During a thunderstorm, the distance between you and the storm varies directly as the time interval between the lightening and thunder.
Suppose thunder from a storm 5400 ft away takes 5 seconds reach you.
Determine the constant of proportionality and write the variation equation for the model.
Sketch the graph. What does the k represent?
If the time interval between the lightening and thunder is 8 sec. How far away is the storm?
y varies inversely as x
y is inversely proportional to x
Formula (Equation to use)
Boyle’s Law – When a sample of gas is compressed at a k onstant temperature the pressure of the gas is inversely proportional to the volume of the gas.
P – pressure
v – volume
k – constant of proportionality
Suppose the pressure of a sample of air that occupies 0.106 m ³ @ 25ºC is 50 kPa. Find the constant of proportionality and write the equation that expresses the inverse proportionality
If the sample expands to a volume of .3m³, find the new pressure
Used when a quantity depends on more than one other quantity. It depends on them jointly.
z varies jointly as x and y
z is jointly proportional to x and y
z is proportional to x and inversely proportional to y
Newton’s Law of Gravitation – Two objects with masses m 1 and m 2 attract each other with a force F that is jointly proportional to their masses and inversely proportional to the square of the distance r between the objects. Express Newton’s Law of Gravitation as an equation.
Examples of Variation
Write an equation that expresses the statement:
R varies directly as t .
v is inversely proportional to z .
y is proportional to s and inversely proportional to t .
R is proportional to j and inversely proportional to the squares of s and t .
Express the statement as a formula. Use the given information to find the constant of proportionality
y is directly proportional to x . If x = 4, then y = 72.
M varies directly as x and inversely as y . If x = 2 and y = 6, then M = 5.
s is inversely proportional to the square root of t . If s = 100, then t = 25.
Hooke’s Law states that the force F needed to keep a spring stretched x units beyond its natural length is directly proportional to x . Here the constant of proportionality is called the spring constant.
Write Hooke’s Law as an equation.
If a spring has a natural length of 10 cm and a force of 40 N is required to maintain the spring stretched to a length of 15 cm, find the spring constant.
What force is needed to keep the spring stretched to a length of 14 cm?
The resistance R of wire varies directly as its length L and inversely as the square of its diameter d .
Write an equation that expresses this joint variation.
Find the constant of proportionality if a wire 1.2 m long and 0.005 m in diameter has a resistance of 140 ohms.
Find the resistance of a wire made of the same material that is 3 m long and has a diameter of 0.008 m.
Modeling Quadratic & Cubic Functions
Define the variable
Find the equation (model)
Answer the question(s) asked
A breakfast cereal company manufactures boxes to package their product. The prototype box has the following shape: Its width is three times its depth and its height is five times its depth. Find a function that models the volume of the box in terms of its depth.
A hockey team plays in an arena with a seating capacity of 15,000 spectators. With the ticket price set at $14, average attendance at recent games has been 9500. A market survey indicates that for each dollar the ticket price is lowered, the average attendance increases by 1000. What price maximizes revenue from ticket sales, and what is the maximum revenue?
A manufacturer makes a metal can that holds 1 L(liter) of oil. What radius minimizes the amount of metal in the can?
A gardener has 140 feet of fencing for her rectangular vegetable garden. Find the dimensions of the biggest area she can fence.