Problem 54 - Precalculus Student Solutions Manual 5th

Question

If p units of an item are sold for \(x\) dollars per unit, the revenue is \(R=p x\). Use this idea to analyze the following problem. Number of Apartments Rented The manager of an 80-unit apartment complex knows from experience that at a rent of \(\$ 300,\) all the units will be full. On the average, one additional unit will remain vacant for each \(\$ 20\) increase in rent over \(\$ 300 .\) Furthermore, the manager must keep at least 30 units rented due to other financial considerations. Currently, the revenue from the complex is \(\$ 35,000 .\) How many apartments are rented? According to the problem, the revenue currently generated is \(\$ 35,000 .\) Substitute this value for revenue into the equation from Exercise \(53 .\) Solve for \(x\) to answer the question in the problem.

Step-by-Step Solution

Answer

70 apartments are rented.

Step 1: Identify the relationship between rent and the number of rented units

Let the number of units rented be represented by the variable \(p\) and the rent by the variable \(x\). Initially, at \(x = 300\), all 80 units are rented, so \(p = 80\). According to the problem, for each \(\$20\) increase in rent, one unit becomes vacant. This relationship can be written as \(p = 80 - \frac{x - 300}{20}\).

Step 2: Simplify the relationship formula

Simplify the formula \(p = 80 - \frac{x - 300}{20}\) to make it easier to work with. We get \(p = 80 - \frac{x - 300}{20}\). Simplify further: \(p = 80 - \frac{x - 300}{20} = 80 - \frac{x}{20} + 15 = 95 - \frac{x}{20}\).

Step 3: Use the revenue formula and given data

The revenue formula is given by \(R = p x\), and the current revenue is \(35,000\). Substitute into the equation: \(35,000 = (95 - \frac{x}{20}) x\).

Step 4: Solve the equation for \(x\)

Solve the equation \(35,000 = 95x - \frac{x^2}{20}\). Multiply both sides by 20 to clear the fraction: \(700,000 = 1900x - x^2\). Rearrange into standard quadratic form: \(x^2 - 1900x + 700,000 = 0\).

Step 5: Use the quadratic formula to find \(x\)

Solve the quadratic equation \(x^2 - 1900x + 700,000 = 0\) using the quadratic formula \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. Here, \(a = 1\), \(b = -1900\), and \(c = 700,000\). Calculate the values: \[x = \frac{1900 \pm \sqrt{1900^2 - 4 \cdot 1 \cdot 700,000}}{2} \]. Simplifying, \[x = \frac{1900 \pm \sqrt{3,610,000 - 2,800,000}}{2} \] \[x = \frac{1900 \pm \sqrt{810,000}}{2}\].

Step 6: Determine the correct value for \(x\)

Compute the square root: \[x = \frac{1900 \pm 900}{2}\]. This gives two solutions: \[x = 1400\] or \[x = 500\]. Since only one of these is realistic, assuming the rent is increased, we choose \[x = 500\].

Step 7: Calculate the number of apartments rented

Use the value of \(x = 500\) and substitute back into the formula for \(p = 95 - \frac{x}{20}\). This gives \[p = 95 - \frac{500}{20} = 95 - 25 = 70\].

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form: \[ax^2 + bx + c = 0\]. They are essential in revenue analysis problems like this one because they allow us to model complex relationships between variables. In this case, the equation \[x^2 - 1900x + 700000 = 0\] helps us determine the rent amount that maximizes revenue. To solve it, we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], which simplifies our problem significantly. By identifying the coefficients \(a = 1\), \(b = -1900\), and \(c = 700,000\), and substituting them into the formula, we find the possible rent values for the apartment complex. Quadratic equations often provide two solutions, and we choose the one that makes sense contextually.

Rental Income

Rental income is the total amount of money collected from renting out apartment units. It's calculated by multiplying the number of units rented by the rent price per unit. In this problem, the revenue formula is given by \(R = px\), where \( p \) is the number of units rented and \ x \ is the rent per unit. We know the total revenue is currently \(35,000\). By substituting this into our equation, we set up our quadratic equation: \(35,000 = (95 - \frac{x}{20}) x\). This mathematical relationship helps us understand how changes in rent impact the total revenue.

Vacancy Rate

Vacancy rate represents the percentage of all available units in a rental property that are vacant (not rented) at a certain point in time. It's an essential factor in revenue analysis. According to the problem, the manager observes that for every \(20 increase in rent over \)300, one additional unit becomes vacant. This implies an inverse relationship between rent and occupancy. The vacancy rate can be calculated using the relationship \(p = 80 - \frac{x - 300}{20}\), simplifying to \(p = 95 - \frac{x}{20}\). Analyzing these variations allows the manager to optimize rental income by finding that balance between maximizing rent and keeping vacancy low. By solving the equation for\( x \), we determine the optimal rental price that keeps enough units occupied to meet financial obligations.