Problem 57 - Precalculus Student Solutions Manual 5th

Question

The manager of a cherry orchard wants to schedule the annual harvest. If the cherries are picked now, the average yield per tree will be \(100 \mathrm{lb},\) and the cherries can be sold for 40 cents per pound. Past experience shows that the yield per tree will increase about 5 lb per week, while the price will decrease about 2 cents per pound per week. How many weeks should the manager wait to get an average revenue of \(\$ 38.40\) per tree?

Step-by-Step Solution

Answer

The manager should wait 4 weeks.

Step 1 - Define variables

Let the number of weeks the manager waits be represented by the variable, \( x \)

Step 2 - Write equations for yield and price per pound

The yield per tree after \( x \) weeks is \( (100 + 5x) \) lb. The price per pound after \( x \) weeks is \( (0.40 - 0.02x) \) dollars.

Step 3 - Write the equation for revenue per tree

The revenue per tree is given by the product of yield and price per pound: \[ \text{Revenue}(x) = (100 + 5x)(0.40 - 0.02x) \]

Step 4 - Set up the equation for the desired revenue

Given the desired revenue is \$38.40 \, the equation is: \[ (100 + 5x)(0.40 - 0.02x) = 38.40 \]

Step 5 - Simplify and solve the equation

Expand and simplify the equation: \[ 100(0.40 - 0.02x) + 5x(0.40 - 0.02x) = 38.40 \] \[ 40 - 2x + 2x - 0.1x^2 = 38.40 \] \ Then simplify further: \[ 40 - 0.1x^2 = 38.40 \ Rightarrow 1.6 = 0.1x^2 \ Rightarrow x^2 = 16 \ Rightarrow x = 4 \].

Key Concepts

variable definition

Variables are symbols that represent unknown values in mathematical problems. In precalculus word problems, defining the right variable is crucial for setting up equations correctly. For example, in our cherry orchard problem, we defined x as the number of weeks the manager waits for the harvest. This variable helps us link the changes in yield and price over time to calculate the desired outcome. Using variables lets us create equations that model real-world situations and find the solutions step by step.

revenue equations

Revenue is the total income generated from selling goods or services. In the context of our cherry orchard problem, revenue depends on both the yield (the amount of cherries) and the price per pound. To find the revenue per tree, we need to multiply the yield per tree by the price per pound. After defining our variable x (weeks waiting), we created equations for yield and price:

• Yield: \( (100 + 5x) \) pounds
• Price: \( (0.40 - 0.02x) \) dollars per pound

We then use these equations to formulate the revenue equation: \[\text{Revenue}(x) = (100 + 5x)(0.40 - 0.02x)\]. This equation allows us to understand how revenue changes as the manager waits.

quadratic equations

Quadratic equations are polynomial equations of the second degree, generally in the form \[ ax^2 + bx + c = 0 \]. These equations often appear in problems involving areas, projectile motion, and, as in our case, revenue. After expanding our revenue equation: \[ (100 + 5x)(0.40 - 0.02x) = 38.40 \], we simplified it to: \[ 40 - 0.1x^2 = 38.40 \]. Solving this quadratic equation involves isolating the variable and taking the square root:

• First, we subtracted 38.40 from both sides to get: \[ 40 - 38.40 = 0.1x^2 \]
• Then, we solved for \ x^2 \ by isolating it: \[ 1.6 = 0.1x^2 \rightarrow x^2 = 16 \rightarrow x = 4 \]

Thus, the manager should wait 4 weeks.

systems of equations

Systems of equations involve solving multiple equations simultaneously to find common solutions. Although our cherry orchard problem was solved with a single quadratic equation, real-world scenarios often require systems of equations to encapsulate multiple relationships. In such problems, we typically use methods like substitution, elimination, or graphical analysis to find solutions. For example, had we needed to account for multiple variables like varying costs or different types of cherries, we might have set up a system of equations. Understanding how to translate word problems into equations and then solving those systems is a key skill in precalculus and helps in tackling more complex problems.