Answer:
(a) The linear formula that models the lean of the renovated tower is:
I = (17 / 300)t + 162
In 2150, the tower will lean 170.44 inches off the perpendicular.
Explanation:
Data:
- Renovators were able to reduce the tower's 1990 tilt by 17 inches.
- The resultant tower leans 162 inches off the perpendicular.
- In 2001, officials forecast that it would take 300 years for the tower to return to its 1990 tilt.
(a)
A linear formula has the form:
y = mx + b
where
- y is the dependent variable
- x is the independent variable
- m is the slope, and
- b is the y-axis interception
In this case, the dependent variable is the tilt of the tower, measured as the number of inches from the perpendicular. Let´s call "I" this variable. And what does it depend on? It depends on the variable time. The tilt of the tower varies over time.
Therefore, the time (in years) is the independent variable. Let´s call "t" this variable.
The slope (m) is the change in the dependent variable for each unit of the independent variable. So, it is the change in the number of inches from the perpendicular, for each year elapsed.
Officials forecast that it would take 300 years for the tower to return to its 1990 tilt. In other words, 300 years to the tower to lean 17 inches. Or, the tower will lean 17 inches in 300 years. That is the slope (m).
m = (17 / 300) inches per year
The y-axis interception (b) is the value of the dependent variable (I) when the independent variable (t) is equal to zero.
Our t=0 occurs when the tower was reopened, in 2001.
At that time, the tower leaned 162 inches off the perpendicular.
b = 162
Then, the linear formula that models the lean of the renovated tower is:
I = (17 / 300)t + 162
or
I = 162 + (17 / 300)t
To predict the lean of the tower in 2150, let´s substitute the independent variable t in the formula for the time elapsed from 2001 (our t=0) to 2150.
Time elapsed = 2150 - 2001 = 149 years
I = 162 + (17 / 300) * 149
I = 170.44
In 2150, the tower will lean 170.44 inches off the perpendicular.