Questions: A manufacturer sells 70 boats per month at 23000 dollars per boat, and each month demand is increasing at a rate of 4 boats per month. What is the fastest the price could drop before the monthly revenue starts to drop? dP/dt= per month

A manufacturer sells 70 boats per month at 23000 dollars per boat, and each month demand is increasing at a rate of 4 boats per month. What is the fastest the price could drop before the monthly revenue starts to drop?

dP/dt=

per month

Transcript text: A manufacturer sells 70 boats per month at 23000 dollars per boat, and each month demand is increasing at a rate of 4 boats per month. What is the fastest the price could drop before the monthly revenue starts to drop? \[ \frac{d P}{d t}=\$ \] $\square$ per month Submit answer Next item

Solution

Solution Steps

Step 1: Define Variables

Let $ P $ be the price per boat and $ t $ be the time in months. The number of boats sold per month is given by $ 70 + 4t $.

Step 2: Revenue Function

The revenue $ R $ as a function of time is defined as: \[ R = P \cdot (70 + 4t) \]

Step 3: Differentiate Revenue

To find the rate of change of revenue with respect to time, we differentiate $ R $ with respect to $ t $: \[ \frac{dR}{dt} = P \cdot \frac{d(70 + 4t)}{dt} + (70 + 4t) \cdot \frac{dP}{dt} \] This simplifies to: \[ \frac{dR}{dt} = 4P + (70 + 4t) \cdot \frac{dP}{dt} \]

Step 4: Set Revenue Derivative to Zero

To find the condition under which revenue does not decrease, we set $ \frac{dR}{dt} = 0 $: \[ 4P + (70 + 4t) \cdot \frac{dP}{dt} = 0 \]

Step 5: Solve for $ \frac{dP}{dt} $

Rearranging the equation gives: \[ (70 + 4t) \cdot \frac{dP}{dt} = -4P \] Thus, we can express $ \frac{dP}{dt} $ as: \[ \frac{dP}{dt} = -\frac{4P}{70 + 4t} \]

Step 6: Analyze the Result

The expression $ \frac{dP}{dt} = -\frac{4P}{70 + 4t} $ indicates the rate at which the price can drop before the revenue starts to decrease. The negative sign indicates a drop in price, and the fraction shows that the rate of price drop is dependent on the current price $ P $ and the number of boats sold, which increases over time.