Question

question #5 is “what is the secret?”.

The secret is that the squares have the largest area when the perimeters are all the same.

1. Does the reasoning you used for answering Question 5 constitute a complete proof? How would you show that no other rectangle with a perimeter of 100 meters will have an area larger than the rectangle you discovered when answering Question 5? Do this in two ways. a. Give a geometric proof using the following figure. 25-a 25+ a 25 25- a 25 25 F. HYPERBOLAS In Section C, you worked with a feasible region that has a curved border That is not uncommon in real-world problems. In this section, you will look at another problem involving a curve 'N gold Number of People In the 19th century, many adventurers traveled to North America to search for gold. A man named Dan Jackson owned some land where gold had been found. Instead of digging for the gold himself, he rented plots of land to the adventurers. The "rent" was to give Dan 50% of any gold found on the plot of land. Dan gave each adventurer four stakes and a rope that was exactly 100 meters long. Each adventurer had to use the stakes and rope to mark off a rectangle with north-south and east-west sides. 1. Did everyone get the same area to dig for gold? Explain your answer. Mathematics in Context Get the Most Out of It 39 Area (in km2) There are many different rectangles you can make that have a perimeter of 100 meters У 100 90 2. a. On graph paper, draw at least five different rectangles with perimeters of 100 meters. Draw your rectangles to scale, using one grid unit to represent five 80 70 60 meters. 50 b. Cut out the rectangles. Tape them on a graph so the lower-left coners lie on the origin. Use the same grid size as in part a, as shown at the right. 40 30 20 10 X 010 20 30 40 50 60 70 80 90 100 Length (in meters) For cach rectangle, only the upper-right vertex will not touch an axis. 3. You should notice that all the upper-right vertices of your rectangles seem to lie on a line. What is the cquation of that line? Use x and y as the dimensions of the rectangles. 4. Calculate the area of each rectangle, and organize all of your information in a table One of the diggers discovered that one kind of rectangle always had the greatest area. He decided to sell the secret to other diggers. 5. What was the secret? Width (in meters)

Answer 1

Proof Suppose side have square of perimeter omm we =2Sm and breddh a lengin re changle of square 25-a Now, cut from the 25-a a 25-a a a a 25-a 25-a 25-a 2S 2 S 25 Thus by rearrange ment Stai'ned parimeter loo 25 a rectangle with faasap we a 25-4 However area Area of square is reduced by a2 is marimum et rectangle be25-a) X (25-a) dimensin je (25-a) m width and (Sta) length

Tts peaimeter 25+a) t a5-412 =100m (25ta) (25-a) 25-a R5 Its a rea 11 2 0of roctangle is perimetar of suare dearly less then area Thue area