$\text { A real estate office handles an apartment complex with } 50 \text { units. When the rent is }$ $\$ 600$ per month, all 50 units are occupied. However, when the rent is $\$ 645$, the average number of occupied units drops to 47 . Assume that the relationship between the monthly rent $p$ and the demand $x$ is linear. Predict the number of units occupied if the rent is raised to $\$ 660$.
A) 43 units
B) 54 units
C) 57 units
D) 49 units
E) 46 units

Question

Quizplus · Accepted Answer

The relationship between the monthly rent $p$ and the demand $x$ is linear, so we can use the two given points to find the slope of the line: $(600, 50)$ and $(645, 47)$. The slope $m$ is $\frac{47 - 50}{645 - 600} = \frac{-3}{45} = -\frac{1}{15}$. The equation of the line can be written as $x = mp + b$, where $b$ is the y-intercept. Using one of the points to solve for $b$, $50 = -\frac{1}{15} \cdot 600 + b$, we find $b = 90$. Thus, the equation relating $p$ and $x$ is $x = -\frac{1}{15}p + 90$. Substituting $p = 660$ into this equation gives $x = -\frac{1}{15} \cdot 660 + 90 = -44 + 90 = 46$.

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