Question

If the dimensions of a prism with surface area 42 m^2 and volume 18 m^3 are enlarged by a factor of 4, what is the surface area and volume of the prism?

Answer 1

Let the original dimensions of the prism be x, y, and z. Then, its surface area and volume are given by:

Surface area = 2xy + 2xz + 2yz = 42
Volume = xyz = 18

We can use these equations to solve for the individual dimensions. First, we can solve for z in terms of x and y from the surface area equation:

2xy + 2xz + 2yz = 42
2z(x + y) = 42 - 2xy
z(x + y) = 21 - xy

Next, we can substitute this expression for z into the volume equation and solve for x and y:

xyz = 18
xy(21 - xy) = 18
21xy - xy^2 = 18
xy^2 - 21xy + 18 = 0
(xy - 2)(xy - 9) = 0

So, either xy = 2 or xy = 9. Since the dimensions of a prism must be positive, we can only use the solution xy = 9. Solving for x and y gives:

xy = 9
x + y + z = 21/xy = 7/3
x = 3, y = 3, z = 1/3

Now, we can scale up the dimensions by a factor of 4 to get the new dimensions of the enlarged prism:

x' = 4x = 12
y' = 4y = 12
z' = 4z = 4/3

The surface area and volume of the enlarged prism are then given by:

Surface area = 2x'y' + 2x'z' + 2y'z' = 576
Volume = x'y'z' = 64

Therefore, the surface area of the enlarged prism is 576 m^2 and the volume is 64 m^3.

Answer 2

SA = 2lw + 2lh + 2wh = 42 m^2 ... (1)

And the volume of the original prism is given by:

V = lwh = 18 m^3 ... (2)

If you enlarge the dimensions of the prism by a factor of 4, then the new dimensions will be 4l, 4w, and 4h. The new surface area of the prism will be:

New SA = 2(4l)(4w) + 2(4l)(4h) + 2(4w)(4h)

= 32lw + 32lh + 32wh

= 32(2lw + 2lh + 2wh)

= 32(42)

= 1344 m^2

So, the surface area of the enlarged prism is 1344 m^2.

Similarly, the new volume of the prism will be:

New V = (4l)(4w)(4h)

= 64lwh

= 64(18)

= 1152 m^3

So, the volume of the enlarged prism is 1152 m^3.

Therefore, the surface area of the prism is 1344 m^2 and the volume of the prism is 1152 m^3 after the dimensions are enlarged by a factor of 4.

*IG:whis.sama_ent