Stewart Precalc 6e Focus on Modeling (Ch 2): Modeling with Functions

26 questions

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Stewart Precalc 6e Focus on Modeling (Ch 2): Modeling with Functions 0/26
1 Area - rectangular building lot · Level 1
A rectangular building lot is three times as long as it is wide. Find a function that models its area \(A\) in terms of its width \(w\).
2 Area - poster · Level 1
A poster is \(10\) inches longer than it is wide. Find a function that models its area \(A\) in terms of its width \(w\).
3 Volume - rectangular box with square base · Level 2
A rectangular box has a square base. Its height is half the width of the base. Find a function that models its volume \(V\) in terms of its width \(w\).
4 Volume - cylinder · Level 2
The height of a cylinder is four times its radius. Find a function that models the volume \(V\) of the cylinder in terms of its radius \(r\).
5 Area - rectangle with fixed perimeter · Level 2
A rectangle has a perimeter of \(20\) ft. Find a function that models its area \(A\) in terms of the length \(x\) of one of its sides.
6 Perimeter - rectangle with fixed area · Level 2
A rectangle has an area of \(16\) m². Find a function that models its perimeter \(P\) in terms of the length \(x\) of one of its sides.
7 Area - equilateral triangle · Level 2
Find a function that models the area \(A\) of an equilateral triangle in terms of the length \(x\) of one of its sides.
8 Surface area - cube · Level 2
Find a function that models the surface area \(S\) of a cube in terms of its volume \(V\).
9 Radius - circle in terms of area · Level 1
Find a function that models the radius \(r\) of a circle in terms of its area \(A\).
10 Area - circle in terms of circumference · Level 2
Find a function that models the area \(A\) of a circle in terms of its circumference \(C\).
11 Surface area - box with fixed volume · Level 3
A rectangular box with a volume of \(60\) ft³ has a square base. Find a function that models its surface area \(S\) in terms of the length \(x\) of one side of its base.
12 Length - shadow under street lamp · Level 3
A woman \(5\) ft tall is standing near a street lamp that is \(12\) ft tall, as shown in the figure. Find a function that models the length \(L\) of her shadow in terms of her distance \(d\) from the base of the lamp.
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13 Distance - two ships · Level 2
Two ships leave port at the same time. One sails south at \(15\) mi/h, and the other sails east at \(20\) mi/h. Find a function that models the distance \(D\) between the ships in terms of the time \(t\) (in hours) elapsed since their departure.
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14 Product - two numbers with fixed sum · Level 1
The sum of two positive numbers is \(60\). Find a function that models their product \(P\) in terms of \(x\), one of the numbers.
15 Area - isosceles triangle · Level 3
An isosceles triangle has a perimeter of \(8\) cm. Find a function that models its area \(A\) in terms of the length of its base \(b\).
16 Perimeter - right triangle · Level 2
A right triangle has one leg twice as long as the other. Find a function that models its perimeter \(P\) in terms of the length \(x\) of the shorter leg.
17 Area - rectangle inscribed in a semicircle · Level 3
A rectangle is inscribed in a semicircle of radius \(10\), as shown in the figure. Find a function that models the area \(A\) of the rectangle in terms of its height \(h\).
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18 Height - cone with fixed volume · Level 2
The volume of a cone is \(100\) in³. Find a function that models the height \(h\) of the cone in terms of its radius \(r\).
19 Optimization - maximizing a product · Level 3
Consider the problem: Find two numbers whose sum is \(19\) and whose product is as large as possible.
(a) Experiment by making a table showing the product of different pairs of numbers that add up to \(19\), and estimate the answer.
(b) Find a function that models the product in terms of one of the two numbers.
(c) Use your model to solve the problem, and compare with your answer to part (a).

Enter your answer directly below each part above.

20 Optimization - minimizing a sum of squares · Level 3
Find two positive numbers whose sum is \(100\) and the sum of whose squares is a minimum.
21 Optimization - fencing a field along a river · Level 3
Consider the problem: A farmer has \(2400\) ft of fencing and wants to fence off a rectangular field that borders a straight river. He does not need a fence along the river (see the figure). What are the dimensions of the field of largest area that he can fence?
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(a) Experiment by drawing several diagrams and estimate the dimensions of the largest possible field.
(b) Find a function that models the area of the field in terms of one of its sides.
(c) Use your model to solve the problem, and compare with your answer to part (a).

Enter your answer directly below each part above.

22 Optimization - dividing a pen · Level 3
A rancher with \(750\) ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle (see the figure).
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(a) Find a function that models the total area of the four pens.
(b) Find the largest possible total area of the four pens.

Enter your answer directly below each part above.

23 Optimization - minimum cost fencing along a road · Level 4
A property owner wants to fence a garden plot adjacent to a road, as shown in the figure. The fencing next to the road must be sturdier and costs \$5 per foot, but the other fencing costs just \$3 per foot. The garden is to have an area of \(1200\) ft².
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(a) Find a function that models the cost of fencing the garden.
(b) Find the garden dimensions that minimize the cost of fencing.
(c) If the owner has at most \$600 to spend on fencing, find the range of lengths he can fence along the road.

Enter your answer directly below each part above.

24 Example - Modeling the Volume of a Box · Level 2
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.
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(a) Find a function that models the volume of the box in terms of its depth.
(b) Find the volume of the box if the depth is \(1.5\) in.
(c) For what depth is the volume \(90\) cubic inches?
(d) For what depth is the volume greater than \(60\) cubic inches?

Enter your answer directly below each part above.

25 Example - Fencing a Garden · Level 2
A gardener has \(140\) feet of fencing to fence in a rectangular vegetable garden.
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(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 \(825\) square feet?
(c) Can she fence a garden with area \(1250\) square feet?
(d) Find the dimensions of the largest area she can fence.

Enter your answer directly below each part above.

26 Example - Minimizing the metal in a can · Level 3
A manufacturer makes a metal can that holds \(1\) L (liter) of oil. What radius minimizes the amount of metal in the can?
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