Response:
Part A
The best choice is;
b. y = 128(0.989)x + 72
Part B
Ella can consume the tea post 7:22
Detailed clarification:
Here, we observe the temperature variation with time represented in an exponential equation as follows;
Let the temperature at time x (measured in minutes) be y
Thus, y = a × mˣ + b
Where:
b = Curve shift or the limiting value of the decreasing exponential function as x → ∞
When y = 200, x = 0
This leads to 200 = a × m⁰ + c = a + c
Here, c represents the graph's shift, which is the temperature increase = final temperature = 72°F
Thus, a = 200 - 72 = 128°F
When y = 197, at x = 2 minutes
Thus, 197 = 128·m² + 72 =
m² = (197 - 72)/128 = 125/128
m = √(125/128) = 0.98821
Therefore, the exponential cooling equation is given by;
y = 128 × (0.98821)ˣ + 72
Hence the best choice is b. y = 128(0.989)x + 72
Part B
When the tea's temperature reaches 172°F, we have;
172 = 128 × (0.989)ˣ + 72
Thus, (0.989)ˣ = (172 - 72)/128 = 100/128 = 25/32
log(0.989)ˣ = log(25/32)
x·log(0.989) = log(25/32)
x = log(25/32)/log(0.989) = 22.32 minutes
Thus, Ella can consume the tea post 7:22.
