Let's break down and solve each of the questions shown in the image.

Question 1:

We are given that the number of items sold $x$ depends on the price $p$ according to the equation: $x = \frac{30,000}{\sqrt{4p + 1}}$ The company is increasing the price by $5 per month, and we are asked to find the rate at which revenue is changing when the price is $180.

Steps:

1. Revenue $R$ is the product of the number of items sold $x$ and the price $p$, i.e., $R = p \cdot x$.
2. Differentiate $R = p \cdot x(p)$ with respect to time $t$, using the product rule and chain rule.
3. You're given that $\frac{dp}{dt} = 5$ (price increases by $5 per month), and $p = 180$.
4. Plug the appropriate values into the derived formula to find $\frac{dR}{dt}$.

Question 2:

The revenue $R$ is given as a function of units sold $x$: $R = 1400x - 3x^2$ We know that sales are increasing at the rate of 40 units per day, i.e., $\frac{dx}{dt} = 40$, and we are asked to find how rapidly the revenue is increasing when 150 units have been sold.

Steps:

1. Differentiate $R = 1400x - 3x^2$ with respect to time $t$, using the chain rule.
2. Plug in $\frac{dx}{dt} = 40$ and $x = 150$ to find $\frac{dR}{dt}$.

Question 3:

This is a related rates problem involving two ships. At noon, ship A is 30 nautical miles west of ship B. Ship A sails west at 16 knots and ship B sails north at 16 knots. We are asked to find how fast the distance between the ships is changing at 4 PM.

Steps:

1. Let the distance between the two ships be $D$, and let the positions of the ships be modeled by the Pythagorean theorem: $D^2 = A^2 + B^2$, where $A$ is the distance traveled by ship A and $B$ is the distance traveled by ship B.
2. Differentiate $D^2 = A^2 + B^2$ with respect to time $t$.
3. You know the speeds of both ships and their positions at 4 PM.
4. Solve for $\frac{dD}{dt}$, which is the rate at which the distance between the ships is changing.

Would you like me to go into detail on any specific question? Here are 5 relative questions that can deepen your understanding:

1. What are the rules for applying the product rule in calculus?
2. How do we apply the chain rule when differentiating implicitly?
3. How does related rates differ from other types of rate problems in calculus?
4. Can you derive the revenue formula from other sales data?
5. How can we interpret the rate of change of the distance between two moving objects?

Tip: When dealing with rates of change, always start by identifying which variables depend on time and use implicit differentiation carefully.