Question

Write a formula that shows the dependence of: the length of the side (a) of a cube on the surface area (S) of the cube

Write the formula for the parabola that has x-intercepts (-3-√2,0) and (-3+√2,0), and y-intercept (0,-5)

Answer 1

Let a = side length of a cube

Let S = surface area of a cube

Area of a square = a²

Since a cube has 6 square sides: S = 6a²

To make a the subject:

S = 6a²

Divide both sides by 6:

`\sf \implies (S)/(6)=a^2`

Square root both sides:

`\sf \implies a=\sqrt{(S)/(6)}`

(positive square root only as distance is positive)

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`\sf x=-3-√(2) \implies (x+[3+√(2)])=0`

`\sf x=-3+√(2) \implies (x+[3-√(2)])=0`

Therefore,

`\sf y=a(x+[3+√(2)]) (x+[3-√(2)])` for some constant a

Given the y-intercept is at (0, -5)

`\sf \implies a(0+3+√(2)) (0+3-√(2))=-5`

`\sf \implies a(3+√(2)) (3-√(2))=-5`

`\sf \implies a(9-2)=-5`

`\sf \implies 7a=-5`

`\sf \implies a=-\frac57`

Substituting found value of a into the equation and simplifying:

`\sf y=-\frac57(x+[3+√(2)]) (x+[3-√(2)])`

`\sf \implies y=-\frac57(x^2+6x+7)`

`\sf \implies y=-\frac57x^2-(30)/(7)x-5`