Length, Width, and Height of a rectangular solid are tripled? (more in additional details) PLEASE HELP ME!!!?

Question

If the length, width, and height of a rectangular solid are all tripled, the lateral area is multiplied by ?, the total area is multiplied by ?, and the volume is multiplied by ?.

Obviously, I have no remote idea, so explanations would be much appreciated along with the answers. Thanks!

Answer 1

a. 9
b. 9
c. 27

Let a = length
b = width
c = height

Then in the enlargement,
3a = length
3b = width
3c = height

The lateral area is the sum of the areas of the faces on the sides (not top or bottom) of the figure. Each of these faces is a rectangle. To find the area of a rectangle, A = l*w
Notice that in the enlargement, the area of EVERY face was increased by 3*3 = 9, so both the lateral area and the total area is multiplied by 9.

To verify, calculate the area of each face -- if you need to, choose random values to use for length, width and height!



For the volume, V = l*w*h
For the original figure,

V = a*b*c

For the enlargement,
V = (3a)*(3b)*(3c) = 27a*b*c

So the volume was multiplied by 27.


In general, when all the dimensions are multiplied by the same factor, the area increases by the square of that factor (3^2 = 9) and the volume increases by the cube of the factor (3^3 = 27).

I hope this helps it all make sense!

Answer 2

Let the Length, Width and Height be denoted by L, W and H respectively.
I suppose you mean total surface area whwn you say lateral area

For this solid Volume, V1=L x W x H
Total Surface Area, A1=2(LxW + WxH + HxL)

For the new Solid
Length=3L
Width=3W
Height=3H
Volume=3L x 3W x 3H, V2=27 x L x W x H=27 V1
Total Surface Area, A2=2(3Lx3W + 3Wx3H + 3Hx3L)
=2(9LxW + 9WxH + 9HxL)
=9 x 2(LxW + WxH + HxL)
=9 A1

Thus the volume will increase 27 times, and lateral area will increase 9 times the original.

In general, for a second degree equation and a factor k, the increase will be k² times and for a third degree the increase will be k³ times, provided k remains constant for all the sides.

If the factors are k1, k2, k3,k4, k5,...... and so on
the multiplying factor would be k1 x k2 x k3 x k4 x k5 ..........

Answer 3

If l, w and h are the length, width and height of a rectangular solid (base is a rectangle), and if each of them is tripled, we have a new solid of 3l, 3w and 3h.

Now lateral area means area of the sides. There are 4 sides and the area is given by 2lh + 2wh (two rectangles are of sides l and h and 2 rectangles are of sides w and h)

Instead of l, we have 3l and instead oh h we have 3h etc.

so, the area is 9 times more.

The total area includes the area of the top and bottom (l x w) and again becomes 9 times more. So, total area also increases 9 times.

Volume of the solid = l x w x h

Volume of the new solid = 3l x 3w x 3h = 27 lwh

So, volume increases 27 times.

Answer 4

let first length=l
width=b
hight=h
after tripled we can write length=3 x l
width= 3 x b
hight=h x3
old area(for any area same case i site an exampel)
=l x b
new area=3 x l x 3x b=9 x l x b=9 x old area
so areas are multiplied by 9.(lateral area,total area all)
for volume
old=l x b x h
new=3 x l x 3 x b x 3 x h=27 x l x b x h=27 x old
so vol. is multiplied with 27

Answer 5

Assume width = 1 then volume = 6, so, width = 27 because volume = 162, because 162/6 = 27. length = 2* width = 54 & height = 3 * width = 81. We know that there are 6 faces & we know the side lengths so we may work out the surface areas of the 6 faces & add them together giving us the answer.

Answer 6

Area is proportional to length^2.
Lateral area and total area are increased to 3^2 or 9 times the original areas.

Volume is proportional to length^3
Volume is increased to 3^3 or 27 times the original volume.

Answer 7

Lateral area is L*B

If each side is tripled then 3L*3B i.e it wud increase 9 times

for total area Sum of area of all 6 sides.
Area of one side has increased 9 times therefore for 6 sides 9*6 i.e 54 times

For volume

L*B*H
3L*3B*3H

which makes 27 times increase in volume

Answer 8

area = 3L (3w) = 9 Lw the area became 9 times
volume = 3L (3w)(3H) = 27 LwH . . volume becomes 27 times