A real estate office handles an apartment complex with 50 units. When the rent per unit is $580 per month, all 50 units are occupied. However, when the rent is $625 per month, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear.
(a) Write the equation of the line giving the demand x in terms of the rent p.
x =
(b) Use this equation to predict the number of units occupied when the rent is $640.
units
(c) Predict the number of units occupied when the rent is $595.
units
Since the equation will be linear, we only need two coordinates to write the equation.
They are given, we can assign the values x = no. of units and y = rent:
x1 = 50, y1 = 580 and x2 = 47, y2 = 625
Find the slope: m = (y2-y1)/(x2-x1)
m = (625-580)/(47 - 50) = 45/-3 = -15 is the slope (m)
a)
Use the point/slope equation
Using the point/slope formula: y - y1 = m(x - x1)
y - 580 = -15(x - 50)
y - 580 = -15x + 750
y = -15x + 750 + 580
y = -15x + 1330 or p(x) = -15x + 1330
(a) To find the equation of the line, we need to determine both the slope and the y-intercept.
Let's use the two data points given:
First point: (rent = $580, occupied units = 50)
Second point: (rent = $625, occupied units = 47)
To find the slope, we use the formula:
slope = (occupied units difference) / (rent difference)
slope = (47 - 50) / (625 - 580)
slope = -3 / 45
slope = -1/15
Now, we can use the point-slope form to write the equation:
y - y1 = m(x - x1)
Using the first point (50, 580), we have:
x1 = 580 (rent)
y1 = 50 (occupied units)
m = -1/15 (slope)
y - 50 = (-1/15)(x - 580)
Rearranging the equation, we get:
y = (-1/15)x + (580/15) + 50
y = (-1/15)x + 38.67 + 50
y = (-1/15)x + 88.67
Thus, the equation of the line is:
x = (-1/15)p + 88.67
(b) To predict the number of units occupied when the rent is $640, we substitute p = 640 into the equation of the line:
x = (-1/15)(640) + 88.67
Calculating the value, we find:
x = -42.67 + 88.67
x = 46
So, the predicted number of units occupied when the rent is $640 is 46 units.
(c) To predict the number of units occupied when the rent is $595, we substitute p = 595 into the equation of the line:
x = (-1/15)(595) + 88.67
Calculating the value, we find:
x = -39.67 + 88.67
x = 49
Therefore, the predicted number of units occupied when the rent is $595 is 49 units.
To find the equation of the line that represents the relationship between the rent and the number of occupied units, we can use the formula for a linear equation:
y = mx + b
Where:
y represents the dependent variable (demand)
x represents the independent variable (rent)
m is the slope of the line
b is the y-intercept (the value of y when x = 0)
To find the slope of the line, we can use the formula:
m = (y2 - y1) / (x2 - x1)
Let's use the given information to find the equation of the line:
Given:
Rent p1 = $580, Units occupied x1 = 50
Rent p2 = $625, Units occupied x2 = 47
Using the formula for slope:
m = (x2 - x1) / (p2 - p1)
m = (47 - 50) / ($625 - $580)
m = -3 / 45
m = -1/15
Now, we have the slope of the line. To find the y-intercept, we can use one of the given data points:
Using x1 = 50 and p1 = $580:
50 = (-1/15)($580) + b
50 = (-580/15) + b
b = 50 + 580/15
b = (750 + 580) / 15
b = 1330 / 15
b = 88.67
(a) The equation of the line giving the demand x in terms of the rent p is:
x = (-1/15)p + 88.67
(b) To predict the number of units occupied when the rent is $640, substitute p = $640 into the equation and solve for x:
x = (-1/15)(640) + 88.67
x = -42.67 + 88.67
x = 46
Therefore, when the rent is $640, approximately 46 units will be occupied.
(c) To predict the number of units occupied when the rent is $595, substitute p = $595 into the equation and solve for x:
x = (-1/15)(595) + 88.67
x = -39.67 + 88.67
x = 49
Therefore, when the rent is $595, approximately 49 units will be occupied.