Variation Functions with Non-Integer Exponents

Direct and Indirect Variation with Fractional Exponents (8.5)

POD

If the weight of a box of tissue varies directly with its volume, and a box of 100 cubic inches weighs 8 ounces, what is the volume of a box that weighs 24 ounces?

POD

If the weight of a box of tissue varies directly with its volume, and a box of 100 cubic inches weighs 8 ounces, what is the volume of a box that weighs 24 ounces?

The equation: w = kv8 = k100k = .08w = .08v

POD

If the weight of a box of tissue varies directly with its volume, and a box of 100 cubic inches weighs 8 ounces, what is the volume of a box that weighs 24 ounces?

w = .08v

24 = .08v

v = 300 in3

POD

The weight of this actual box of Kleenex in class is calculated at 11.4 ounces.

What is its volume in cubic inches?

Does this fit our model?

What are we measuring with weight and volume?

Word problems again today

We’ll do the same sort of work today.

The difference is exponents will not be integers like those we’ve used: 1, 2, -1, and -2.

So, we’ll see equations like y = x3/2 or y = 1/x1/3.

The Method

The Method to determine equations for these scenarios is the same as the one we’ve used.

What are those steps again?

Use the Method

The surface area of an egg’s shell is directly proportional to the 2/3 power of the mass of the egg.

If a 60 gr. chicken egg has an area of 28 cm2, find the surface area of a 1600 gr. ostrich egg.

Use the Method

The surface area of an egg’s shell is directly proportional to the 2/3 power of the mass of the egg.

If a 60 gr. chicken egg has an area of 28 cm2, find the surface area of a 1600 gr. ostrich egg.

Variables: x is mass in grams y is surface area in square cm.

k = 1.827 The equation: y = 1.827x2/3

Use the Method

The surface area of an egg’s shell is directly proportional to the 2/3 power of the mass of the egg.

If a 60 gr. chicken egg has an area of 28 cm2, find the surface area of a 1600 gr. ostrich egg.

y = 1.827(1600)2/3

= 250 cm2

Use the Method

The surface area of an egg’s shell is directly proportional to the 2/3 power of the mass of the egg.

If a lizard egg has an area of .6 cm2, what is its mass?

Use the Method

The surface area of an egg’s shell is directly proportional to the 2/3 power of the mass of the egg.

If a lizard egg has an area of .6 cm2, what is its mass? y = 1.827x2/3

.6 = 1.827x2/3

.328 = x2/3

x = .3283/2 = .19 gr.

Use the Method again

In July 1975, Scientific American reported a study that showed the base diameter of a tree’s trunk varies directly with the 3/2 power of its height.

If a young sequoia 5 meters tall has a base diameter of 14.5 centimeters, find the equation to express this model.

Use the Method again

If a young sequoia 5 meters tall has a base diameter of 14.5 centimeters, find the equation to express this model.

In this scenario, the diameter (in cm.) is the dependent variable and height (in meters) is the independent variable.

d = kh3/2 d = 1.30h3/2

14.5 = k53/2

y = 1.30

Use the Method again

What would this tree’s diameter be when it’s 50 meters tall, according to this model?

If the tree is ten times as tall, is the diameter 10 times as long? More? Less? Why?

d = 1.30h3/2

Use the Method again

What would this tree’s diameter be when it’s 50 meters tall, according to this model?

If the tree is ten times as tall, is the diameter 10 times as long? More? Less? Why?

d = 1.30h3/2

d = 1.30(50)3/2

d = 459.6 cm

The diameter is more than 10 times as long, because the exponent for x is greater than 1. The diameter grows at a faster rate than height.

Use the Method again

The General Sherman sequoia in California is the largest known sequoia. Its base is 985 cm. Calculate its height based on this model.

d = 1.30h3/2

Use the Method again

The General Sherman sequoia in California is the largest known sequoia. Its base is 985 cm. Calculate its height based on this model.

d = 1.30h3/2

985 = 1.30h3/2

757.7 = h3/2

h = 757.72/3 = 83.1 m.