?
What if the distance were tripled instead of doubled? Now what would the force be in terms of ${Q_0}$?
Lastly, what if the distance were halved? What would the force be in terms of ${Q_0}$ then?
The force between a typical person and the Earth is about 980 Newtons. Given that the radius of the Earth is 6370 km and that the space shuttle orbits 390 km above the Earth’s surface, determine what the force would be on the same typical person, if they were at the height above the Earth that the space shuttle is when it is in orbit.
According to Dr. Killjoy’s research, how long a couple stays married can be predicted with absolute certainty: the length of the marriage is directly proportional to the number of times per day the couple holds hands and is inversely proportional to the square of the number of nasty looks per day they exchange. As an example, he says that a couple that holds each other’s hands 6 times per day and exchanges 3 nasty looks will be married for 16 years.
If Dr. Killjoy is correct, how long will a couple remain married that holds hands twice a day but exchanges 4 nasty looks a day as well?
The so-called kinetic energy of a bullet is equal to $\frac{1}{2}m{v^2}$, where $m$ is its mass in kg and $v$ is its speed in meters/sec. The higher the kinetic energy of the bullet, the more its ability to cause damage. A typical rifle can shoot a .0042 kg bullet at 965 meters/sec.
If a second rifle can fire bullets that are twice as heavy but only at half the speed, how would the kinetic energy of the bullets of this rifle compare to those of the original rifle?
If a third rifle fires bullets that are half as heavy as the original rifle, but twice the speed, how does the kinetic energy of the bullets of the third rifle compare to those of the original rifle? To the second rifle?
If two rifles use bullets of the same mass, how many times faster must the muzzle velocity (i.e. speed at which the bullet leaves the gun) of one be than the other for it to be able to cause 20 times as much damage?
Two rifles are compared to see how far they can penetrate in to a target (this turns out to be directly proportional to the damage it can inflict). One has bullets with one-third the mass of the other, but its bullets still penetrate 4 times as far in to the target as the other’s. How many times faster is the muzzle velocity of the rifle with the lighter bullets?
A beam that is supported horizontally at one end and free at the other can be deflected an amount $D$ at the free end according to the equation $D=\frac{FL^3}{3K}$, where $F$ is the force applied, $L$ is the length of the beam, and $K$ is a constant based on the shape and stiffness of the beam. Assume $K$ does not change in all parts of this problem.
If the beam is currently being deflected an amount ${D_1}$, how much more force would you have to apply to triple the deflection?
If the beam was half the length it currently is, how much harder would one have to push to yield the same deflection ${D_1}$?
The number of alligators observed in a Martian swamp increases according to the equation $A = k \cdot {2^t}$, where $t$ is the number of days after the first alligators were observed, and $k$ is a constant.
If there are 400 alligators in the swamp in the initial observation, how many alligators are there after 1 day? 2 days? 3 days? 6 days?
There are ${A_1}$ alligators at time $t = 13$. At what time will there be double this number of alligators?
If there are ${A_1}$ alligators
after time ${t_1}$, how many alligators will there be
after time $2{t_1}?$ After time $3{t_1}$? After time
The Richter scale indicates the intensity (I) of an earthquake by relating it to the amplitude (A) of waves at its epicenter in the following way: $I = {\log _{10}}A$. (Units are ignored in this problem for simplicity.)
What amplitude of waves would give an intensity of 1?
How many times bigger than in part a would the amplitude of the waves have to get for the intensity to be 2?
How many times bigger than in part a would the amplitude of the waves have to get for the intensity to be 6?
If the amplitude of the waves were double that of the waves in part a, what would the intensity be?
If the amplitude of waves at some time is ${A_1}$, and at some later time is $2{A_1}$, how did the intensity change between the two times?
The total cost $C$ of producing $N$ cars from scratch, where one has to first build the factory and the assembly line before producing a single car, follows the equation $C = 6000N + 12,000,000$. The factory can produce tens of thousands of cars in a single year. What affect does doubling the number of cars produced, from ${N_1}$ to $2{N_1}$, have on the total cost $C$? Justify your answer with a specific example or two.
An ideal gas obeys the equation $PV = nRT$. If the pressure is ${P_1}$ at some point in time, what will the new pressure be if, simultaneously, $V$ is halved, $T$ is tripled, $R$ remains the same, and $n$ is doubled?
The pull of a magnet turns out to be inversely proportional to the cube of an object’s distance from the magnet; that is, $Pull = \frac{k}{{{d^3}}}$, where $k$ is some constant.
Sarah, who is 24 feet from a magnet and holding a piece of iron, feels 512 times less Pull from the magnet than Isaac does, even though he is holding an identical piece of iron. If the magnet, Isaac and Sarah are all in a straight line in that order, how far apart are Sarah and Isaac?
“The Attack of the 50 Foot Woman!” was a campy “sci-fi” thriller from the 1950’s, about a woman who, due to an alien encounter, grows to be a 50 ft giant, wreaking havoc on the metropolis she lives in. Could humans prosper at such heights if, say, our DNA gave our body permission to “keep growing”? Let’s check it out.
