Answer:
Step-by-step explanation:
Here we have the temperature variation with time given in an exponential equation as follows;
Let the temperature at time x (in minutes) = y
Therefore, y = a × mˣ + b
Where:
b = Shift of the curve or the limit value of the decreasing exponential function as x → ∞
When y = 200, x = 0
Therefore. 200 = a × m⁰ + c = a + c
We note that c is the shift of the graph, the value upon which temperature increases = final temperature = 72°F
Hence a = 200 - 72 = 128°F
When y = 197, x = 2 minutes
Therefore, 197 = 128·m² + 72 =
m² = (197 - 72)/128 = 125/128
m = √(125/128) = 0.98821
Hence the exponential equation of cooling is presented in the following equation;
y = 128 × (0.98821)ˣ + 72
Therefore, y = 128(0.989)x + 72
When the temperature is 172°F we have;
170 = 128 × (0.989)ˣ + 72
∴ (0.989)ˣ = (170 - 72)/128
= 98/128
= 0.7656
log(0.989)ˣ = log(0.7656)
x·log(0.989) = log(0.7656)
x = log(0.7656)/log(0.989)
= 24.15 minutes
