Problem 42 - Precalculus

Question

Imagine that you own a grove of orange trees, and suppose that from past experience you know that when 100 trees are planted, each tree will yield approximately 240 oranges per year. Furthermore, you've noticed that when additional trees are planted in the grove, the yield per tree decreases. Specifically, you have noted that the yield per tree decreases by about 20 oranges for each additional tree planted. (a) Let \(y\) denote the yield per tree when \(x\) trees are planted. Find a linear equation relating \(x\) and \(y\) Hint: You are given that the point (100,240) is on the line. What is given about \(\Delta y / \Delta x ?\) (b) Use the equation in part (a) to determine how many trees should be planted to obtain a yield of 400 oranges per tree. (c) If the grove contains 95 trees, what yield can you expect from each tree?

Step-by-Step Solution

Answer

(a) Equation: \( y = -20x + 2240 \); (b) 92 trees; (c) Yield is 340 oranges per tree.

Step 1: Initial Setup for Part (a)

We know the point (100, 240) is on the line, where 100 is the number of trees planted, and 240 is the yield per tree. The problem also states that when additional trees are planted, the yield per tree decreases by 20 oranges. This implies that the slope \( \Delta y / \Delta x = -20 \).

Step 2: Formulate Linear Equation

Using the point (100, 240) and slope \(-20\), we use the point-slope form of a line equation: \( y - y_1 = m(x - x_1) \). Substitute \( m = -20 \), \( x_1 = 100 \), and \( y_1 = 240 \) to get: \( y - 240 = -20(x - 100) \). Simplify to find the linear equation: \( y = -20x + 2240 \).

Step 3: Solve Part (b)

To find the number of trees \( x \) for a yield \( y = 400 \), substitute \( y = 400 \) into the equation \( y = -20x + 2240 \). Solve for \( x \): \( 400 = -20x + 2240 \). Subtract 2240 from both sides: \( -1840 = -20x \). Divide by -20: \( x = 92 \).

Step 4: Solve Part (c)

For the grove with 95 trees, substitute \( x = 95 \) into the equation \( y = -20x + 2240 \) to find the yield per tree: \( y = -20(95) + 2240 \). Calculate: \( y = -1900 + 2240 = 340 \).

Key Concepts

Slope of a Line

The concept of the slope of a line is fundamental in understanding how two variables are related linearly. In a linear equation, the slope represents the rate at which one variable changes with respect to another. In our orange grove problem, the slope tells us how much the yield per tree (\(y\)) changes when the number of trees (\(x\)) changes.

The slope is calculated as the change in yield per tree divided by the change in the number of trees. This is represented as:

• \(\Delta y / \Delta x = \text{slope}\)
• In our problem, the slope is given as \(-20\), meaning for every additional tree planted, the yield per tree decreases by 20 oranges.

Understanding the slope helps us predict yield decreases and assists in planning how many trees to plant for optimal results.

Point-Slope Form

The point-slope form is a way to write the equation of a line when you know a point on the line and the slope. This form is especially useful when you're solving problems related to change, like the orange grove scenario.

The point-slope form equation is:

• \(y - y_1 = m(x - x_1)\)
• \(m\) is the slope.
• \( (x_1, y_1) \) is a specific point on the line.

For our exercise, the point given is (100, 240) and the slope is \(-20\). Plugging these into the equation, we transform it into: \(y - 240 = -20(x - 100)\). This equation can be simplified to get our linear equation: \(y = -20x + 2240\).

This simplification helps us use the equation in various scenarios, like predicting yields based on the number of trees planted.

Problem Solving

Problem-solving with linear equations often involves interpreting the problem context and applying mathematical concepts appropriately. In our orange grove problem, we took an observation (how yield decreases with more trees) and translated it into a mathematical model using a linear equation.

Let's examine how this equation aids in practical problem-solving:

• In Part (b), we used the equation to find the number of trees needed to achieve a certain yield per tree. By setting the yield \(y = 400\) and solving for \(x\), we determined that \(92\) trees should be planted.
• In Part (c), the equation helped predict the yield from 95 trees by substituting \(x = 95\) to calculate \(y = 340\).

Thus, solving problems like these requires understanding how to set up, interpret, and manipulate linear equations to find meaningful solutions.