The number of square units, A, in the area covered by a bacteria culture is increasing at a rate of 20% every 7 days. If the bacteria culture initially covers an area of 10 square centimeters, which equation can be used to find the number of square units in the area covered by the bacteria culture after d days?

Options

A. `A=10(1.20)^((d)/(7))`
B. `A=10(1.20)^(7d)`
C. `A=10(0.80)^((d)/(7))`
D. `A=(12)^(7d)`

Answer

To find the equation that represents the area covered by the bacteria culture after \( d \) days, we can follow these steps:

### Step 1: Identify the initial area
The initial area covered by the bacteria culture is given as \( 10 \) square centimeters.

### Step 2: Determine the growth rate
The bacteria culture increases at a rate of \( 20\% \) every \( 7 \) days. We can express this growth rate as a multiplier:
\[ 1 + \frac{20}{100} = 1.2 \]
This means that after \( 7 \) days, the area will be multiplied by \( 1.2 \).

### Step 3: Set up the equation for growth
After every \( 7 \) days, the area becomes:
\[ \text{Area after 7 days} = 10 \times 1.2 \]
After \( 14 \) days, it becomes:
\[ \text{Area after 14 days} = 10 \times (1.2)^2 \]
Continuing this pattern, after \( n \) periods of \( 7 \) days, the area can be expressed as:
\[ \text{Area after } n \text{ periods} = 10 \times (1.2)^n \]

### Step 4: Relate the number of days to the number of periods
To find the number of periods \( n \) in terms of \( d \) days, we note that each period is \( 7 \) days long. Therefore, the number of periods \( n \) is:
\[ n = \frac{d}{7} \]

### Step 5: Substitute \( n \) into the area equation
Substituting \( n \) into our area equation gives:
\[ \text{Area after } d \text{ days} = 10 \times (1.2)^{\frac{d}{7}} \]

### Final Equation
Thus, the equation that can be used to find the number of square units in the area covered by the bacteria culture after \( d \) days is:
\[ A(d) = 10 \times (1.2)^{\frac{d}{7}} \]