Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The demand function for a certain product is \(D=3500-x^{2}, 0 \leq x \leq 59.16,\) where \(x\) is the price per unit in dollars and \(D\) is in thousands of units demanded. Determine the number of units demanded if the price per unit is \(\$ 32\).

Short answer

2476 thousand units

Step-by-step solution

- Identify the given function

The demand function given is \(D = 3500 - x^2\).Here, \(x\) is the price per unit in dollars, and \(D\) is the number of units demanded in thousands.

- Substitute the given price into the function

We need to determine the number of units demanded when the price per unit is \(\$ 32\).Substitute \(x = 32\) into the demand function:\[D = 3500 - (32)^2\]

- Perform the calculation

Calculate the value of \(32^2\):\[32^2 = 1024\]Now substitute back into the function:\[D = 3500 - 1024\]Simplify:\[D = 2476\]

Key concepts

price per unit

To understand the demand function, you need to know what 'price per unit' means. The term refers to the cost for one unit of the product. In our problem, the variable \(x\) represents this price. For example, if \(x = 32\), it means each unit costs \$32. Remember, this is crucial because changes in the price per unit directly affect the number of units demanded. A higher price will usually result in lower demand, and vice-versa.

units demanded

Units demanded indicates how many units of the product people want to buy. In the demand function given, \(D = 3500 - x^2\)\, the output \(D\) represents the number of units demanded, but it's measured in thousands. So, if \(D = 2476\)\, it means 2,476,000 units are demanded. When substituting a price into the demand function, you calculate the resulting units demanded.

substitution in algebra

Substitution in algebra is a process where you replace a variable with a given value to simplify the expression or solve an equation. In our example, the substitution step involves replacing \(x\) with 32 in the demand function \(D = 3500 - x^2\). This yields \(D = 3500 - (32)^2\). Substitution lets you see the concrete impact of a specific price on the units demanded. Bullet points to make it clear:

• Identify the variable in the function.
• Replace it with the given value.
• Simplify the expression step-by-step.

This step is essential for solving the problem accurately.

demand curve

A demand curve graphically represents the relationship between the price per unit and the units demanded. For our function \(D = 3500 - x^2\), as the price \(x\) increases, the units demanded \(D\) decreases. Plotting this on a graph helps visualize this inverse relationship.

Key points:

• The y-axis typically represents units demanded.
• The x-axis shows the price per unit.
• Each point on the curve shows a price and the corresponding demand.

This curve helps predict how changes in price can affect demand, aiding in strategic pricing decisions.