**Word problems**

The application of problems of the derivative aim to form an expression for the question using a single variable.

**Example 1**

A rectangular frame is formed from iron wire, the surface area of which is desired to be as large as possible. 40 cm of iron wire is used. What are the dimensions of the frame?

Solution

Draw the situation and form an expression for the area.

The area of a rectangle is* A = xy*. In order to derive this, there must be only one variable in it. The circumference is *40 cm*, so

*2x + 2y = 40 *

and we can solve for y

* y = 20-x*

So we get expression for the area

The graph of the area expression is a downward-opening parabola, so it gets its maximum value at its peak, that is, at the zero point of the derivative.

The surface area of the frame is largest when the side length is 10 cm. The frame is square.

**Example 2**

Liisa-Petter's railway company Vetelät runoilijat (Flappy poets), VR for short, planned to change ticket prices. The basic ticket cost was €10 and there were 1200 passengers per day. The consultancy firm Kaikki rahat putsataan (All Money is Cleaned), or KRP for short, was led by its director Klaus-Heidi, who had conducted a study showing that a one euro price increase reduced passenger by 50 per day. Correspondingly, the reduction in the price by one euro increased passengers by 50 per day. In this case, what would be the price of the ticket so that the value of a day's ticket sale is at its maximum?

Solution

Generate an expression for the daily sale.

Let the price increases of the euro be denoted with the variable *x*, then the price of the ticket is *10 + x *and the number of passengers is *1200-50x*

Daily sale

The graph of the daily ticket sales expression is a downward-opening parabola and it gets its maximum value at the peak, that is at the zero point of the derivative.

When the ticket price is €10 + €7 = €17, the value of the day sale is the highest. In this case, there are 850 passengers and the value of the daily sale is €14450 .

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