A pentagonal prism has dimensions that are four times the dimensions of a similar pentagonal prism. So its volume is _____times the volume of the smaller prism.
A) 64
B) 8
C)16
D)4
Would it be A) 64?
Next Question: If the dimensions of a cylinder are doubled the surface area will be _______? Would it be doubled?
Thank you for the help
Volumes are (linear ratio)³ and
areas are (linear ratio)²
1. V = 2.83*r^2*h.
V1/V2 = 2.83*(4r)^2*4h/(2.83*r^2*h)
V1/V2 = 16r^2*4h/(r^2*h = 64.
2. A1 = 2pi*r^2 + 2pi*r*h
A1 = 2pi*r(r+h)
A2/A1 = (2pi*2r(2r+2h))/(2pi*r(r+h)
A2/A1 = (4pi*r(2r+2h))/(2pi*r(r+h)
A2/A1 = 2(2(r+h)/(r+h) = 4.
2. A shorter Method
A1 = 2pi*r(r+h). Let r = 1, and h = 2.
A1 = 6.28*1(1+2) = 18.84 Sq. Units.
Double r, and h.
A2 = 6.28*2(2+4) = 75.36 Sq. Units.
A2/A1 = 75.36/18.84 = 4.
64, Quadrupled
Yes, you are correct. The volume of a three-dimensional shape is directly proportional to the cube of the scale factor. In this case, since the dimensions of the larger pentagonal prism are four times the dimensions of the smaller prism, the scale factor is 4. Therefore, the volume of the larger prism will be 4^3 = 64 times the volume of the smaller prism. So, the answer is A) 64.
For the second question, the surface area of a cylinder is directly proportional to the square of the scale factor. Since the dimensions of the cylinder are doubled, the scale factor is 2. Therefore, the surface area of the larger cylinder will be 2^2 = 4 times the surface area of the smaller cylinder. So, the answer is not doubled, it is actually four times the original surface area.