In a rectangular parallelepiped, the perimeter of the two side faces is 24cm 32cm.

| August 11, 2021 | education

In a rectangular parallelepiped, the perimeter of the two side faces is 24cm 32cm. Calculate the volume of the parallelepiped with the largest lateral surface.

Let Paa1d1d = 24 cm, Paa1d1d = 32 cm.

Let us denote the dimensions of the parallelepiped through, X., Y, Z.

AB = X cm, HELL = Y cm, AA1 = Z cm.

Then 2 * (X + Z) = 24 cm.

2 * (Y + Z) = 32 cm.

In a system of two equations, we express X through Y.

X + Z = 12.

Y + Z = 16.

Let us subtract the first equation from the second.

Y – X = 4.

Y = X + 4.

Then S side = 2 * (X + Y) * Z = 2 * (X + X + 4) * Z = 4 * (X + 2) * Z.

Z = 12 – X, then: 4 * (X + 2) * (12 – X).

-4 * X2 + 48 * X – 8 * X + 24.

-4 * X2 + 40 * X + 96.

-X2 + 10 * X + 24 = 0.

X1 = -2.

X2 = 12.

Let’s define the derivative of the quadratic equation.

-2 * X + 10 = 0.

Let’s define the critical points.

X = 10/2 = 5.

X = 5 lies in the interval (0; 12).

Determine the sign of the derivative at the point X = 0 and X = 6.

Y ‘(0) = 10.

Y ‘(6) = -2.

The derivative changes sign from + to -, which means X = 5 maximum point.

Then Y = X + 4 = 5 + 4 = 9 cm.

Z = 16 – Y = 16 – 9 = 7 cm.

Then V = 5 * 9 * 7 = 315 cm3.

Answer: The volume of the parallelepiped is 315 cm3.



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