Volume and mass

by Ludger, Dorian, Finn-Niklas, Marie, Anna, Nico, Miran

… of chocolate solids

The problem

A super chocolate should have a volume of 250 cm³. Choose a form of a solid and find suitable values for the measurements.What about the weight of this piece of chocolate?

Ideas 1. a cube2. a rectangular prism3. a prism with 4 edges4. a prism with 3 edges 5. a squared pyramid6. a cylinder7. a cone8. a sphere 9. a truncated cone

1. A chocolate cube

= 6,299605 cm cmThe cube will have a lenght of 6,3 cm.

2. Rectangular chocolate prisms a) We choose a= 15 cm and b= 10 cm

Then we can calculate the height of the chololate bloc: V=a*b*c

250 cm³ = 15cm*10 cm*c , c= 1cm1,66666 cm The chocolate bloc will have a height of 1,7 cm.

b) We choose a =18 cm and b = 3cm Then we can calculate the height of the chololate bloc: V=a*b*c 250 cm³ = 18cm*3 cm*c , ccm The chocolate bloc will have a height of 4,63 cm.

18 3

4,6

1,6

1015

There are many more combinations. Choose two values, then you can calculate the third.

3. Chocolate prisms (4 edges)

1. A parallolopid:

Formula:

eca

hV

)2

(

cmca

cmcmca

cm

3

216

³250122

5,2

Chosen:a =6 cmh =2,5 cme =12 cmCalculated: b = 10,6 cm

e

a

hc

1. A trapeziod:

Formula:

chaV

³2505,73

265

²3

133

³2505,7

cmcmcmcm

cmha

cmcmha

Chosen:a =6 cmc =7,5 cmCalculated: b =6,6 cm

4. Chocolate prisms (3 edges)Chosen:h =8 cmH =10 cmCalculated: g =6,25 cm

Chosen:h =4 cmH =18 cmCalculated: g =4,17 cm

cmh

cmcmHh

cmHhg

Hhg

VFormula

Example

25,6

³250²2:)8(

³2502

2:

:1

cmh

cmh

cmcmh

cmHhg

Hhg

VFormula

Example

7

³25036

³250²182:)4(

³2502

2:

:2

There are many more combinations. Choose two values, then you can calculate the third.

H=10 cm

H=18 cm

g= 6,25 cm

g= 4,17 cm

h

h

5. Squared pyramides of chocolateExample 1:

Example 2:

30

25|:75025

3|250²53

1

²31

h

h

h

haV Chosen:a =5 cmCalculated: h =30 cm

1,7

|50²

15|:750²15

3|25015²3

1

²31

a

a

a

a

haV Chosen:h =15 cmCalculated: a 7,1 cm

h=15 cm

A=7,1 cm

h=15 cm

a=5 cm

There are many more combinations. Choose the value of a or h, then you can calculate the third.other one.

6. Chocolate cylindersChosen:h =5 cmCalculated: r 4cm

cmr

cmr

cmr

cmcmr

cmhr

hrVFormula

Example

99,3

|²92,15²

|:²50

³2505

³250

:

:1

2

2

2

2

cmr

cmr

cmr

cmcmr

cmhr

hrVFormula

Example

427885401,2

|²894629522,5²

|:²51851852,18

³2505,13

³250

:

:1

2

2

2

2

Chosen:h =13,5 cmCalculated: r 2,4cm

There are many more combinations. Choose one value r or h, then you can calculate the second one.

h= 13,5 cm

r= 2,4 cm

h= 5 cm

r= 4 cm

7. Chocolate conesChosen:h =10 cmCalculated: r 4,9 cm

Chosen:r =3 cmCalculated: h 26,5 cm

cmh

cmcmh

mcmh

cmhr

hrV

5,26

)9(|:³7509

3|³2503

9

³2503

²3

²

There are many more combinations. Choose one value r or h, then you can calculate the second one.

r=4,9 cmr=4,9 cm

r= 3cm

h=10 cm

h=10 cm

8. A chocolate sphere

r=3,9 cm

9. Chocolate as a truncated cone: Help by a facebook posting?

How to find suitable measurements? 1

Ludger: “It seems to be difficult. We have three unknown variables, the height, the radius at the bottom and the upper radius and we have only one information, the volume has to be 250 cm³! Let’s take 5 cm for the height. I’ll look for a formula to calculate the volume of a truncated cone!“

)(3

1 22 rrRRhV

How to find suitable measurements? 2

Ludger: “I assume for the height 5 cm. For R I could try out 5 cm or 4 cm. Let’s start!”

h=5cm, R=5 cm:

74648,225

7468,47255

5

3250525

250)525(53

1

2

2

2

2

rr

rr

rr

rr

h=5cm, R= 4 cm:

74648,314

7468,47164

5

3250416

250)416(53

1

2

2

2

2

rr

rr

rr

rr

Further calculationsLudger (grade 8): “So far, ok! But we did not yet learn how to solve these equations.”

Dorian (grade 9):”I will show you how to solve this quadratic equation by using a formula.”

88,788,2

3849,52

5

7468,225,22

5

7468,222

5

2

5

07468,225

21

2/1

22/1

2

2/1

2

rr

r

r

r

rr

9789,79789,3

9789,52

7468,3142

7468,2222

4

07468,314

21

2/1

2/1

22/1

2

rr

r

r

r

rr

Measurements for these truncated cones of 250 cm³:

5,8cm

10cm

5cm

7,96cm

8 cm

5cm

This a cylinder! We got the same value for the radius of a cylinder.

Other assumptions for height and R lead to much more other truncated cones and values.

4. The mass of the super chocolate

The density of choclate ist 1,3 g/cm³.

So a solid of 250 cm³- full of chocolate- has a mass of 1,3g/cm³*250cm³ =325 g.

We found the density of chololate here.