1.7 Modeling with Functions
Helpful Geometry Formulas:
Area of a rectangle: _________
Perimeter of a rectangle: __________
Area of a circle: _________
Circumference of a circle: ________
Area of a triangle: _________
Area of a trapezoid: _________
Volume of a cube: _________
Volume of a right circular cylinder: _________
Volume of a right circular cone: __________
Volume of a sphere: __________
Surface area of a sphere: _______
Surface area of a right circular cylinder:
_____________________
Aug 6­3:00 PM
Examples Write a mathematical expression for the quantity
described verbally.
1.) 8 more than 6% of a number
2.) A number x decreased by 7 and
then quadrupled
3.) A salary after a 4% decrease
4.) A salary after a 4% increase
5.) The revenue when each item
sells for $4.75.
6.) The total cost is $35,000 increased
by 5% plus $10.75 for each item
produced.
Aug 6­3:06 PM
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Example Write the area A of a circle as a function of its:
a.) radius r
b.) diameter d
c.) circumference C
Aug 6­3:15 PM
Examples Write the specified quantity as a function of the
specified variable. Draw a picture!
1.) A circle is inscribed in a square. Write a function for the area
of the circle in terms of the side of the square.
2.) A cube is contained in a sphere. Write a function for the
surface area of the sphere in terms of the side s of the cube.
Aug 6­3:17 PM
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3.) A right circular cylinder has a height of 10 cm and base with
radius r. Write functions for the volume V in terms of the:
a.) radius r
b.) diameter d
c.) circumference C
Aug 6­3:22 PM
Examples Write an equation and solve.
1.) A tabletop is rectangular. Its length is 2 feet less
than twice its width. The perimeter is 16 feet. What
are the dimensions of the tabletop?
Benzene
2.) A beaker contains 400 mL of a 15% benzene solution.
How much 40% solution must be added to produce a 35%
benzene solution?
Aug 6­3:24 PM
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A Maximum Value Problem
A square of side x inches is cut out of each corner of an 8 in. by
15 in. piece of cardboard and the sides are folded up to form an
open top box.
a.) Write the volume V of the box as a function of x.
b.) Find the domain V as a function of x.
c.) Graph V as a function of x over the domain. What dimensions
yield the maximum volume?
Aug 6­3:31 PM
*Turn to page 157 for regression types.
Examples Which type of regression is likely to give the most
accurate model for the scatter plot?
1.)
3.)
2.)
4.)
Aug 6­3:43 PM
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Sep 13­8:27 AM
Your graphing calculator will display the r value, which is called
the ____________ ___________.
It will also display the r2 value, which is called the __________
___ ______________.
In either case, a useful rule of thumb is: the closer the absolute
value of this number is to 1, the better the curve fits the data.
Turn to page 156 and let's look at Example 6.
Aug 6­7:04 PM
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Example The table below lists the number of Americans (in thousands) who
are expected to be over 100 years old for selected years.
a.) Make a scatter plot of the data.
b.) Find the quadratic regression equation.
c.) Does the value of r2 suggest that the quadratic model is appropriate?
d.) Find the linear regression equation.
e.) Does the value of r suggest that the linear model is appropriate?
f.) Use both curves to predict the number of Americans who were over 100
in 2008. Compare predictions.
Aug 6­7:19 PM
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