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Chapter 3: Problem 18

A large wholesale nursery sells shrubs to retail stores. The cost \(C(x)\) and revenue \(R(x)\) equations (in dollars) for \(x\) shrubs are $$ C(x)=15 x+12,000 \quad \text { and } \quad R(x)=18 x $$ a. Find the equilibrium point. b. Explain the meaning of the coordinates for the equilibrium point.

Short Answer

Expert verified
The equilibrium point is (4,000, 72,000). It means the nursery breaks even when selling 4,000 shrubs.

Step by step solution

01

- Define the equilibrium point

The equilibrium point is where the cost equals the revenue. This means we need to find the value of x such that the cost function equals the revenue function, or mathematically when: C(x) = R(x)
02

- Set up the equation

Given the cost equation: C(x) = 15x + 12,000 And the revenue equation: R(x) = 18x Set them equal to each other to find the equilibrium point: 15x + 12,000 = 18x
03

- Solve for x

To solve for x, subtract 15x from both sides of the equation: 12,000 = 18x - 15x This simplifies to: 12,000 = 3x Now, divide both sides by 3 to find x: x = 4,000
04

- Find the corresponding revenue or cost

Using the value of x = 4,000, plug it back into either the cost or revenue equation to find the dollar value. We鈥檒l use the revenue equation: R(4,000) = 18(4,000) = 72,000 So, the coordinates for the equilibrium point are (4,000, 72,000).
05

- Interpret the equilibrium point

The equilibrium point (4,000, 72,000) means that when 4,000 shrubs are sold, the cost to produce those shrubs is equal to the revenue generated from selling them. At this point, the wholesale nursery neither makes a profit nor a loss.

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Key Concepts

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

Cost Function
The cost function represents the total cost incurred by a business to produce a certain quantity of goods. In this exercise, the cost function is given by \( C(x) = 15x + 12,000 \) where:
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