The diagonal of a regular quadrangular prism is 6 cm, and the side surface is 32 cm2.

| May 9, 2021 | education

The diagonal of a regular quadrangular prism is 6 cm, and the side surface is 32 cm2. Find in the volume of the prism.

Let the length of the side of the square, at the base of the prism, be equal to X cm, and the height of the prism equal to Y cm.

Then S side = 32 = 4 * X * Y.

X * Y = 32/4 = 8.

Y = 8 / X. (1).

The diagonal of the prism is: DB12 = X2 + X2 + Y2 = 2 * X2 + Y2 = 36.

Substitute (1) into the last equality.

36 = 2 * X2 + 64 / X2.

2 * X4 – 36 * X2 + 64 = 0.

X4 – 18 * X2 + 32 = 0.

Let X2 = Z, then.

Z2 – 18 * Z + 32 = 0.

Let’s solve the quadratic equation.

Z1 = 2.

Z2 = 16.

Then X1 = √2, X2 = 4.

Y1 = 8 / √2 = 4 * √2 cm.

Y2 = 8/4 = 2 cm.

When X = √2.

Sbn = 2 cm2, then V = 2 * 4 * √2 = 8 * √2 cm3.

When X = 4.

Sbn = 16 cm2, then V = 16 * 2 = 32 cm3.

Answer: The volume of the prism is 8 * √2 cm3 or 32 cm3.



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