Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose the output of a certain country is given by $$f(x, y)=100 x^{3 / 5} y^{2 / 5}$$ billion dollars if \(x\) billion dollars are spent for labor and \(y\) billion dollars are spent on capital. Find the output if the country spent $$\$ 32$$ billion on labor and \(\$ 243\) billion on capital.

Step-by-Step Solution

Answer

The output of the country when they spend \(\$32\) billion on labor and \(\$ 243\) billion dollars on capital is \(\$ 7200\) billion.

Step 1: Plug in the values of \(x\) and \(y\) into the function \(f(x, y)\)

Let \(x = 32\) billion and \(y = 243\) billion dollars. The output function is given by: $$f(x, y)=100x^{3/5}y^{2/5}.$$ We need to plug the values of \(x\) and \(y\) into the function: $$f(32, 243)=100(32)^{3/5}(243)^{2/5}.$$

Step 2: Simplify the exponents in the function

First, we need to simplify \((32)^{3/5}\) and \((243)^{2/5}\): $$f(32, 243)=100(32)^{3/5}(243)^{2/5},$$ $$=100(2^5)^{3/5}(3^5)^{2/5}.$$ Using the property of exponents \(a^{bc}=(a^b)^c\), we can simplify the above equation as: $$f(32, 243)=100 (2^{5\cdot (3 / 5)})(3^{5\cdot (2 / 5)}),$$ $$= 100 (2^3)(3^2).$$

Step 3: Calculate the final output

Now that we have simplified the exponents, we can calculate the final output: $$f(32, 243)=100(8)(9),$$ $$=100\cdot 72,$$ $$= 7200.$$ The output of the country when they spend \(\$32\) billion on labor and \(\$ 243\) billion dollars on capital is \(\$ 7200\) billion.

Key Concepts

Output Function

An output function is a mathematical model used to represent the output of an economy based on certain inputs like labor and capital. It helps to predict the amount of goods or services produced in financial terms depending on how much is invested in each input. In our exercise, the output function is denoted as \( f(x, y) = 100x^{3/5}y^{2/5} \), where \( x \) represents labor expenditure in billion dollars, and \( y \) signifies capital expenditure in the same unit. This function allows us to systematically evaluate how changes in spending on labor and capital can influence the total output of the economy in a monetary format. Understanding such models is crucial because they form the basis for making informed policy and investment decisions in economic planning.

Exponents

Exponents are an essential part of mathematical equations that help us understand repeated multiplication. In our context, the output function involves exponents, specifically \( x^{3/5} \) and \( y^{2/5} \), indicating fractional powers of the inputs. These exponents signify how changes in the inputs, labor and capital, will affect the output. A foundational idea is that an exponent like \( 3/5 \) or \( 2/5 \) can be interpreted as taking a fifth root and then cubing or squaring, respectively. For instance, \( (32)^{3/5} \) can be solved by finding the fifth root of 32 and then raising the result to the power of three. Understanding these operations is key to manipulating and simplifying expressions involving exponents.

Simplification

Simplification involves reducing a complex expression into its simplest form while retaining its value. In our exercise, simplification happens when we solve expressions like \( (32)^{3/5} \) and \( (243)^{2/5} \). We use properties of exponents, such as \( a^{bc} = (a^b)^c \), to break down these expressions into more manageable calculations.In solving \( (32)^{3/5} \), notice that \( 32 = 2^5 \). Hence, \( (2^5)^{3/5} \) simplifies to \( 2^3 \) which equals 8. Similarly, \( 243 = 3^5 \) leads to \( (3^5)^{2/5} \) simplifying to \( 3^2 \), which equals 9. Simplifying these expressions allows us to efficiently calculate the overall output without tedious computations.

Calculating Output

Calculating output involves substituting the simplified expressions back into the original output function to find the final economic output. After simplifying the exponents, we need to compute the expression:

• \( 100 \times 8 \times 9 \)

This multiplication results from putting the values back into the output function \( f(x, y) = 100x^{3/5}y^{2/5} \) with the simplified forms \( 2^3 = 8 \) and \( 3^2 = 9 \).Thus, the output becomes \( 100 \cdot 72 = 7200 \). This final result tells us that when \( \\(32 \) billion is spent on labor and \( \\)243 \) billion on capital, the country's production translates to an economic output of \( \$7200 \) billion, effectively showing how model simplification aids in outcome determination.