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Smath 5346 UNofficial release

Berechnen Sie den Durchmesser d und die Höhe h eines Zylinders mit gegebenem Volumen V und gegebener Oberfläche A

Compute the diameter 'd' and the height 'h' of a cylinder of given volume 'V' and given surface area 'A' Source: Re-published from Martin Handbook.

Define the cylinder A 10 m 2^* : Surface area ≡ V 1 m 3^* : Volume ≡ 21 line

Transport informations data surface area A volume V 21 mat :

System of equations f A 2 π 4 / d 2^* (* π d * h * + ≡ V π 4 / d 2^* h * ≡ 21 mat :

(embedded assignment) f d 1 m * ≡ h 1 m * ≡ 21 mat FindRoot 0.4109 m * 7.5411 m * 21 mat

Startwerte mit passenden Maßeinheiten vorgeben und Gleichungssystem auflösen.

Provide initial guesses with appropriate units and solve for the unknowns.

Sind die Anfangswerte in Gleichungsform gegeben, dann werden die Variablen mit den Lösungswerten definiert.

If the initial guesses are given as equalities name=value, then the solution is assigned to these names (embedded assignment).

solved variables d h 21 mat 0.4109 m * 7.5411 m * 21 mat

Sanity check of the "Given/Find Cylinder" data d h 21 mat 2 π 4 / d 2^* (* π d * h * + π 4 / d 2^* h * 21 mat 13 mat surface area A volume V 21 mat 0.4109 m * 7.5411 m * 21 mat 10 m 2^* 1 m 3^* 21 mat 13 mat

C 6 : Circle ≡

L 0 : Line ≡

Φ vector equalities Φ C x 2^y 2^+ ≡ L x y + ≡ 21 mat :

init x 1 ≡ y 1 ≡ 21 mat :

Does not support assignment Φ init FindRoot 1.732052 - 1.732052 21 mat

Assign the solutions sol x y 21 mat : 1.732052 - 1.732052 21 mat

C x 2 ^ - sqrt C x 2 ^ - sqrt - L x - sol 1 el sol 2 el . 14 black sol 2 el sol 1 el . 14 black 2 5 mat 4 1 sys

x y find Φ init FindRoot :

x y find 1.7321 - 1.7321 21 mat

Solving a user system ===================================

b 1 : c π : 21 line

f a0 0 ≡ a1 c ≡ a1 a2 + a3 + 0 ≡ a1 2 a2 * + 3 a3 * + b ≡ 41 mat :

init a0 1 ≡ a1 1 ≡ a2 1 ≡ a3 1 ≡ 41 mat :

f init FindRoot 0 3.1416 7.2832 - 4.1416 41 mat

a0 a1 a2 a3 41 mat 0 3.1416 7.2832 - 4.1416 41 mat

a0 a1 a2 a3 sol f init FindRoot :

a0 a1 a2 a3 sol 0 3.1416 7.2832 - 4.1416 41 mat

Solving quaternion ======================================

α 0.5 :

β 1.5 :

The Quaternion system f p q + r - 1012 -^≡ p 2^q 2^+ r 2^+ 1 ≡ β p * α r * + 1012 -^≡ 31 mat :

init p 1012 -^≡ q 1012 -^≡ r 1012 -^≡ 31 mat :

Sanity check vs Mathcad 11 f init FindRoot 0.1961 - 0.7845 0.5883 31 mat

p q r 31 mat 0.1961 - 0.7845 0.5883 31 mat

Quaternion p α - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / q 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / α β + (* r 12 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 1 - 2 α 2^* 2 α * β * + 2 β 2^* + sqrt / β * 33 mat :

Next: evaluate numeric

modulo 2 β 2^* 2 β * α * + 2 α 2^* + sqrt :

α 0.5 :

β 1.5 :

Quaternion 2 col 0.1961 - 0.7845 0.5883 31 mat

V α - modulo / α modulo / α β + (modulo / α β + (- modulo / α modulo / α - modulo / 32 mat :

V 0.1961 - 0.1961 0.7845 0.7845 - 0.1961 0.1961 - 32 mat

Next: stack a 'p' value

p 0.6 :

q Quaternion 2 col p stack :

q 0.1961 - 0.7845 0.5883 0.6 41 mat

Nex: the unit vector

q 0.1961 - 0.7845 0.5883 0.6 41 mat :

q q q norme / : 0.1682 - 0.6727 0.5045 0.5145 41 mat

k1 0.5 : k2 0.25 : k3 0.1 : 31 line

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

init γ1 1012 -^≡ γ2 1012 -^≡ 21 mat :

f init FindRoot 0.5006 0.3494 21 mat

Does NOT clear, does NOT override

Clear(...) - (v1,v2,...,vn) - Clear values and definitions of variables. If the variable name is one of the built-in constants, the default value will be restored.

k1 k1 Clear :

k2 k2 Clear :

k3 k3 Clear :

k1 10.052 : k2 14.836 : k3 4.982 : 13 mat

π 1.23456789 :

π π Clear :

init γ1 0 ≡ γ2 0 ≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

π 1

f init FindRoot #

Copy/paste in blank sheet ... or => ...

k1 10.052 : k2 14.836 : k3 4.982 : 13 mat

init γ1 1099 -^≡ γ2 1099 -^≡ 21 mat :

f γ1 γ2 + k1 k2 + k3 + ≡ γ1 γ2 * k1 k2 * k1 k3 * + ≡ 21 mat :

f init FindRoot #

... or => renew the system

α1 10.052 : α2 14.836 : α3 4.982 : 13 mat

init β1 1099 -^≡ β2 1099 -^≡ 21 mat :

f β1 β2 + α1 α2 + α3 + ≡ β1 β2 * α1 α2 * α1 α3 * + ≡ 21 mat :

f init FindRoot 19.818 10.052 21 mat