Number of apartments rented
The manager of an 80 unit apartment complex knows from experience that at a rent of £¤300 per unit, all the units will be full.On thw average, an additional unit will remain wacant for each £¤20 increase in rent over £¤300. Furthermore, the manager must keep at least 30 units rented due to toher financial considerations. Currently, the revenue from the complex is £¤35000. How many apartments are rented?
a) Suppose that x represents the number of £¤20 increases over £¤300. Represent the number of apartment units that will be rented in terms of x
b)Represent the rent per unit in terms of x
c)Use the answer in (a) and (b) to write an expression that defines the revenue generated when there are x £¤20 increases over £¤300.
d)According to the problem, the revenue currently generated in £¤35000. Write a quadratic equation in standard form
e)Solve the equation in (d) and answer the question in the problem
I will do the question the way I would do it.
I will let you sort out which part becomes your a) , b) , etc
let x be as defined above.
number of units rented = 80 - x
rent for each unit = 300+20x
(80 - x)(300+20x) = 35000
24000 +1300x - 20x^2 = 35000
-20x^2 + 1300x -11000 = 0
x^2 - 65x + 550 = 0
(x - 55)(x - 10) = 0
x = 55 or x = 10
number of units rented
= 80-55 = 25
or
= 80-10 = 70
but it said that at least 30 must be rented, so
there are 70 units rented
check:
if x=10, rent = 300+10(20) = 500
units rented = 70
revenue = 70 x 500 = 35000
a) Let's represent the number of £¤20 increases over £¤300 as x.
Since the number of rented units decreases by 1 for each £¤20 increase in rent over £¤300, we can express the number of rented units as follows:
Number of rented units = 80 - x
b) The rent per unit can be represented as £¤300 + £¤20 * x.
c) The revenue generated can be calculated by multiplying the number of rented units by the rent per unit. So the expression that defines the revenue generated is:
Revenue = (80 - x) * (£¤300 + £¤20 * x)
d) According to the problem, the revenue currently generated is £¤35000. We can write a quadratic equation in standard form by setting the revenue expression equal to £¤35000:
(80 - x) * (£¤300 + £¤20 * x) = £¤35000
e) To solve the equation, we need to simplify and rearrange it into standard quadratic form, which is ax^2 + bx + c = 0. However, this equation is not quadratic, so we'll expand and rearrange it:
(80 - x) * (£¤300 + £¤20 * x) = £¤35000
Multiplying the terms:
80 * (£¤300 + £¤20 * x) - x * (£¤300 + £¤20 * x) = £¤35000
Expanding further:
£¤24000 + £¤1600 * x - £¤300 * x - £¤20 * x^2 = £¤35000
Rearranging to standard form:
-£¤20 * x^2 + (£¤1600 - £¤300) * x + (£¤24000 - £¤35000) = 0
-£¤20 * x^2 + £¤1300 * x - £¤11000 = 0
Now, you can solve this quadratic equation using methods like factoring, completing the square, or using the quadratic formula to find the values of x.