Using the second derivative test, \( R''=-20 \lt 0 \) (for any value of \( p \)), so the critical number is a local maximum. Now, we can find the maximum of this function by finding critical numbers. Substituting this into our revenue equation, we get an equation for revenue involving only one variable: \ Simplifying to slope-intercept form gives the demand equation \ Using the point-slope form of the line, we can write the equation relating price and quantity: \ You may notice that the second step in that calculation corresponds directly to the statement of the problem: the attendance drops 50 people for every $5 the price increases. Let \(x\) and \(y\) be the dimensions of the enclosure, with \(y\) being the length of the side made of blocks. The fourth side will be built of cement blocks, at a cost of $14 per running foot.įind the dimensions of the least costly such enclosure. Three sides of the enclosure will be built of redwood fencing, at a cost of $7 per running foot. The manager of a garden store wants to build a 600 square foot rectangular enclosure on the store's parking lot in order to display some equipment. Find the dimensions of the least costly such enclosure.įirst, translate line by line into math and a picture: Text The fourth side will be built of cement blocks, at a cost of $14 per running foot. This will give us our constraint equation. For the garden store, the fixed area relates the length and width of the enclosure. The constraint equations are always equations, so they will have equals signs. Equations that relate the variables in this way are called constraint equations. If there is an equation that relates the variables we can solve for one of them in terms of the others, and write the objective function as a function of just one variable. In the garden store example again, the length and width of the enclosure are both unknown. In many cases, there are two (or more) variables in the problem. Look at the garden store example the cost function is the objective function. The objective function can be recognized by its proximity to est words (greatest, least, highest, farthest, most, …). The function we're optimizing is called the objective function (or objective equation). The process of finding maxima or minima is called optimization. Find the dimensions of the least costly such enclosure. The manager of a garden store wants to build a 600 square foot rectangular enclosure on the store’s parking lot in order to display some equipment. §2: Calculus of Functions of Two Variables.§2: The Fundamental Theorem and Antidifferentiation.§11: Implicit Differentiation and Related Rates.§6: The Second Derivative and Concavity.Here are the instructions how to enable JavaScript in your web browser. For full functionality of this site it is necessary to enable JavaScript.
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