Precalculus – Spring 2005. Chapter 2.7. Modeling With Functions. Modeling. Modeling = a function that describes the dependence of one quantity on another Example : number of bacteria in a certain culture increases with time
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3. Set up the Model. Express the function in the language of algebra by writing it as a function of the single variable chosen in Step 2.
4. Use the Model. Use the function to answer the question posed in the problem. (To find a maximum or a minimum, use the algebraic or graphical methods learned)
A breakfast cereal company manufactures boxes to package their product. For aesthetic reasons, the box must have the following proportions: Its width is 3 times its depth and its height is 5 times its depth.
If the depth is 1 in., then the width is 3 in. and the height is 5 in.
So in this case the volume is V = 1*3*5= =15in3.
Notice: the greater the depth the greater the volume.
Step 1. Volume = depth * width * height
Step 2. x = depth of the box
width = 3x
height = 5x
Step 3. V(x) = x*3x*5x = 15 x3
Step 4. (b) V(1.5) = 15(1.5)3 = 50.625 in3
(c) V(x) = 90, so 15 x3 = 90, so x = 1.82 in
(d) V(x) > 60, so 15x3 > 60, so x > 1.59 in.
A gardener has 140 feet of fencing to fence in a rectangular vegetable garden.
(a) Find a function that models the area of the garden she can fence.
(b) For what range of widths is the area greater than 825 ft2?
(c) Can she fence a garden with area 1250 ft2?
(d) Find the dimensions of the largest area she can fence.