Answer :
Let's break down the problem step-by-step:
### Step 1: Identify the Initial Amount
In the exponential function [tex]\( g(t) = 240(0.75)^t \)[/tex], the initial amount [tex]\( a \)[/tex] is the coefficient of the base. So, the initial amount [tex]\( a \)[/tex] is 240.
### Step 2: Determine the Rate of Decay
The rate of decay is found from the base of the exponent in the form [tex]\( (b)^t \)[/tex]. Here, the base is [tex]\( 0.75 \)[/tex].
- To find the rate of decay as a percentage, we calculate [tex]\( (1 - 0.75) \times 100 \)[/tex].
- [tex]\( 1 - 0.75 = 0.25 \)[/tex].
- Converting 0.25 to a percentage gives us [tex]\( 0.25 \times 100 = 25\%\)[/tex].
Therefore, the rate of decay [tex]\( r \)[/tex] is 25%.
### Step 3: Evaluate the Function When [tex]\( t = 3 \)[/tex]
To find [tex]\( g(3) \)[/tex], substitute [tex]\( t = 3 \)[/tex] into the function:
[tex]\[ g(3) = 240 \times (0.75)^3 \][/tex]
Calculating this gives:
[tex]\[ g(3) = 240 \times 0.421875 = 101.25 \][/tex]
### Step 4: Round the Result
We need to round [tex]\( 101.25 \)[/tex] to the nearest tenth:
- Rounded to the nearest tenth, [tex]\( 101.25 \)[/tex] becomes 101.2.
### Summary
- The initial amount [tex]\( a \)[/tex]: 240
- The rate of decay [tex]\( r \)[/tex]: 25%
- The value of the function at [tex]\( t = 3 \)[/tex]: 101.2
### Step 1: Identify the Initial Amount
In the exponential function [tex]\( g(t) = 240(0.75)^t \)[/tex], the initial amount [tex]\( a \)[/tex] is the coefficient of the base. So, the initial amount [tex]\( a \)[/tex] is 240.
### Step 2: Determine the Rate of Decay
The rate of decay is found from the base of the exponent in the form [tex]\( (b)^t \)[/tex]. Here, the base is [tex]\( 0.75 \)[/tex].
- To find the rate of decay as a percentage, we calculate [tex]\( (1 - 0.75) \times 100 \)[/tex].
- [tex]\( 1 - 0.75 = 0.25 \)[/tex].
- Converting 0.25 to a percentage gives us [tex]\( 0.25 \times 100 = 25\%\)[/tex].
Therefore, the rate of decay [tex]\( r \)[/tex] is 25%.
### Step 3: Evaluate the Function When [tex]\( t = 3 \)[/tex]
To find [tex]\( g(3) \)[/tex], substitute [tex]\( t = 3 \)[/tex] into the function:
[tex]\[ g(3) = 240 \times (0.75)^3 \][/tex]
Calculating this gives:
[tex]\[ g(3) = 240 \times 0.421875 = 101.25 \][/tex]
### Step 4: Round the Result
We need to round [tex]\( 101.25 \)[/tex] to the nearest tenth:
- Rounded to the nearest tenth, [tex]\( 101.25 \)[/tex] becomes 101.2.
### Summary
- The initial amount [tex]\( a \)[/tex]: 240
- The rate of decay [tex]\( r \)[/tex]: 25%
- The value of the function at [tex]\( t = 3 \)[/tex]: 101.2
