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Chapter 4: Problem 47

South Wind Vineyards uses 820 acres to plant Chardonnay and Riesling grapes. The vintner knows the profits will be greatest by planting 140 more acres of Chardonnay than Riesling. How many acres of each type of grape should be planted?

Short Answer

Expert verified
480 acres of Chardonnay and 340 acres of Riesling should be planted.

Step by step solution

01

Define Variables

Let the number of acres planted with Riesling grapes be denoted as \( R \). Let the number of acres planted with Chardonnay grapes be denoted as \( C \).
02

Formulate Equations

We know that together, the total acreage is 820. This gives us the first equation: \[ C + R = 820 \]. We also know that 140 more acres are planted with Chardonnay than with Riesling. This gives us the second equation: \[ C = R + 140 \].
03

Substitute for Chardonnay Acreage

Substitute \( C = R + 140 \) from the second equation into the first equation: \[ (R + 140) + R = 820 \].
04

Solve for Riesling Acreage

Combine the terms and solve for \( R \): \[ 2R + 140 = 820 \], then subtract 140 from both sides: \[ 2R = 680 \], and finally, divide by 2: \[ R = 340 \].
05

Find Chardonnay Acreage

Substitute \( R = 340 \) back into the second equation to find \( C \): \[ C = 340 + 140 \], which ends up being \[ C = 480 \].
06

Confirm Solution

Verify the solution by adding the Chardonnay and Riesling acres: \[ 480 + 340 = 820 \], which matches the total given in the problem.

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Key Concepts

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

Variable Definition
In algebra, defining variables is the first and a crucial step for solving equations. Variables serve as placeholders for unknown quantities. In our problem, we defined two variables:
  • Let \(\( R \)\) be the number of acres planted with Riesling grapes.
  • Let \(\( C \)\) be the number of acres planted with Chardonnay grapes.
By defining these variables clearly, it becomes easier to set up equations that represent the relationships described in the problem. Using simple, explicit definitions for variables avoids confusion and sets a firm foundation for solving the equations.
Simultaneous Equations
Simultaneous equations are sets of equations with multiple variables that are solved together because they share these variables. In this problem, we formulated two key equations:
  • The total acreage: \ C + R = 820 \lit
  • 140 more acres are planted with Chardonnay than with Riesling: \ C = R + 140 \lit
These equations capture the constraints and dependencies specified in the problem. By solving them together, we can find values for both \( C \) and \( R \) that satisfy both conditions simultaneously.
Substitution Method
The substitution method is a powerful algebraic technique for solving simultaneous equations. It involves solving one equation for one variable and then substituting that solution into the other equation. Here鈥檚 how we used it:
First, solve the second equation for \( C \): \( C = R + 140 \).
Then substitute this expression in place of \( C \) in the first equation:

\[ (R + 140) + R = 820 \]
Solving this substituted equation gives the value of \( R \). Once \( R \) is known, we substitute it back into the second equation to find \( C \). With this method, breaking the problem into manageable steps can simplify even complex equations.
Algebraic Problem-Solving
Algebraic problem-solving involves using well-defined steps to transform complex real-world problems into solvable mathematical expressions. For this vine planting problem:
  • Define the variables: \( R \) and \( C \).
  • Formulate the relationships as equations: \( C + R = 820 \), \( C = R + 140 \).
  • Use the substitution method to solve for one variable.
  • Substitute back to find the other variable.
  • Verify the solution.
This systematic approach ensures that you don't skip essential steps and can effectively solve for unknowns.

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