The problem describes an office tower where the ground floor has an area of $1330.0 \ m^2$ and each subsequent floor is reduced by 9% in area. We need to find the area of the 6th floor.

1. Problem Description

The problem describes an office tower where the ground floor has an area of $1330.0 \ m^2$ and each subsequent floor is reduced by 9% in area. We need to find the area of the 6th floor.

2. Solution Steps

The area of each floor can be calculated using the formula for exponential decay. Since the area is reduced by 9% each floor, the remaining area is $100\% - 9\% = 91\% = 0.91$ of the previous floor's area.

The general formula for the area of the $n$th floor, $A_n$, is given by:

$A_n = A_0 * (1 - r)^n$

where $A_0$ is the area of the ground floor, $r$ is the rate of reduction (as a decimal), and $n$ is the floor number.

In this case, $A_0 = 1330.0 \ m^2$, $r = 0.09$, and we want to find the area of the 6th floor, so $n = 6$.

$A_6 = 1330.0 * (1 - 0.09)^6$

$A_6 = 1330.0 * (0.91)^6$

$A_6 = 1330.0 * 0.568800068721$

$A_6 = 756.50409149993$

Rounding to one decimal place, $A_6 = 756.5 \ m^2$.

3. Final Answer

The area of the 6th floor is approximately $756.5 \ m^2$.