Question 693482: if you multiply the length of both sides of a recatngle by 5, how much times bigger will the area of the new reactangle be? if you multiply the length of all three sides of a of a rectangular prism by 8, how much times bigger will the volume of the new prism be?
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! I will separate each question by placing them under two different sections. The first will attempt to define the first question, and the latter for the latter question.
1. Rectangle
If you multiply the length of both sides of a rectangle by 5, how much times bigger will the area of the new rectangle be?
This question deals with comparison. We need to know between what things does the comparison take place. The problem wants to know how many times bigger a new area would be given an original area. So, we need to define two different areas. Let A1 be the original area and let A2 be the magnified area. Starting with A1, we can say it has sides x and y. The area of a rectangle is the length times the width, so the area of A1 would be xy. Now we need to let the sides of the rectangle be multiplied by 5 to get a new rectangle. The new sides would now be 5x and 5y accordingly. The formula remains the same for the area, which the new area is 5x(5y) or 25xy. Since we have defined what the A1 and A2 are, we need to return to the original question. Comparing the original area and the new area, how many more times is the new area to the original area?
A1 = xy
A2 = 25xy or 25(xy) (25 and xy are being multiplied together)
A2 = 25(A1) (since A1 = xy we can just plug A1 in for xy).
We can see that the new area, A2, is 25 times greater than A1. This would be true for any natural value you plug in for x or y (as long as it constitutes a rectangle):
Letting x and y equal 10 and 15 respectively,
A1 = 10(15) = 150
A2 = 25(10)(15) = 3750
3750 is 25 times greater than 150.
2. Rectangular Prism
If you multiply the length of all three sides of a of a rectangular prism by 8, how much times bigger will the volume of the new prism be?
We can follow the same method as the rectangle problem, but we need to keep in mind that we are dealing with volume instead of area. Let us use V1 and V2 to represent the original volume and the new volume respectively. The formula for the volume of a rectangular prism is either lwh or Bh depending on your perspective. (The "B" in Bh is the area of the base of the prism and the "h" is the height of the prism.) So, letting the sides of prism 1, which has V1, equal x, y, and z, we have V1 = xyz. V2 would equal 8x(8y)(8z), or 512xyz. What is the comparison?
V1 = xyz
V2 = 512xyz or 512(xyz)
V2 = 512(V1).
We can see that V2 has a volume 512 times greater than V1. Again, this will remain the same for any natural value input:
Letting x = 5, y = 9, and z = 17,
V1 = 765
V2 = 512(765) = 391,680
391,680 is 512 times greater than 765.
In retrospect, we could also use a shortcut method. When dealing with area in this circumstance (that is, each side is being multiplied by the same number), we could just square the number the number of times the area is supposed to be greater. In other words, since we are trying to make the new area be 5 times greater than that of the original, we could just square 5 to get the answer 25. The same is with the volume: 8^3 = 512.
|
|
|
