91Ó°ÊÓ

Calculate the consumers'surplus at the indicated unit price \(\bar{p}\) for each of the demand equations. $$q=50-3 p ; \bar{p}=10$$

Calculate the consumers'surplus at the indicated unit price \(\bar{p}\) for each of the demand equations. $$q=20-0.05 p^{2} ; \bar{p}=5$$

Complete the given table with the values of the 3 -unit moving average of the given function. HINT [See Example 3.] $$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \boldsymbol{s}(\boldsymbol{x}) & 2 & 9 & 7 & 3 & 2 & 5 & 7 & 1 \\ \hline \bar{s}(\boldsymbol{x}) & & & & & & & & \\ \hline \end{array} $$

Calculate the consumers'surplus at the indicated unit price \(\bar{p}\) for each of the demand equations. $$q=500 e^{-0.5 p}-50 ; \bar{p}=1$$

Profit Your monthly profit on sales of Avocado Ice Cream is rising at an instantaneous rate of \(10 \%\) per month. If you currently make a profit of \(\$ 15,000\) per month, find the differential equation describing your change in profit, and solve it to predict your monthly profits. HINT [See Example 3.]

Newton's Law of Cooling For coffee in a ceramic cup, suppose \(k \approx 0.05\) with time measured in minutes. (a) Use Newton's Law of Cooling to predict the temperature of the coffee, initially at a temperature of \(200^{\circ} \mathrm{F}\), that is left to sit in a room at \(75^{\circ} \mathrm{F}\). (b) When will the coffee have cooled to \(80^{\circ} \mathrm{F}\) ? HIIIT [See Example 4.]

Newton's Law of Cooling For coffee in a paper cup, suppose \(k \approx 0.08\) with time measured in minutes. (a) Use Newton's Law of Cooling to predict the temperature of the coffee, initially at a temperature of \(210^{\circ} \mathrm{F}\), that is left to sit in a room at \(60^{\circ} \mathrm{F}\). (b) When will the coffee have cooled to \(70^{\circ} \mathrm{F}\) ? HINI [See Example 4.]

Market Saturation You have just introduced a new flat-screen monitor to the market. You predict that you will eventually sell 100,000 monitors and that your monthly rate of sales will be \(10 \%\) of the difference between the saturation value of 100,000 and the total number you have sold up to that point. Find a differential equation for your total sales (as a function of the month) and solve. (What are your total sales at the moment when you first introduce the monitor?)

Electric Circuits The flow of current \(i(t)\) in an electric circuit without capacitance satisfies the linear differential equation $$ L \frac{d i}{d t}+R i=V(t) $$ where \(L\) and \(R\) are constants (the inductance and resistance respectively) and \(V(t)\) is the applied voltage. (See figure.) If the voltage is supplied by a 10 -volt battery and the switch is turned on at time \(t=1\), then the voltage \(V\) is a step function that jumps from 0 to 10 at \(t=1: V(t)=5\left[1+\frac{|t-1|}{t-1}\right]\). Find the current as a function of time for \(L=R=1 .\) Use a grapher to plot the resulting current as a function of time. (Assume there is no current flowing at time \(t=0 .\) [Use the following integral formula: \(\int\left[1+\frac{|t-1|}{t-1}\right] e^{t} d t=\) \(\left.\left[1+\frac{|t-1|}{t-1}\right]\left(e^{t}-e\right)+C .\right]\)

Vales My financial adviser has predicted that annual sales of Frodo T-shirts will continue to decline by \(10 \%\) each year. At the moment, \(I\) have 3,200 of the shirts in stock and am selling them at a rate of 200 per year. Will I ever sell them all?