A regular heptagonal prism has a volume of 52m^3. If each one of its edges is doubled in size, what would be its new volume? I know the answer its 416m^3 but I don't know how to solve it. Thanks!! :)

Question

jack1 · Answer

formula for volume of a prism = area of base x ht

so what is happening to the base area if you double one of the sides?

anonymous · Answer

it increases

jack1 · Answer

yeah... but by a factor of...?
ie whats the formula for area of a heptagon?

anonymous · Answer

its 5 x apothem x perimeter?

jack1 · Answer

yeah, that's one formula
but in this case i'd go with the regular polygon formula

Area = (1/4 x s^2 x N) / (tan [180/N])

anonymous · Answer

but I don't get how do you know by what factor does it increases is it 2 because it says that it double the sides or do you count the sides? How do you know?

jack1 · Answer

now it looks complex, but it's really simple and it works for all regular polygons
the key is s = side length; and N = number of sides
so for a heptagon: 7 sides

Area = (1/4 x s^2 x 7) / (tan [180/7])

Area = (7/4 x s^2 ) / (tan [180/25.71])

Area = (1.75 x s^2) / (0.657)

therefore
Area of heptagon = 2.66 * s^2

jack1 · Answer

so assume a side length of 10 and ht of 10 for a sec:
Area of base = 2.66 x 10^2 = 2.66 x 100
ht remains the same, so volume = 2660

now if you double side length:
new side length of 2x10 
so new Area of base = 2.66 x (2x10)^2
                              = 2.66 x 20^2
                              = 2.66 x 400
                              = 1064

and ht of 10 remains the same, so volume = 10640

as u can see, increasing side length by 2times increased volume by 4times (2^2)
so volume varies with the square of side length

jack1 · Answer

ie if you increased side length to 4 times the original size, the change in volume would be 16 times greater (4x4 = 16)

jack1 · Answer

from your question it states "if each one of the edges doubles in size", therefore side length doubles

anonymous · Answer

oh I get it so if it 3 times the original size the change in volume would be 9 (3X3=9)

jack1 · Answer

yep, u got it
so volume varies with (side length) sqrd

jack1 · Answer

Mathematically from Q:
so original volume of 52: (from q)
assume height of 4
original side length would be
Area x ht = volume
2.66 x s^2 x 4 = 52
2.66 x s^2 = 13
s^2 = 4.8872180451
s =  2.2107

now if you double the side length of the heptagon:
s = 2.2107 x 2    = 4.4114

Volume = area of base x ht
            = 2.66 x s^2 x 4
            = 2.66 x (4.4114)^2 x 4
            = 207.01 ish which is roughly 208 which is 4 times the volume of the original

anonymous · Answer

ok

jack1 · Answer

aw damm, ëach one of its edges is doubled in size
so the height is also doubled
so :
            = 2.66 x s^2    x  8
            = 2.66 x (4.4114)^2    x   8
            = 414.11 ish which is roughly 416 (as i've been rounding) which is 8 times the volume of the original

jack1 · Answer

so summation:
double height = 2x volume
double side length of the base = 4x volume
do both = 4x2x volume = 8x volume

anonymous · Answer

oh ok it make sense now! Thank you so much now I think I get it, thank you I really didn't know how to solve it

jack1 · Answer

all good, sorry it took so long dude

anonymous · Answer

It's ok I didn't mind waiting but thank you for helping me :)

jack1 · Answer

nah any time dude, good luck and slaters ;D

anonymous · Answer

thanks! later :D

directrix · Answer

Theorem: If two solids are similar, the cube of the scale factor of the two solids is equal to the ratio of the volumes.

 (1/2)^3 = 52/x where x is the new volume and 1/2 is the scale factor.

 1/8 = 52/x

 x = 8 * 52 = ?

anonymous · Answer

thanks Directrix :)

directrix · Answer

You are welcome.