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The table below shows the cost and revenue functions in dollars for manufacturing [tex]\( q \)[/tex] items of a product.
[tex]\[
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$q$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
$R(q)$ & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 \\
\hline
$C(q)$ & 4 & 8 & 12 & 15 & 16 & 18 & 24 & 29 \\
\hline
\end{tabular}
\][/tex]

Find the marginal profit at [tex]\( q = 2 \)[/tex].



Answer :

GinnyAnswer
To solve for the marginal profit at [tex]\( q = 2 \)[/tex] using the given data, we follow these steps:

1. Understand the Marginal Profit Formula:
The marginal profit at a certain quantity [tex]\( q \)[/tex] is defined as the change in revenue minus the change in cost when producing one additional unit. Mathematically, if [tex]\( P(q) \)[/tex] represents profit at quantity [tex]\( q \)[/tex], then the marginal profit at [tex]\( q \)[/tex] can be calculated as:
[tex]\[ MP(q) = \Delta R(q) - \Delta C(q) \][/tex]
where [tex]\( \Delta R(q) \)[/tex] is the change in revenue and [tex]\( \Delta C(q) \)[/tex] is the change in cost:
[tex]\[ \Delta R(q) = R(q+1) - R(q) \][/tex]
[tex]\[ \Delta C(q) = C(q+1) - C(q) \][/tex]

2. Identify [tex]\( q = 2 \)[/tex] in the table:
At [tex]\( q = 2 \)[/tex]:
[tex]\[ R(q) = 10 \quad \text{and} \quad R(q + 1) = 15 \][/tex]
[tex]\[ C(q) = 12 \quad \text{and} \quad C(q + 1) = 15 \][/tex]

3. Calculate the changes in revenue and cost:
[tex]\[ \Delta R(2) = R(3) - R(2) = 15 - 10 = 5 \][/tex]
[tex]\[ \Delta C(2) = C(3) - C(2) = 15 - 12 = 3 \][/tex]

4. Compute the marginal profit:
[tex]\[ MP(2) = \Delta R(2) - \Delta C(2) = 5 - 3 = 2 \][/tex]

Thus, the marginal profit at [tex]\( q = 2 \)[/tex] is [tex]\( 2 \)[/tex] dollars.

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