A casino hotel manager has approximated demand for two-room suites to be $$ p=3000-\frac{q^{2}}{24} $$, where $$ p $$ is the price (in dollars) and $$ q $$ is the quantity demanded. Use implicit differentiation to find and interpret $$ \frac{d q}{d p} $$ when $$ q=4 $$. Choose the correct answer below and fill in the answer box to complete your answer. A. $$ \frac{d q}{d p}=\square $$, which means that if the price is increased by $$ \$ 1 $$, the quantity demanded decreases by approximately $$ \left|\frac{d q}{d p}\right| $$ suites. B. $$ \frac{d q}{d p}=\square $$, which means that if the price is increased by $$ \$ 1 $$, the quantity demanded increases by approximately $$ \left|\frac{d q}{d p}\right| $$ suites.

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UpStudy AI · Accepted Answer

To find $$ \frac{d q}{d p} $$ using implicit differentiation, we start with the demand equation:

$$
p = 3000 - \frac{q^2}{24}
$$

We will differentiate both sides with respect to $$ p $$ and $$ q $$.

1. Differentiate the left side with respect to $$ p $$:
   $$
   \frac{d}{dp}(p) = 1
   $$

2. Differentiate the right side with respect to $$ p $$ using the chain rule:
   $$
   \frac{d}{dp}\left(3000 - \frac{q^2}{24}\right) = 0 - \frac{1}{24} \cdot 2q \cdot \frac{dq}{dp} = -\frac{q}{12} \cdot \frac{dq}{dp}
   $$

Now we set the derivatives equal to each other:
$$
1 = -\frac{q}{12} \cdot \frac{dq}{dp}
$$

Next, we solve for $$ \frac{dq}{dp} $$:
$$
\frac{dq}{dp} = -\frac{12}{q}
$$

Now we need to evaluate $$ \frac{dq}{dp} $$ when $$ q = 4 $$:
$$
\frac{dq}{dp} = -\frac{12}{4} = -3
$$

Thus, we have:
$$
\frac{dq}{dp} = -3
$$

### Interpretation:
This means that if the price is increased by $$ \$ 1 $$, the quantity demanded decreases by approximately $$ 3 $$ suites.

Now, we can fill in the answer box:

A. $$ \frac{d q}{d p} = -3 $$, which means that if the price is increased by $$ \$ 1 $$, the quantity demanded decreases by approximately $$ \left| -3 \right| = 3 $$ suites.
$$ \frac{dq}{dp} = -3 $$, which means that if the price increases by \$1, the quantity demanded decreases by approximately 3 suites.

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