Question
Question
A scientist is observing a particular species of bacteria growing in a petri dish. The number of bacteria can be represented by the function n(t)=50(1.2)t
, where n is the number of bacteria after t hours.
When graphing this situation, why is this graphed in quadrant 1?
A scientist is observing a particular species of bacteria growing in a petri dish. The number of bacteria can be represented by the function n(t)=50(1.2)t
, where n is the number of bacteria after t hours.
When graphing this situation, why is this graphed in quadrant 1?
Answers
Answered by
GPT-4o mini
The function \( n(t) = 50(1.2)^t \) represents the growth of the bacteria with respect to time \( t \). In this scenario, \( t \) typically represents time in hours, and \( n(t) \) represents the number of bacteria.
The reason this situation is graphed in Quadrant I (where both \( x \) and \( y \) values are positive) is due to the following points:
1. **Positive Time Values**: In this context, time \( t \) is generally considered to start from 0 and increase (i.e., \( t \geq 0 \)). This means we are only looking at the right side of the \( y \)-axis.
2. **Positive Number of Bacteria**: The expression \( n(t) = 50(1.2)^t \) results in a positive number of bacteria as long as \( t \) is non-negative. Specifically, when \( t = 0 \), \( n(0) = 50(1.2)^0 = 50 \), indicating that there are 50 bacteria present initially. As \( t \) increases, since \( 1.2^t \) is always greater than 0 for non-negative \( t \), \( n(t) \) will also always be positive.
Since both the values of \( t \) (time) and \( n(t) \) (number of bacteria) are non-negative, the graph of this function will reside entirely in Quadrant I, where both axes represent positive values. Quadrant I is characterized by positive \( x \) (horizontal) values and positive \( y \) (vertical) values, which correctly reflects the situation described.
The reason this situation is graphed in Quadrant I (where both \( x \) and \( y \) values are positive) is due to the following points:
1. **Positive Time Values**: In this context, time \( t \) is generally considered to start from 0 and increase (i.e., \( t \geq 0 \)). This means we are only looking at the right side of the \( y \)-axis.
2. **Positive Number of Bacteria**: The expression \( n(t) = 50(1.2)^t \) results in a positive number of bacteria as long as \( t \) is non-negative. Specifically, when \( t = 0 \), \( n(0) = 50(1.2)^0 = 50 \), indicating that there are 50 bacteria present initially. As \( t \) increases, since \( 1.2^t \) is always greater than 0 for non-negative \( t \), \( n(t) \) will also always be positive.
Since both the values of \( t \) (time) and \( n(t) \) (number of bacteria) are non-negative, the graph of this function will reside entirely in Quadrant I, where both axes represent positive values. Quadrant I is characterized by positive \( x \) (horizontal) values and positive \( y \) (vertical) values, which correctly reflects the situation described.