To determine the function that describes the difference in the number of components assembled per day by an experienced employee and a new employee, follow these steps:
1. Write down the given functions:
- [tex]\( N(t) = \frac{10t}{t + 4} \)[/tex]
- [tex]\( E(t) = \frac{20t}{t + 3} \)[/tex]
2. Set up the difference function [tex]\( D(t) \)[/tex]:
[tex]\[ D(t) = E(t) - N(t) \][/tex]
3. Substitute the given functions into the difference function:
[tex]\[ D(t) = \frac{20t}{t + 3} - \frac{10t}{t + 4} \][/tex]
4. Find a common denominator to combine the fractions:
The common denominator for [tex]\((t + 3)\)[/tex] and [tex]\((t + 4)\)[/tex] is [tex]\((t + 3)(t + 4)\)[/tex].
5. Rewrite each fraction with the common denominator:
[tex]\[ D(t) = \frac{20t(t + 4)}{(t + 3)(t + 4)} - \frac{10t(t + 3)}{(t + 3)(t + 4)} \][/tex]
6. Combine the fractions:
[tex]\[ D(t) = \frac{20t(t + 4) - 10t(t + 3)}{(t + 3)(t + 4)} \][/tex]
7. Expand the numerators:
[tex]\[ 20t(t + 4) = 20t^2 + 80t \][/tex]
[tex]\[ 10t(t + 3) = 10t^2 + 30t \][/tex]
8. Subtract the numerators:
[tex]\[ (20t^2 + 80t) - (10t^2 + 30t) = 20t^2 + 80t - 10t^2 - 30t = 10t^2 + 50t \][/tex]
9. Write the final expression for [tex]\( D(t) \)[/tex]:
[tex]\[ D(t) = \frac{10t^2 + 50t}{(t + 3)(t + 4)} \][/tex]
Next, simplify the numerator [tex]\( 10t^2 + 50t \)[/tex]:
[tex]\[ D(t) = \frac{10t(t + 5)}{(t + 3)(t + 4)} \][/tex]
After simplifying, we test it against the given options:
A. [tex]\( D(t) = \frac{10(2t + 13)}{(t + 3)(t + 4)} \)[/tex]
B. [tex]\( D(t) = \frac{10t(2t - 13)}{t + t} \)[/tex]
C. [tex]\( D(t) = \frac{10t(2t + 13)}{t + 3} \)[/tex]
D. [tex]\( D(t) = \frac{10t(2t - 13)}{(t + 3)(t + 4)} \)[/tex]
By examining the given differential equations, we can confirm that none of these equations match the simplified expression.
Thus, the answer is:
```
None of the given options match.
```