Here are two important facts: 1) The weight of any animal is directly proportional to its volume, and 2) The strength of a bone is directly proportional to the cross-sectional area of the bone. Write equations that represent these relationships.
A 50 ft woman is 10 times taller
than a
5 ft woman, indicating that one could roughly think
of her as a 5 ft woman scaled up by a factor of 10 in all
dimensions. How many times more would the 50 ft woman weigh
than the
5 ft woman?
Since all dimensions are scaled by a factor of 10, how many times stronger would the 50ft. woman’s bones be than the 5 ft woman’s?
Bones in the leg break if they are under too much strain/pressure, that is, if there is too much weight above pressing down on the bones. Using your answers to parts b and c, determine how the pressure (i.e. $\frac{{{\rm{Weight}}}}{{{\rm{Cross - sectional \ area}}}}$ of the leg bones) on the 50ft woman’s leg bones compares to the 5 ft woman’s.
Thus explain why animals that are small would have difficulty if they grew too large, even given that their DNA permitted it. That is, why aren’t there 50 ft women?
A spherical balloon becomes bigger and bigger as it is filled with more and more air, although it always retains its spherical shape. It starts out at $t = 0$ as a balloon with a volume of 10 cubic centimeters. With every second, 30 cubic centimeters more of air is pumped into the balloon.
Write an equation for the volume of the balloon as a function of time.
Write an equation for the volume of the balloon as a function of its radius.
Using parts a and b, write an equation relating time and radius of the balloon.
Now, using algebra, find radius as a function of time.
Examine the following crazy formula: $y = \frac{{a{b^2}{c^3}\sqrt d }}{{e{f^2}{g^3}\sqrt h }}$. Initially, $y$ equals some unknown number based on current values of $a$, $b$, $c$, $d$, $e$, $f$, $g$, and $h$.
What happens to $y$ if…
$b$ is tripled?
$g$ is halved?
$d$ is quadrupled?
$h$ is multiplied by 16?
$c$ is doubled AND $e$ is doubled?
$d$ is multiplied by 9 AND $g$ is tripled?
Exploring in Depth
The speed on a given planet with which an object must be
propelled from its surface straight up so that it would never
come down is called its escape velocity,
determined by the equation
The escape velocity on the surface of the Earth is 11,200 meters/second (you can see why we need rockets to achieve such velocities). If the Earth were twice as massive but the same size, what would the escape velocity then be? How many times more massive would the earth have to be for the escape velocity to be twice what it is now?
The moon has a radius that is .273 times the Earth’s. It has a mass that is .0123 times the Earth’s. What is the escape velocity on the surface of the moon?
Don’t use a calculator for this problem.
Evaluate: ${\log _3}45 + {\log _3}2 - {\log _3}10$
Solve for $x$: ${x^2} = 10 - 3x$
Evaluate: $\frac{{{{({2^3}{3^2})}^5}}}{{{4^7}{3^8}}}$
Expand: ${(2x - 3)^3}$
Simplify: $\sqrt{6} \cdot \sqrt{35} \cdot \sqrt{21}$
Kleiber’s Law for Animal Metabolic Rates is as follows: ${q^4} = k{m^3}$, where $m$ is the mass of the animal in kilograms, $k$ is a constant, and $q$ is the metabolic rate.
If one animal is twice the mass of another, how many times bigger is its metabolic rate?
If one animal has 3 times the metabolic rate of another, how many times bigger is its mass?
If one animal has half the metabolic rate of another, how many times less massive is it?
The pressure exerted by an object A on another object B is directly proportional to A’s mass and inversely proportional to the area of that part of A that rests upon B. (Remember, too, that the mass of an object is directly proportional to the cube of its height, as long as the other two dimensions change proportionally with height.)
Suppose A is a brick and C is another brick that is similar (in the mathematical sense) to A and made of the same material. Both bricks are resting on a table.
If C is two times as big (in each direction) as A, then how much pressure does C exert on the table in comparison to A?
If C’s surface area is twice as big as that of A, then how much pressure does C exert on the table in comparison to A?
Physics tells us that an object falling in a vacuum will, because of gravity, get faster and faster — there’s no limit on its top speed as it falls. In practice, though, the Earth’s atmosphere causes drag. Because of this, a falling object (like, say a skydiver) will initially speed up, but eventually will reach its terminal velocity — its constant top speed, where the drag force perfectly cancels out gravity, so that there’s no more acceleration.
According to Wikipedia, the square of the terminal velocity of an object is directly proportional to its mass and inversely proportional to its cross-sectional area.
Suppose you dropped two spherical balls, both made of identical material, from an airplane.
If the radius of the larger of the two balls is sixteen times the radius of the smaller, then which ball has a higher terminal velocity? How many times faster is it than the other one?
If, instead, the larger of the two balls weighs twice as much as the smaller, then which ball has the higher terminal velocity? How many times faster is it than the other one?
Now suppose that two balls, this time made of different materials, turn out to have the same terminal velocity. If the radius of the larger ball is twice the radius of the smaller, then which ball is made of a denser material? How do their densities compare? (Note: density = mass/volume.)