Select the correct answer.

A company manufactures computers. Function [tex]\( N \)[/tex] represents the number of components that a new employee can assemble per day. Function [tex]\( E \)[/tex] represents the number of components that an experienced employee can assemble per day. In both functions, [tex]\( t \)[/tex] represents the number of hours worked in one day.

[tex]\[
\begin{array}{l}
N(t) = \frac{50t}{t+4} \\
E(t) = \frac{70t}{t+3}
\end{array}
\][/tex]

Which function describes the difference in the number of components assembled per day by the experienced and new employees?

A. [tex]\( D(t) = \frac{10t(2t-13)}{(t+3)(t+4)} \)[/tex]

B. [tex]\( D(t) = \frac{10t(2t-13)}{t+4} \)[/tex]

C. [tex]\( D(t) = \frac{10t(2t+13)}{(t+3)(t+4)} \)[/tex]

D. [tex]\( D(t) = \frac{10t(2t+13)}{t+3} \)[/tex]



Answer :

To solve this problem, we need to find the difference between the number of components that an experienced employee and a new employee can assemble per day, represented by [tex]\( E(t) - N(t) \)[/tex].

Given:
[tex]\[ N(t) = \frac{50t}{t + 4} \][/tex]
[tex]\[ E(t) = \frac{70t}{t + 3} \][/tex]

First, we'll find [tex]\( D(t) = E(t) - N(t) \)[/tex]:

[tex]\[ D(t) = \frac{70t}{t + 3} - \frac{50t}{t + 4} \][/tex]

To subtract these fractions, we need a common denominator. The common denominator for [tex]\( \frac{70t}{t + 3} \)[/tex] and [tex]\( \frac{50t}{t + 4} \)[/tex] is [tex]\( (t + 3)(t + 4) \)[/tex].

Next, we'll rewrite both fractions with this common denominator:

[tex]\[ E(t) = \frac{70t (t + 4)}{(t + 3)(t + 4)} \][/tex]
[tex]\[ N(t) = \frac{50t (t + 3)}{(t + 3)(t + 4)} \][/tex]

Now, we can subtract the numerators:

[tex]\[ D(t) = \frac{70t (t + 4) - 50t (t + 3)}{(t + 3)(t + 4)} \][/tex]

Simplify the numerator:

[tex]\[ D(t) = \frac{70t^2 + 280t - 50t^2 - 150t}{(t + 3)(t + 4)} \][/tex]
[tex]\[ D(t) = \frac{70t^2 - 50t^2 + 280t - 150t}{(t + 3)(t + 4)} \][/tex]
[tex]\[ D(t) = \frac{20t^2 + 130t}{(t + 3)(t + 4)} \][/tex]

Factor the numerator:

[tex]\[ D(t) = \frac{10t(2t + 13)}{(t + 3)(t + 4)} \][/tex]

Thus, the correct function that describes the difference, [tex]\( D(t) \)[/tex], is:

[tex]\[ D(t) = \frac{10t(2t + 13)}{(t + 3)(t + 4)} \][/tex]

Therefore, the correct answer is:
C. [tex]\( D(t) = \frac{10t(2t + 13)}{(t + 3)(t + 4)} \)[/tex]