Answer :
Let's go through each part of the question step-by-step:
### a. Reasonableness of Linear Function
It is reasonable to say that the number of occupied apartments is a linear function of rent because the data shows a consistent pattern: as the rent increases, the number of occupied apartments decreases at a relatively constant rate. This linear relationship suggests that for every fixed increase in rent, there is a proportional decrease in the number of occupied apartments.
### b. Slope of the Linear Function
Let [tex]\( A \)[/tex] be the number of occupied apartments and [tex]\( R \)[/tex] the monthly rent charged (in dollars). If we model the number of occupied apartments as a linear function of rent, [tex]\( A = f(R) \)[/tex], we can find the slope of this function.
The formula for the slope [tex]\( m \)[/tex] of a linear function given two points [tex]\((R_1, A_1)\)[/tex] and [tex]\((R_2, A_2)\)[/tex] is:
[tex]\[ m = \frac{\Delta A}{\Delta R} = \frac{A_2 - A_1}{R_2 - R_1} \][/tex]
Using the data points (650, 203) and (900, 168), the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{168 - 203}{900 - 650} = \frac{-35}{250} = -0.14 \][/tex]
The slope [tex]\( -0.14 \)[/tex] means that for every [tex]$1 increase in rent, the number of occupied apartments decreases by 0.14 units. ### c. Formula for \(A = f(R)\) and Reasonable Domain To determine the linear formula for \( A \) as a function of \( R \), we use the point-slope form of a linear equation: \[ A = mR + b \] We know the slope \( m = -0.14 \) and we can solve for \( b \) using one of the data points, for example (650, 203): \[ 203 = -0.14 \cdot 650 + b \] \[ 203 = -91 + b \] \[ b = 203 + 91 = 294 \] Thus, the formula for \( A = f(R) \) is: \[ A = -0.14R + 294 \] A reasonable domain for the function would be the range of rents provided in the data, $[/tex]650 \leq R \leq [tex]$900, because this is the range over which we have observed values. Predicting outside this range without further data could be less reliable. ### d. Occupied Apartments and Revenue at \$[/tex]1000 Rent
If the rent [tex]\( R \)[/tex] were increased to \[tex]$1000, we can use the linear formula to predict the number of occupied apartments: \[ A = -0.14 \cdot 1000 + 294 \] \[ A = -140 + 294 \] \[ A = 154 \] The total revenue \( TR \) collected would be: \[ TR = R \cdot A \] \[ TR = 1000 \cdot 154 \] \[ TR = 154000 \] Thus, if the rent is increased to \$[/tex]1000, the apartment manager should expect 154 occupied apartments and a total monthly revenue of \$154,000.
### e. Reason for Data Collection
The apartment manager is interested in the data collected to make informed decisions about rent pricing. By understanding the relationship between rental prices and occupancy, the manager can optimize rent levels to balance between maximizing revenue and maintaining high occupancy rates. The collected data helps to identify the point at which increases in rent begin to reduce occupancy rates significantly, enabling strategic planning for pricing and revenue management.
By carefully analyzing these relationships, the manager can ensure that they are setting rents at levels that achieve their financial goals while also keeping the apartments as occupied as possible.
### a. Reasonableness of Linear Function
It is reasonable to say that the number of occupied apartments is a linear function of rent because the data shows a consistent pattern: as the rent increases, the number of occupied apartments decreases at a relatively constant rate. This linear relationship suggests that for every fixed increase in rent, there is a proportional decrease in the number of occupied apartments.
### b. Slope of the Linear Function
Let [tex]\( A \)[/tex] be the number of occupied apartments and [tex]\( R \)[/tex] the monthly rent charged (in dollars). If we model the number of occupied apartments as a linear function of rent, [tex]\( A = f(R) \)[/tex], we can find the slope of this function.
The formula for the slope [tex]\( m \)[/tex] of a linear function given two points [tex]\((R_1, A_1)\)[/tex] and [tex]\((R_2, A_2)\)[/tex] is:
[tex]\[ m = \frac{\Delta A}{\Delta R} = \frac{A_2 - A_1}{R_2 - R_1} \][/tex]
Using the data points (650, 203) and (900, 168), the slope [tex]\( m \)[/tex] is:
[tex]\[ m = \frac{168 - 203}{900 - 650} = \frac{-35}{250} = -0.14 \][/tex]
The slope [tex]\( -0.14 \)[/tex] means that for every [tex]$1 increase in rent, the number of occupied apartments decreases by 0.14 units. ### c. Formula for \(A = f(R)\) and Reasonable Domain To determine the linear formula for \( A \) as a function of \( R \), we use the point-slope form of a linear equation: \[ A = mR + b \] We know the slope \( m = -0.14 \) and we can solve for \( b \) using one of the data points, for example (650, 203): \[ 203 = -0.14 \cdot 650 + b \] \[ 203 = -91 + b \] \[ b = 203 + 91 = 294 \] Thus, the formula for \( A = f(R) \) is: \[ A = -0.14R + 294 \] A reasonable domain for the function would be the range of rents provided in the data, $[/tex]650 \leq R \leq [tex]$900, because this is the range over which we have observed values. Predicting outside this range without further data could be less reliable. ### d. Occupied Apartments and Revenue at \$[/tex]1000 Rent
If the rent [tex]\( R \)[/tex] were increased to \[tex]$1000, we can use the linear formula to predict the number of occupied apartments: \[ A = -0.14 \cdot 1000 + 294 \] \[ A = -140 + 294 \] \[ A = 154 \] The total revenue \( TR \) collected would be: \[ TR = R \cdot A \] \[ TR = 1000 \cdot 154 \] \[ TR = 154000 \] Thus, if the rent is increased to \$[/tex]1000, the apartment manager should expect 154 occupied apartments and a total monthly revenue of \$154,000.
### e. Reason for Data Collection
The apartment manager is interested in the data collected to make informed decisions about rent pricing. By understanding the relationship between rental prices and occupancy, the manager can optimize rent levels to balance between maximizing revenue and maintaining high occupancy rates. The collected data helps to identify the point at which increases in rent begin to reduce occupancy rates significantly, enabling strategic planning for pricing and revenue management.
By carefully analyzing these relationships, the manager can ensure that they are setting rents at levels that achieve their financial goals while also keeping the apartments as occupied as possible.
