Sure! Let's solve this discrete exponential growth problem step by step.
We are given:
- The rent on the ground floor (floor 0) is [tex]$1,340.00.
- The rent on the 6th floor is $[/tex]1,600.03.
- The rent increases by a constant percentage per floor.
- We need to find the rate of increase per floor and round it to the nearest percent.
Let's denote:
- [tex]\( r_0 \)[/tex] as the rent on the ground floor.
- [tex]\( r_6 \)[/tex] as the rent on the 6th floor.
- [tex]\( n \)[/tex] as the number of floors difference.
- [tex]\( rate \)[/tex] as the rate of increase per floor.
Given:
[tex]\( r_0 = 1340.00 \)[/tex]
[tex]\( r_6 = 1600.03 \)[/tex]
[tex]\( n = 6 \)[/tex]
The formula for exponential growth is:
[tex]\[ r_6 = r_0 \times (1 + rate)^n \][/tex]
We want to find [tex]\( rate \)[/tex]. Rearranging the formula to solve for [tex]\( rate \)[/tex]:
[tex]\[ (1 + rate)^n = \frac{r_6}{r_0} \][/tex]
[tex]\[ 1 + rate = \left(\frac{r_6}{r_0}\right)^{\frac{1}{n}} \][/tex]
[tex]\[ rate = \left(\frac{r_6}{r_0}\right)^{\frac{1}{6}} - 1 \][/tex]
Plugging in the numbers:
[tex]\[ rate = \left(\frac{1600.03}{1340.00}\right)^{\frac{1}{6}} - 1 \][/tex]
From the solved problem, we know the numerical result is approximately:
[tex]\[ rate \approx 0.0299999917 \][/tex]
To convert this rate into a percentage, we multiply by 100:
[tex]\[ rate\_percentage \approx 0.0299999917 \times 100 \approx 2.999999170126766 \][/tex]
Rounding to the nearest percent:
[tex]\[ rate\_rounded \approx 3\% \][/tex]
Therefore, the rate at which the rent increases per floor is:
B) 3% per floor.
