Area and Perimeter of Quadrilaterals Practice Problems
Area and Perimeter of Quadrilaterals Practice
Suppose a sign manufacturing company gets tired of making rectangular billboards, and decides to put up a trapezoidal billboard instead. The top and the bottom of the billboard are horizontal, but neither of the other sides is vertical. The big sign measures 20 meters across the top edge, and 30 meters across the bottom edge. Two different companies want to advertise on the billboard, and both of them insist on having portions of equal height. What is the length of the line that divides the spaces allotted to the two advertisements? Does this represent a fair division of the sign area?
The line segment that divides the two portions is the median of the sign. Its length, therefore, is the average of 20 meters and 30 meters, which, as you should be able to guess right away, is 25 meters. Whether or not this represents a fair split of the sign area can be debated. The advertiser on the bottom gets more area than the advertiser on the top, but the ad on top is likely to be the one that drivers in passing cars and trucks look at first. By the time drivers are finished with the ad on the top, they might be passing the sign.
Refer back to Problem 3-4. Suppose the whole billboard is 15 meters high. Recall that it is a trapezoidal billboard, measuring 20 meters along the top edge and 30 meters along the bottom. The sign is divided by a median, horizontally placed midway between the top and the bottom. What fraction of the total billboard surface area, as a percentage, does the advertiser with the top half get?
The length of the median, as determined in Problem 3-4, is 25 meters, which is the average of the lengths of the bottom and the top. Thus m = 25. We are given that h = 15. The total interior area of the sign, call it A total , is therefore:
A total = 25 meters × 15 meters
= 375 meters squared
The area of the top half is found by considering the trapezoid in which m forms the base. We must use the more complicated formula—the one involving the arithmetic mean, above—in order to find the interior area of this smaller trapezoid. Let’s call this area A top . The base length of this trapezoid is 25 meters, while the length of the top is 20 meters. The height is 7.5 meters, half the height of the whole sign. Thus, A top is found by this calculation:
A top = [(25 meters + 20 meters)/2] × 7.5 meters
= (45 meters/2) × 7.5 meters
= 22.5 meters × 7.5 meters
= 168.75 meters squared
The fraction of the total area represented by the top portion of the sign is the ratio of A top to A total . That is 168.75 meters squared divided by 375 meters squared, or 0.45. Therefore, the top advertiser gets 45 percent of the total interior area of the sign.
Suppose the billboard is a rectangle rather than a trapezoid, measuring 25 meters across both the top and the bottom. Suppose the sign is 15 meters tall, and is to be split into upper and lower portions, one for each of two different advertisers, Top Inc. and Bottom Inc. Suppose that the executives of Bottom Inc. demand that Top Inc. only get 45 percent of the total area of the sign because of Top Inc.’s more favorable viewing position. How far from the bottom of the sign should the dividing line be placed?
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