Direct Variation and Proportion Objective: Write and apply direct variation and proportion problems.
Section 2.8 Modeling Using Variation. Direct Variation.
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Transcript of Section 2.8 Modeling Using Variation. Direct Variation.
![Page 1: Section 2.8 Modeling Using Variation. Direct Variation.](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649f435503460f94c63aa1/html5/thumbnails/1.jpg)
Section 2.8Modeling Using Variation
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Direct Variation
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Example
The volume of a sphere varies directly as the cube of the radius. If the volume of a sphere is 523.6 cubic inches when the radius is 5 inches, what is the radius when the volume is 33.5 cubic inches.
r
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Example
The pressure, P, of a gas in a spray container varies inversely as the volume, V, of the container. If the pressure is 6 pounds per square inch when the volume is 4 cubic inches, what is the volume when the pressure is down to 3 pounds per square inch?
![Page 8: Section 2.8 Modeling Using Variation. Direct Variation.](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649f435503460f94c63aa1/html5/thumbnails/8.jpg)
Combined Variation
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We can combine two variation equations:
Which says that the monthly sales, S, varies directly
as the advertising budget A.
and
Which says that the monthly sales, S, vary inversely
as the price of
S kA
kS
P
the product, P.
To get
![Page 10: Section 2.8 Modeling Using Variation. Direct Variation.](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649f435503460f94c63aa1/html5/thumbnails/10.jpg)
Example
The TIXY calculator leasing company has determined that leases L, vary directly as its advertising budget and inversely as the price/month of the calculator rentals. When the TIXY company spent $500 on advertising on the internet and charge $30/month for the rentals, their monthly rental income was $4000. Write an equation of variation that describes this situation. Determine the monthly leases if the amount of advertising is increased to $2000.
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Joint Variation
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Joint variation is a variation in which a variable
varies directly as the product of two or more other
variables. Thus, the equation y=kxz is read
"y varies jointly as x and z."
Joint variation plays
1 22
a critical role
in Isaac Newton's formula for gravitation:
The formula states that the force of gravitation, F, between
two bodies varies jointly as the pro
mmF G
d
1
2
duct of their masses, m
and m , and inversely as the square of the distance between
them d. G is the gravitational constant. The formula indicates
that gravitational force exists between any two objects in the
universe, increasing as the distance between the bodies decreases.
dm1
m2
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Example
The volume of a model square based pyramid, V, various jointly as its height, h, and the square of its side, s ,of the square base. A model pyramid that has a side of the square base that is 6 inches, and the height is 10 inches, has a volume of 120 cubic inches. Find the volume of a pyramid with a height of 9 inches and a square base of 5 inches.
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(a)
(b)
(c)
(d)
The surface area of a sphere is directly proportional
to the square of its radius. If the surface area of a sphere
is 201 square inches when the radius is 4 inches, find the
surface area when the radius is 5 inches?
64 square inches
314 square inches
100 square inches
12 square inches
r
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(a)
(b)
(c)
(d)
The pressure, p, of a gas in a container varies
inversely as the volume, V. If the pressure is
12 pounds per square inch when the volume is
5 cubic inches, what is the volume when the pressure
is 10 pounds per square inch.
4 cubic inches
8 cubic inches
7 cubic inches
6 cubic inches