Questions: PARTICIPATION ACTIVITY #23 A coffee shop determines that the daily profit on scones obtained by charging s dollars per scone is P(s)=-20 s^2+140 s-30. The coffee shop currently charges 3.25 per scone. Find P'(3.25), the rate of change of profit when the price is 3.25. P'(3.25)= Number per scone.

PARTICIPATION ACTIVITY #23

A coffee shop determines that the daily profit on scones obtained by charging s dollars per scone is P(s)=-20 s^2+140 s-30. The coffee shop currently charges 3.25 per scone. Find P'(3.25), the rate of change of profit when the price is 3.25.
P'(3.25)=
Number
per scone.

Transcript text: PARTICIPATION ACTIVITY \#23 A coffee shop determines that the daily profit on scones obtained by charging $s$ dollars per scone is $P(s)=-20 s^{2}+140 s-30$. The coffee shop currently charges $\$ 3.25$ per scone. Find $P^{\prime}(3.25)$, the rate of change of profit when the price is $\$ 3.25$. $P^{\prime}(3.25)=\$$ Number $\square$ per scone.

Solution

Solution Steps

To find the rate of change of profit when the price is $3.25 per scone, we need to calculate the derivative of the profit function $ P(s) = -20s^2 + 140s - 30 $ and then evaluate it at $ s = 3.25 $. The derivative $ P'(s) $ represents the rate of change of profit with respect to the price of a scone.

Step 1: Define the Profit Function

The profit function for the coffee shop is given by

\[ P(s) = -20s^2 + 140s - 30 \]

where $ s $ is the price charged per scone.

Step 2: Calculate the Derivative

To find the rate of change of profit with respect to the price, we compute the derivative $ P'(s) $:

\[ P'(s) = \frac{d}{ds}(-20s^2 + 140s - 30) = -40s + 140 \]

Step 3: Evaluate the Derivative at $ s = 3.25 $

Next, we evaluate the derivative at $ s = 3.25 $:

\[ P'(3.25) = -40(3.25) + 140 = -130 + 140 = 10 \]