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Chapter 2: Problem 61

The global stockpile of plutonium for military applications between \(1990(t=0)\) and 2003 \((t=13)\) stood at a constant 267 tons. On the other hand, the global stockpile of plutonium for civilian use was $$ 2 t^{2}+46 t+733 $$ tons in year \(t\) over the same period. a. Find the function \(f\) giving the global stockpile of plutonium for military use from 1990 through 2003 and the function \(g\) giving the global stockpile of plutonium for civilian use over the same period. b. Find the function \(h\) giving the total global stockpile of plutonium between 1990 and 2003 . c. What was the total global stockpile of plutonium in 2003 ?

Short Answer

Expert verified
a. The function for military stockpile is \(f(t) = 267\), and for civilian stockpile, it is \(g(t) = 2t^2 + 46t + 733\). b. The function for total stockpile is \(h(t) = f(t) + g(t) = 2t^2 + 46t + 1000\). c. The total global stockpile of plutonium in 2003 was 1936 tons.

Step by step solution

01

Define the function for military stockpile (f)

Since the military stockpile remains constant at 267 tons, the function for military stockpile is a constant function: $$ f(t)=267 $$
02

Define the function for civilian stockpile (g)

The given function for the civilian stockpile is already in the form of a quadratic function: $$ g(t)=2t^2+46t+733 $$
03

Find the function for total stockpile (h)

To find the function for the total stockpile, we just need to add the functions for military and civilian stockpiles: $$ h(t)=f(t)+g(t)=267+(2t^2+46t+733) $$ Simplifying this expression gives us the function h: $$ h(t)=2t^2+46t+1000 $$
04

Find the total global stockpile in 2003

To find the total global stockpile in 2003, we need to evaluate the function h for t=13, since 2003 is represented by year 13: $$ h(13)=2(13)^2+46(13)+1000 $$ Calculating the value: $$ h(13)=2(169)+598+1000=338+598+1000=1936 $$ So, the total global stockpile of plutonium in 2003 was 1936 tons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Functions
Quadratic functions are fundamental to algebra and appear in a variety of real-world situations, including our example of tracking the global stockpile of plutonium for civilian use. Formally, a quadratic function is any function that can be described by an equation of the form \( f(x) = ax^2 +bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a \eq 0 \).

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \). In the case of the plutonium stockpile, the function \( g(t) = 2t^2 + 46t + 733 \) represents the amount of plutonium over time, with \( t \) being the number of years since 1990.

Why Are Quadratic Functions Important?

Understanding the behavior of quadratic functions is crucial in predicting outcomes and trends over time for various scenarios. As with the plutonium stockpile, a quadratic function shows how the amount changes at an accelerating rate, which informs decision-making and resource management in practical situations.
Constant Functions
In contrast to quadratic functions, constant functions are the epitome of simplicity in the world of mathematics. They are represented by an equation of the form \( f(x) = c \), where \( c \) is a constant value. No matter what value of \( x \) is chosen, the function's value remains the same. In the provided exercise, the military plutonium stockpile is described by the constant function \( f(t) = 267 \), indicating that the stockpile remained at a steadfast 267 tons from 1990 through 2003.

Constant functions are graphically represented by a horizontal line on the coordinate plane, reflecting their static nature.

Significance of Constant Functions

These functions are powerful in their simplicity for providing a baseline in comparisons and indicating stability over time. In policy or strategic planning, knowing a quantity that remains constant is just as important as understanding one that changes.
Function Evaluation
The process of function evaluation involves finding the output of a function given a particular input. It is a fundamental skill when working with any type of function, such as quadratic or constant functions. To evaluate a function, you substitute the input value (often \( x \)) into the function and simplify the equation to find the output.

In the context of the plutonium stockpile problem, function evaluation is used to determine the total stockpile in a specific year. For instance, evaluating the function \( h(t) \) for \( t = 13 \) gives us the total amount of plutonium stockpiled in 2003. The calculation involves replacing \( t \) with 13 and simplifying the expression to determine the total quantity.

Practical Application of Function Evaluation

By evaluating functions, we can make informed decisions based on the specific values at certain points in time. In logistics, engineering, science, and economics, this skill is indispensable for effective planning and analysis.

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Most popular questions from this chapter

In Exercises 51 and 52 , determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If the book value \(V\) at the end of the year \(t\) of an asset being depreciated linearly is given by \(V=-a t+b\) where \(a\) and \(b\) are positive constants, then the rate of depreciation of the asset is \(a\) units per year.

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If the profit function is given by \(P(x)=a x^{2}+b x+c\), where \(x\) is the number of units produced and sold, then the level of production that yields a maximum profit is \(-\frac{b}{2 a}\) units.
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