Function Model Construction and Application
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AP Precalculus › Function Model Construction and Application
A company's profit $$P$$ (in thousands of dollars) is modeled by $$P(x) = -2x^3 + 15x^2 - 24x + 10$$, where $$x$$ is the number of years since 2020. What is the company's profit in 2023?
$$7$$ thousand dollars
$$25$$ thousand dollars
$$19$$ thousand dollars
$$13$$ thousand dollars
Explanation
In 2023, $$x = 3$$. Substituting: $$P(3) = -2(27) + 15(9) - 24(3) + 10 = -54 + 135 - 72 + 10 = 19$$ thousand dollars. Choice A uses $$x = 2$$. Choice B uses $$x = 1$$. Choice D uses $$x = 4$$.
A rational function models the concentration $$C$$ (in mg/L) of a medication in the bloodstream $$t$$ hours after injection: $$C(t) = \frac{120t}{t^2 + 4}$$. What is the concentration after 2 hours?
$$30$$ mg/L
$$20$$ mg/L
$$25$$ mg/L
$$15$$ mg/L
Explanation
Substituting $$t = 2$$: $$C(2) = \frac{120(2)}{2^2 + 4} = \frac{240}{4 + 4} = \frac{240}{8} = 30$$ mg/L. Choice A incorrectly computes $$\frac{120}{8}$$. Choice B uses wrong denominator calculation $$(2^2 + 8)$$. Choice C uses wrong numerator calculation $$\frac{200}{8}$$.
A population of bacteria grows according to $$P(t) = \frac{5000t}{t + 10}$$, where $$t$$ is time in hours. What happens to the population as $$t$$ approaches infinity?
The population grows without bound
The population approaches 500 bacteria
The population approaches 50 bacteria
The population approaches 5000 bacteria
Explanation
As $$t \to \infty$$, $$P(t) = \frac{5000t}{t + 10} \to \frac{5000t}{t} = 5000$$. The horizontal asymptote is $$y = 5000$$. Choice A divides 5000 by 10 incorrectly. Choice C ignores the rational function behavior. Choice D uses wrong calculation entirely.
The temperature $$T$$ (in °F) in a building $$t$$ hours after midnight is modeled by $$T(t) = \frac{t^2 + 38t + 400}{t + 10}$$ for $$0 \leq t \leq 24$$. What is the temperature at 2:00 AM?
$$37$$ °F
$$39$$ °F
$$40$$ °F
$$42$$ °F
Explanation
At 2:00 AM, $$t = 2$$. Substituting: $$T(2) = \frac{2^2 + 38(2) + 400}{2 + 10} = \frac{4 + 76 + 400}{12} = \frac{480}{12} = 40$$ °F. Choice A uses wrong arithmetic in numerator. Choice B uses $$t = 1$$. Choice D uses wrong denominator calculation.
A ball is thrown upward from a height of 6 feet with an initial velocity of 48 feet per second. The height function is $$h(t) = -16t^2 + 48t + 6$$. When does the ball hit the ground?
$$t = 3.25$$ seconds
$$t = 2.87$$ seconds
$$t = 3.12$$ seconds
$$t = 3.00$$ seconds
Explanation
The ball hits ground when $$h(t) = 0$$: $$-16t^2 + 48t + 6 = 0$$. Using the quadratic formula: $$t = \frac{-48 \pm \sqrt{48^2 + 4(16)(6)}}{-32} = \frac{-48 \pm \sqrt{2688}}{-32}$$. Taking the positive root: $$t \approx 3.12$$ seconds. Choices B, C, and D result from computational errors in the quadratic formula.
The resistance $$R$$ (in ohms) of a wire is given by $$R(L) = \frac{0.5L}{A}$$, where $$L$$ is length in meters and $$A$$ is cross-sectional area in square millimeters. For a wire with area $$A = 2$$ square millimeters, what is the resistance when $$L = 100$$ meters?
$$100$$ ohms
$$50$$ ohms
$$200$$ ohms
$$25$$ ohms
Explanation
Substituting $$L = 100$$ and $$A = 2$$: $$R(100) = \frac{0.5(100)}{2} = \frac{50}{2} = 25$$ ohms. Choice B forgets to divide by area. Choice C uses wrong coefficient calculation. Choice D doubles the correct answer.
A rectangular swimming pool is being designed with a perimeter of 80 feet. If the length is $$l$$ feet, what function represents the area $$A$$ of the pool?
$$A(l) = l(80 - l)$$ for $$0 < l < 80$$
$$A(l) = l(40 - l)$$ for $$0 < l < 40$$
$$A(l) = l(20 - l)$$ for $$0 < l < 20$$
$$A(l) = 2l(40 - l)$$ for $$0 < l < 40$$
Explanation
Perimeter: $$2l + 2w = 80$$, so $$w = 40 - l$$. Area: $$A(l) = l(40 - l)$$. Domain: $$0 < l < 40$$ since width must be positive. Choice B doesn't divide perimeter correctly. Choice C uses wrong perimeter division. Choice D incorrectly doubles the area formula.
The efficiency $$E$$ (as a percentage) of a solar panel depends on temperature $$T$$ (in °C) according to $$E(T) = \frac{-T^2 + 20T + 1200}{T + 30}$$. What is the efficiency when the temperature is 10°C?
$$35.0%$$
$$27.5%$$
$$32.5%$$
$$30.0%$$
Explanation
Substituting $$T = 10$$: $$E(10) = \frac{-(10)^2 + 20(10) + 1200}{10 + 30} = \frac{-100 + 200 + 1200}{40} = \frac{1300}{40} = 32.5%$$. Choice A uses wrong numerator calculation. Choice B uses $$T = 20$$. Choice D uses wrong denominator calculation.
A company's daily production cost is $$C(x) = 0.01x^3 - 0.6x^2 + 15x + 500$$ dollars for producing $$x$$ units. If they want to minimize cost per unit, what function represents cost per unit?
$$\frac{C(x)}{x} = 0.01x - 0.6 + \frac{15}{x} + \frac{500}{x}$$
$$\frac{C(x)}{x} = 0.01x^3 - 0.6x^2 + 15x + 500$$
$$\frac{C(x)}{x} = 0.01x^2 - 0.6x + 15 + 500$$
$$\frac{C(x)}{x} = 0.01x^2 - 0.6x + 15 + \frac{500}{x}$$
Explanation
Cost per unit is $$\frac{C(x)}{x} = \frac{0.01x^3 - 0.6x^2 + 15x + 500}{x} = 0.01x^2 - 0.6x + 15 + \frac{500}{x}$$. Choice B doesn't divide by $$x$$. Choice C doesn't properly handle the constant term division. Choice D incorrectly divides each coefficient by $$x$$.
A rectangular garden has a length that is 4 meters more than twice its width. If the width is $$w$$ meters, which function represents the area $$A$$ of the garden in square meters?
$$A(w) = w^2 + 8w$$
$$A(w) = 2w^2 + 4w$$
$$A(w) = 2w^2 + 8w$$
$$A(w) = w^2 + 4w$$
Explanation
The length is $$2w + 4$$ meters. The area is length times width: $$A(w) = w(2w + 4) = 2w^2 + 4w$$. Choice B incorrectly uses $$w + 4$$ for length. Choice C incorrectly uses $$2w + 8$$ for length. Choice D incorrectly uses $$w + 8$$ for length.