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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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## Chapter 2.7.

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**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 • Goal: model the phenomenon by finding the precise function that relates the bacteria population to the elapsed time**Guidelines for Modeling with Functions**• Expressthe Model inWords. Identify the quantity you want to model and express it, in words, as a function of the other quantities in the problem. • Choose the Variable. Identify all the variables used to express the function in Step 1. Assign a symbol, such as x, to one variable and express the other variables in terms of this symbol.**Guidelines for Modeling with Functions**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)**Example: Modeling the Volume of a Box**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. • Find a function that models the volume of the box in terms of its depth. • Find the volume of the box if the depth is 1.5 in. • For what depth is the volume 90 in3? • For what depth is the volume greater than 60 in3?**Thinking about the problem**Let’s experiment: 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. 3x 5x**Solution**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.**Homework**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.

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