Question: Select the correct answer.
A company manufactures computers. Function $N$ represents the number of components that a new employee can assemble per day. Function $E$ represents the number of components that an experienced employee can assemble per day. In both functions, $t$ represents the number of hours worked in one day.
$$N(t) = \frac{50t}{t+4}$$
$$E(t) = \frac{70t}{t+3}$$
Which function describes the difference of the number of components assembled per day by the experienced and new employees?
$$D(t) = \frac{10t(2t-13)}{(t+3)(t+4)}$$
$$D(t) = \frac{10t(2t-13)}{t+4}$$
$$D(t) = \frac{10t(2t+13)}{(t+3)(t+4)}$$
$$D(t) = \frac{10t(2t+13)}{t+3}$$
1. **State the problem:** We need to find the function $D(t)$ that represents the difference between the number of components assembled by an experienced employee $E(t)$ and a new employee $N(t)$ in one day.
2. **Given functions:**
$$N(t) = \frac{50t}{t+4}$$
$$E(t) = \frac{70t}{t+3}$$
3. **Difference function:**
$$D(t) = E(t) - N(t) = \frac{70t}{t+3} - \frac{50t}{t+4}$$
4. **Find common denominator and subtract:**
$$D(t) = \frac{70t(t+4)}{(t+3)(t+4)} - \frac{50t(t+3)}{(t+3)(t+4)}$$
5. **Expand numerators:**
$$70t(t+4) = 70t^2 + 280t$$
$$50t(t+3) = 50t^2 + 150t$$
6. **Subtract numerators:**
$$D(t) = \frac{70t^2 + 280t - (50t^2 + 150t)}{(t+3)(t+4)} = \frac{70t^2 + 280t - 50t^2 - 150t}{(t+3)(t+4)}$$
7. **Simplify numerator:**
$$D(t) = \frac{(70t^2 - 50t^2) + (280t - 150t)}{(t+3)(t+4)} = \frac{20t^2 + 130t}{(t+3)(t+4)}$$
8. **Factor numerator:**
$$D(t) = \frac{10t(2t + 13)}{(t+3)(t+4)}$$
9. **Conclusion:** The difference function is
$$D(t) = \frac{10t(2t + 13)}{(t+3)(t+4)}$$
This matches the third option.
**Final answer:**
$$D(t) = \frac{10t(2t + 13)}{(t+3)(t+4)}$$