Question

Apartment Rental \(\quad\) A real estate office handles an apartment complex with 50 units. When the rent is \(\$ 580\) per month, all 50 units are occupied. However, when the rent is \(\$ 625,\) the average number of occupied units drops to \(47 .\) Assume that the relationship between the monthly rent \(p\) and the demand \(x\) is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving the demand \(x\) in terms of the rent \(p\). (b) Linear extrapolation Use a graphing utility to graph the demand equation and use the trace feature to predict the number of units occupied if the rent is raised to \(\$ 655 .\) (c) Linear interpolation Predict the number of units occupied if the rent is lowered to \(\$ 595 .\) Verify graphically.

Short answer

The linear equation describing the relation between the rent and the demand is \(x = -\frac{1}{15}p + 88.33\). If the rent is increased to \$655, approximately 45 units will be occupied. However, if the rent is lowered to \$595, the demand will increase to about 49 units.

Step-by-step solution

Formulate The Linear Equation

First of all, we are going to derive a linear equation. The problem states a linear relationship between the rent and the occupied units. Start by identifying two points from the problem. For \(p=\$580\), \(x=50\) and for \(p=\$625\), \(x=47\). Now, you can use the formula of a slope \(m=\frac{(y_2-y_1)}{(x_2-x_1)}\) where \(x\) stands for price \(p\) and \(y\) for demand \(x\). You'll get \(m=\frac{(47-50)}{(625-580)} = -\frac{1}{15}\). Now that you have the slope, you need to find the y-intercept. Use the point-slope form of linear equations, \(y - y_1 = m(x - x_1)\), using one of the points (\(p = 580\), \(x = 50\)) for \(x_1\) and \(y_1\), and the slope \(m = -\frac{1}{15}\) to find \(b = y - mx = 50 - -\frac{1}{15}*580 = 88.33\). Finally, the linear equation which describes the relationship between rent and demand is \(x = -\frac{1}{15}p + 88.33\)

Linear extrapolation using the linear equation

Linear extrapolation means estimating a value outside the known data points. Use the equation from step 1 to predict the number of occupied units if the rent is raised to \$655. To do that, you substitute the given rent into the linear equation: \(x = -\frac{1}{15}*655 + 88.33 = 44.67\). Therefore, if the rent is raised to \$655, the number of units occupied will drop to around 45.

Linear interpolation using the linear equation

Linear interpolation means predicting a value between two known values. Use the equation from step 1 to predict the number of occupied units if the rent is lowered to \$595. By substituting \$595 into the linear equation, you get the number of occupied units: \(x = -\frac{1}{15}*595 + 88.33 = 49\). Therefore, if the rent is lowered to \$595, the number of units occupied will increase to 49.