Combined variation functions - a function with more than one independent variable.

Joint Variation - a type of combined variation where the independent variables are multiplied.

z depends on x and y z depends jointly on x and y

Composite functions - functions in which one variable depends on a second, and that second variable depends on a third.

z depends on y, and y depends on x.

If you have two dependent variables each dependent on the same independent variable, you just have two separate variation functions.

y depends on x and z depends on x

Functions with more than one variable

1. y and z both depend on x

y = k1xn and z = k2xn

2. z depends on x and y (Combined Variation Functions)

z = kxnym (Joint Variation Function)

z = k(xn + yn) (Non-Joint Combined Variation Function)

z = kxn/yn (Non-Joint Combined Variation Function)

Note: Almost always, when you translate word problems from English into math, "and" means "plus" or "added to". But in joint variation, "and" just means "both of these are together on the same side of the fraction”, and you multiply. If you are supposed to add two variables, they'll say "varies as the sum of x and y.

4. z depends on y, and y depends on x (Composite Function)

z = k1yn and y = k2xn

A bridge is held up by columns 20 inches in diameter and of varying lengths. From a Strength of Materials text you find that as you put more and more weight on a column, it can collapse either by buckling of by crushing. The load which will buckle a column is directly proportional to the fourth power of it diameter, and inversely proportional to the square of its length. The load which will crush a column varies directly with the square of its diameter, and is independent of it length.

a. Write an equation expressing the number of tons which will buckle a column in terms of the length and diameter of the column. Then write the

particular equation if a 2” diameter column 3’ long will buckle under a load of 4 tons.

b. Write another general equation expressing the number of tons that will crush a column in terms of its diameter. Then write the particular

equation if a 2” diameter column is crushed by a load of 5 tons.

c. Calculate the number of tons needed to buckle 20” diameter bridge columns which are 20’, 30’, and 50’ long.

d. Calculate the number of tons that will crush a 20” diameter column from the bridge that is 20’ long.

e. Draw a graph of the number of tons a bridge column will support versus length of column in the domain from 0 through 50 feet. Remember that the number of tons is either the buckling or crushing load, whichever is less.

f. What conclusions can you make about the way in which long columns collapse and the way in which short columns collapse?

g. At approximately what length will a bridge collapse by both buckling and crushing? Explain.

When a tornado moves over a house, the sudden decrease in air pressure creates a force that can lift the roof. The force exerted on the roof equals the pressure times the area of the roof. But, if the joint between the roof and the walls is strong enough, the roof will not be lifted off. The force a roof will withstand varies directly with the length of the roof-to-walls joint (i.e. the perimeter of the room.) The area of the room varies directly with the square of the perimeter of the room.

a. Write the three general equations expressing the information.

b. Write a new equation to show that the force a roof will withstand is directly proportional to the 1/2 power of the roofs area.

c. The critical pressure for a room is the pressure at which the force exerted by the tornado just equals the force of the roof will withstand. Combine the necessary equations to show how that the critical pressure varies inversely with the square root of the roofs area.

Some physical and behavioral characteristics of mammals can be explained by comparing the way their surface area and mass are related. Assume that

i. the mass of an animal is directly proportional to the cube of its length.

ii. the area of its skin is directly proportional to the square of its length.

iii. the rate at which it loses heat to its surroundings is directly proportional to its skin area, and

iv. the amount of food it must eat per day is directly proportional to the rate of heat loss.

a. Write the general equations for each of the four functions above.

b. By appropriate substitutions, derive a general equation expressing amount of food per day in terms of the animal’s mass. Tell how food consumption varies with mass.

c. Divide both members of the equation in part b by the animal’s mass to get an equation for the fraction of its mass an animal consumes per day. Tell how this fraction varies with mass.

d. The smallest mammal, a shrew, eats 3 times its mass each day. A shrew has a mass of about 2 grams. Find the proportionality constant for part c.

e. What fraction of its mass would a 6000 kilogram elephant eat each day?