Problem 10 - Applied Calculus

Question

A demand curve is given by \(75 p+50 q=300,\) where \(p\) is the price of the product, in dollars, and \(q\) is the quantity demanded at that price. Find \(p\) - and \(q\) -intercepts and interpret them in terms of consumer demand.

Step-by-Step Solution

Answer

The \(p\)-intercept is 4, and the \(q\)-intercept is 6, indicating the minimum price and max demand, respectively.

Step 1: Identify the Form of the Equation

The given demand curve equation is \(75p + 50q = 300\). This is a linear equation in the variables \(p\) (price) and \(q\) (quantity). Our task is to find the intercepts of this equation.

Step 2: Find the p-intercept

To find the \(p\)-intercept, set \(q = 0\) in the equation. This implies:\[75p + 50(0) = 300\] \[75p = 300\] Divide both sides by 75:\[p = \frac{300}{75} = 4\] Thus, the \(p\)-intercept is at \(p = 4\).

Step 3: Find the q-intercept

To find the \(q\)-intercept, set \(p = 0\) in the equation. This implies:\[75(0) + 50q = 300\] \[50q = 300\] Divide both sides by 50:\[q = \frac{300}{50} = 6\] Thus, the \(q\)-intercept is at \(q = 6\).

Step 4: Interpretation of the Intercepts

In the context of consumer demand, the \(p\)-intercept of \(p = 4\) means that when the quantity demanded \(q\) is zero, the price \(p\) is \(4. This represents the minimum price at which the dealer will consider selling the product, as demand ceases below \)4. The \(q\)-intercept of \(q = 6\) means that when the price \(p\) is zero, the quantity demanded \(q\) is 6 units. This implies that at no cost, the maximum need or want for the product is 6 units.

Key Concepts

Linear Equation

A linear equation in the context of demand curves is a mathematical representation that shows the relationship between two variables: price (\(p\)) and quantity (\(q\)). In our example, the demand curve is represented by the equation:

• \[75p + 50q = 300\]

This equation is linear because if you graph it, the result will be a straight line. Linear equations are easy to work with and are often used in demand analysis because they can be easily interpreted.
The coefficients 75 and 50 express how much the price and quantity will change in relation to changes in demand.
This linearity implies that an increase in price by one unit will have a predictable impact on the quantity demanded, and vice versa.

Price-Intercept

The price-intercept of a demand curve is found where the quantity demanded is zero. Essentially, it's the price point where consumers will stop buying the product altogether. In our example, to find the price-intercept, we set \(q = 0\) in the equation and solve for \(p\):

• \[75p + 50(0) = 300\]
• \[75p = 300\]
• \[p = \frac{300}{75} = 4\]

Thus, the price-intercept is 4.
This means that when the price is 4 dollars, no consumers are willing to buy the product.
In other words, 4 dollars is the lowest price at which sellers still want to sell their product, as demand ceases below this price.

Quantity-Intercept

The quantity-intercept is the point on the demand curve where the price is zero. When price is zero, it means the product is free. The quantity-intercept tells us the maximum amount of the product consumers would demand if it were free. To find the quantity-intercept, you set \(p = 0\) in the equation and solve for \(q\):

• \[75(0) + 50q = 300\]
• \[50q = 300\]
• \[q = \frac{300}{50} = 6\]

So, the quantity-intercept is 6.
This tells us that at a price of zero, consumers would demand 6 units of the product.
It highlights the natural limit of consumer interest regardless of price.

Consumer Demand

Consumer demand represents how much of a product consumers are willing to buy at various prices. It's driven by factors like consumer preference, income, and the price of alternatives.
In a demand curve, consumer demand is illustrated by a downward-sloping line, indicating that as price decreases, quantity demanded tends to increase.
The equation

• \[75p + 50q = 300\]

shows this inverse relationship between price and quantity. As you solve for different points on the curve, you can see how changes in price affect the quantity desired by consumers. These concepts help in understanding how markets function and how pricing strategies can be developed.
Grasping consumer demand is key in making informed economic decisions.