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A. 980 m2 B. 7 m2 C. 1,024 m2 D. 324 m2
Since you have similar trapezoids, the lengths of the sides are proportional. The areas are proprotional to the square of the scale factor.
Here is a simple example that explains my point. |dw:1370836887635:dw|
Notice the rectangle on the right has both the length and width that are double the ones on the left. The lengths on the right are in a 2:1 ratio to the lengths on the left. So far ok?
Why would it be a 2:1 ?
Like, what makes it a 2:1.
I'm only talking about my example so far. Don't think of your problem yet. My example is simpler, so follow my example. Then we'll turn to your problem.
I purposely made the rectangle on the right double the length and double the width of the rectangle on the left. So we have a scale factor for the side lengths of 2:1. The rectangle on the right is 2 times longer and 2 times wider.
Now this is what you need to understand to be able to solve your problem.
With the right rectangle having the length and width 2 times larger than the left rectangle, what happened to the area of the right rectangle compared to the area of the left rectangle? How many times larger is the right rectangle's area than the left rectangle's area?
It doubles, doesn't it?
If the rectangle is 2 times larger than the one on the left, the area should be 2 times larger.
No. Look again. I wrote the areas of the rectangles below them. What are the areas of the rectangles?
Left = 2cm^2 Right = 8cm^2
Correct. 8 cm^2 is 4 times 2 cm^2
So you see that when the dimensions of this ploygon are increased by 2 times, the area of the polygon is increased by 4 times. 4 is the same as 2^2. If you increase the dimensions of a polygin by n times, the area of the polygon increases n^2 times.
Ohh, okay! so when the shape is increased by 2, the area increases by 4?
Did you put the two rectangles together in this picture?
Look at the newest rectangle. The dimensions are now increased by 3 times compared to the original 1 cm by 2 cm rectangle. The new area is 18 cm^2 which is 9 times the original area of 2 cm^2. Why is it 9 times? It's because the increase in area is the square of the increase of the length and width. Since the increase in length and width was 3 times, the increase in area is 3^2 times, or 9 times. Sure enough, 18 cm^2 is 9 times 2 cm^2.
So is the actual problem I posted up, similar to this one?
Ok. Now lets get back to your problem. Now it becomes a simple problem. 32 m is how many times 18 m? Express it as a fraction.
Excellent. The ratio of the larger length to the smaller length is 32/18 = 16/9.
What does that mean now as far as the ratio of the larger area to the smaller area?
Not the square root, the square.
(16/9)^2 = 256/81 The larger area is larger by a factor of 256/81 which is the square of the ratio of the larger length to the smaller length.
OH! Wow... I forgot you had to square it..
Since the small area is 310 m^2, the large area is 310 * (256/81) m^2
I got 979 61/81
Now round off to the nearest whole number.
Ha, awesome thank you.
I'm gonna make another question.
Keep in mind, in a figure, if you double the side, you multiply the area by 2^2=4, and the volume by 2^3 = 8.