A Petri dish is filled with 250 bacterial cultures. The number of bacteria in the dish triples every hour.
recursive formula: gn = 250 · 3n−1
explicit formula: gn = gn−1 · 3
a. Predict the number of bacterial cultures in the dish after 8 hours.
b. Does this sequence represent a function? Yes or No
c. Explain your reasoning.
a. To predict the number of bacterial cultures in the dish after 8 hours, we can use the recursive formula.
Plugging in n = 8 into the recursive formula, we get:
g8 = 250 · 3^(8-1)
g8 = 250 · 3^7
g8 = 250 · 2187
g8 = 546,750
Therefore, there would be 546,750 bacterial cultures in the dish after 8 hours.
b. Yes, this sequence represents a function.
c. A function is a relation between a set of inputs and a set of outputs, where each input is associated with exactly one output. In this case, the input is the number of hours and the output is the number of bacterial cultures. The number of bacterial cultures in the dish after each hour can be uniquely determined based on the input (number of hours) using the recursive or explicit formula. Therefore, the sequence represents a function.
a. To predict the number of bacterial cultures in the dish after 8 hours, we can use the recursive formula:
gn = 250 · 3^(n-1)
Let's substitute n = 8 into the formula:
g8 = 250 · 3^(8-1)
g8 = 250 · 3^7
g8 = 250 · 2187
g8 = 546,750
Therefore, the number of bacterial cultures in the dish after 8 hours is predicted to be 546,750.
b. Yes, this sequence represents a function.
c. A function is a mathematical relationship in which each input has a unique output. In this case, each value of n (representing the number of hours) corresponds to a unique value of gn (the number of bacterial cultures). The recursive formula and explicit formula both provide a clear relationship between the input (n) and output (gn), showing that each hour will result in a specific number of bacterial cultures. Therefore, this sequence represents a function.