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Question : Find the dimensions that produce the maximum floor area for a one-story house : 2151724

Graph the function on the indicated domain, and use the capabilities of your calculator to find the location and value of the indicated absolute extremum.

20) f(x) = (x - 8)(x + 3); [0, ∞)

Minimum

A) -30.25 at x = 2.5

B) -30.21 at x = 2.7

C) -30.16 at x = 2.2

D) -29.89 at x = 1.9

21) f(x) = x(x - 9)^{2/3}; (-∞, ∞)

Minimum

A) 4.2 at x = 9.3

B) 5.4 at x = 8.5

C) 0 at x = 9.0

D) No absolute minimum

22) f(x) = (x^{3} - 4x + 1/x^{4} + x^{2} + 5); [-4, 1]

Maximum

A) 0.6 at x = -0.8

B) -0.3 at x =0.9

C) -0.2 at x = 0.6

D) -0.2 at x = -3.9

Find the absolute extrema if they exist as well as where they occur.

23) f(x) = -3x^{4} + 20x^{3} - 36x^{2} + 5

A) Absolute maximum of -14 at x = 1; no absolute minima

B) No absolute extrema

C) Absolute maximum of 5 at x = 0; no absolute minima

D) Absolute maximum of -27 at x = 2; no absolute minima

24) f(x) = (x - 1/x^{2} + 5x + 10)

A) Absolute minimum of - 1 at x = -3; no absolute maxima

B) No absolute extrema

C) Absolute minimum of - (5/6) at x = -4; absolute maximum of (1/15) at x = 5

D) Absolute minimum of - 1 at x = -3; absolute maximum of (1/15) at x = 5

25) f(x) = 2 - x - 4/x, x > 0

A) Absolute maximum of 6 at x = -2; absolute minimum of 2 at x = 0

B) Absolute minimum of -2 at x = 2; no absolute maximum

C) Absolute maximum of -3 at x = 1; no absolute minimum

D) Absolute maximum of -2 at x = 2; no absolute minimum

26) f(x) = 10xlnx

A) No absolute minimum or maximum

B) Absolute maximum of 2,202,646.58 at x = e^{-10}; no absolute minimum

C) Absolute minimum of 3.6788 at x = e^{-1}; no absolute maximum

D) Absolute minimum of 0 at x = -10; no absolute maximum

Solve the problem.

27) P(x) = -x^{3} + (27/2)x^{2} - 60x + 100, x ≥ 5 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 4 hundred thousand

B) 5 hundred thousand

C) 4.5 hundred thousand

D) 5.5 hundred thousand

28) P(x) = -x^{3} + 12x^{2} - 36x + 400, x ≥ 3 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 2 hundred thousand

B) 3 hundred thousand

C) 6 hundred thousand

D) 7 hundred thousand

29) P(x) = -x^{3} + 24x^{2} - 144x + 50, x ≥ 2 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 4 hundred thousand

B) 10 hundred thousand

C) 12 hundred thousand

D) 2 hundred thousand

30) P(x) = -x^{3} + 12x^{2} - 21x + 100, x ≥ 4 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 7 hundred thousand

B) 4 hundred thousand

C) 13 hundred thousand

D) 10 hundred thousand

31) P(x) = -x^{3} + 15x^{2} - 48x + 450, x ≥ 3 is an approximation to the total profit (in thousands of dollars) from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must be sold to maximize profit.

A) 8 hundred thousand

B) 5 hundred thousand

C) 3 hundred thousand

D) 10 hundred thousand

32) The graph gives the profit P(x) as a function of production level. Use graphical optimization to estimate the production level that gives the maximum profit per item produced.

A) 5 units

B) 4 units

C) 3 units

D) 6 units

33) In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year during the years 1990 - 2000 are given in the figure. Consider the closed interval [1990, 2000].

A(1990, 170) D(1993, 282) G(1996, 188) L(1999, 238)

B(1991, 204) E(1994, 211) H(1997, 258) M(2000, 270)

C(1992, 255) F(1995, 143) K(1998, 247)

Give all relative maxima and minima on the interval and the years when they occur.

A) Relative maxima of 282 in 1993, 258 in 1997, 270 in 2000

Relative minima of 170 in 1990, 143 in 1995, 238 in 1999

B) Relative maxima of 282 in 1993 and 258 in 1997

Relative minima of 143 in 1995 and 238 in 1999

C) Relative maxima of 282 in 1993 and 258 in 1997

Relative minima of 170 in 1990, 143 in 1995, 238 in 1999

D) Relative maxima of 282 in 1993, 258 in 1997, 270 in 2000

Relative minima of 143 in 1995 and 238 in 1999

34) In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year during the years 1990 - 2000 are given in the figure. Consider the closed interval [1990, 2000].

A(1990, 171) D(1993, 280) G(1996, 188) L(1999, 236)

B(1991, 204) E(1994, 211) H(1997, 257) M(2000, 271)

C(1992, 255) F(1995, 142) K(1998, 247)

Give the absolute maximum and minimum on the interval and the years when they occur.

A) Absolute maximum of 257 in 1997

Absolute minimum of 171 in 1990

B) Absolute maximum of 280 in 1993

Absolute minimum of 171 in 1990

C) Absolute maximum of 280 in 1993

Absolute minimum of 142 in 1995

D) Absolute maximum of 271 in 2000

Absolute minimum of 142 in 1995

35) S(x) = -x^{3} - 9x^{2} + 165x + 1300, 5 ≤ x ≤ 20 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon.

A) 20°C

B) 6°C

C) 5°C

D) 19°C

36) S(x) = -x^{3} - 3x^{2} + 72x + 900, x ≥ 2 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon.

A) 6°C

B) 8°C

C) 4°C

D) 2°C

37) Researchers have discovered that by controlling both the temperature and the relative humidity in a building, the growth of a certain fungus can be limited. The relationship between temperature and relative humidity, which limits growth, can be described by

R(T) = -0.00005T^{3} + 0.318T^{2} - 1.6572T + 97.086,

0 ≤ T ≤ 46,

where R(T) is the relative humidity (in %) and T is the temperature (in °C). Find the temperature at which the relative humidity is minimized.

A) 6.61°C

B) 3.61°C

C) 2.61°C

D) 4237.39°C

38) The velocity of a particle (in (ft/s)) is given by v = t^{2} - 3t + 7, where t is the time (in seconds) for which it has traveled. Find the time at which the velocity is at a minimum.

A) 1.5 sec

B) 3 sec

C) 7 sec

D) 3.5 sec

39) Find the dimensions that produce the maximum floor area for a one-story house that is rectangular in shape and has a perimeter of 151 ft.

A) 75.5 ft × 75.5 ft

B) 37.75 ft × 37.75 ft

C) 37.75 ft × 151 ft

D) 12.58 ft × 37.75 ft

40) An architect needs to design a rectangular room with an area of 94 ft^{2}. What dimensions should he use in order to minimize the perimeter?

A) 9.7 ft × 9.7 ft

B) 18.8 ft × 94 ft

C) 9.7 ft × 23.5 ft

D) 23.5 ft × 23.5 ft

41) A piece of molding 151 cm long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area?

A) 30.2 cm × 30.2 cm

B) 37.75 cm × 37.75 cm

C) 12.29 cm × 12.29 cm

D) 12.29 cm × 37.75 cm