Problem: Write an article based on this summary: Find the first derivative of the revenue function. Set the derivative equal to 0. Solve for the number of items at the 0 value. Calculate the maximum price. Combine the results to calculate maximum revenue. Summarize the results.

Answer: In calculus, the derivative of any function is used to find the rate of change of that function. The maximum value of a given function occurs when the derivative equals zero. So, to maximize the revenue, find the first derivative of the revenue function. Suppose the revenue function, in terms of number of units sold, is R(q)=500q150q2{\displaystyle R(q)=500q-{
rac {1}{50}}q^{2}}. The first derivative, therefore, is: R(q)=500250q{\displaystyle R^{\prime }(q)=500-{
rac {2}{50}}q} For a review of derivatives, see the wikiHow article on how to Take Derivatives. When the derivative is zero, the graph of the original function is at either a peak or a trough. This will be either the maximum or minimum value. For some higher level functions, there may be more than one solution to the zero derivative, but not a basic price-demand function. R(q)=500250q{\displaystyle R^{\prime }(q)=500-{
rac {2}{50}}q} 0=500250q{\displaystyle 0=500-{
rac {2}{50}}q} Use basic algebra to solve the derivative for the number of items to sell where the derivative is equal to zero. This will give you the number of items that will maximize the revenue. 0=500250q{\displaystyle 0=500-{
rac {2}{50}}q} 250q=500{\displaystyle {
rac {2}{50}}q=500} 150q=250{\displaystyle {
rac {1}{50}}q=250} q=50250{\displaystyle q=50*250} q=12,500{\displaystyle q=12,500} Using the optimal number of sales from the derivative calculation, you can enter that value into the original price formula to find the optimal price. p=500150q{\displaystyle p=500-{
rac {1}{50}}q} p=50015012,500{\displaystyle p=500-{
rac {1}{50}}12,500} p=500250{\displaystyle p=500-250} p=250{\displaystyle p=250} After you have found the optimal number of sales and the optimal price, multiply them to find the maximum revenue. Recall that R=pq{\displaystyle R=p*q}. The maximum revenue for this example, therefore, is: R=pq{\displaystyle R=p*q} R=(250)(12,500){\displaystyle R=(250)(12,500)} R=3,125,000{\displaystyle R=3,125,000} Based on these calculations, the optimal number of units to sell is 12,500, at the optimal price of $250 each. This will result in a maximum revenue, for this sample problem, of $3,125,000.

Problem: Write an article based on this summary: Collect some hot water in a jar. Dissolve a crystal chemical in hot water until the solution is saturated. Filter the solution. Place a small amount of the saturated solution on a plate. Suspend the seed crystal in a clean jar. Pour the saturated solution into the container with crystal seed. Refresh the solution every two weeks. Coat the surface of the crystal with a lacquer.

Answer:
Collect approximately 100mL (.44 cups) of hot water. The water doesn't need to be boiling. Tap water will work fine. Just leave the tap on for a minute until the water is very hot. To get the purest crystals, begin with a hot solution and allow it to cool slowly by leaving it at room temperature. This slow process will allow the crystal to reject any impurities as the molecules are arranged. The solution will be saturated when it can no longer dissolve more crystals. You will probably need about 60 grams (2.1oz.) of chemical. Add more until the crystal no longer dissolve. There are a number of chemicals you can select depending on the color of crystal you want to make. Some of the chemicals, like borax, may be around your house or at the local grocery store. Other ingredients are available to order online. Borax can be used to produce clear crystals. Copper sulfate can be used to create bright blue crystals. Purple chrome alum can be used to create purple crystals. Pouring the solution through a filter will help to get rid of any excess crystals that could not be dissolved in the solution. Take a plastic funnel and place a coffee filter in the funnel. Then pour the solution into a separate container through the funnel. Now you should have a clear solution with no crystals floating around in the liquid. Leave it for 24 hours. Once you have some small crystals scrape them off the plate with a butter knife. You can use these small crystals as seeds to produce larger crystals. You can poke two small holes in the lid and tie one end around the seed crystal and tie the other end to the top of the jar. The container should be clean so the crystals attach to your seed crystal and not the container. Pick any container that has a good cover that seals. Or make a cover by suspending your crystal from a flat piece of cardboard. Evaporation will help speed up the formation of crystals. Wait as the solution evaporates and your crystal grows larger. Be patient. It can take weeks to grow a large crystal. Keep the object in a cool, shaded area. Make sure it is someplace where the container won't be moved or disturbed too often. Over time, small crystals will begin to form in the solution, so it will be necessary to put your seed crystals into a new container. Remove the seed crystals and place them on a paper towel. Pour the solution through a coffee filter into the new container. Scrape out any crystals in the old container. You can save them and use them as seed crystals for the next time you want to grow a crystal. If you need more solution, dissolve some more crystal powder in hot water and add the solution to your new container. To protect the crystal from being destroyed by exposure to moisture, cover the crystal with a coat of lacquer. Use a brush to apply an even coat over the entire surface. You can also try coating the surface with a layer of nail polish to protect it.