What is the volume of a rectangular parallelepiped with diagonals equal to √5cm; √10cm; √13cm.

| May 13, 2021 | education

For the solution, we denote the lengths of the edges of the parallelepiped by X, Y, Z.

Then in a right-angled triangle АА1B,

X ^ 2 + Y ^ 2 = (√13) ^ 2. (one)

In a right-angled triangle АA1D.

X ^ 2 + Z ^ 2 = (√5) ^ 2. (2)

In a right-angled triangle ABD.

Z ^ 2 + Y ^ 2 = (√10) ^ 2. (3)

Let us solve the system of three equations by the substitution method.

Subtract equation 2 from equation 1.

X ^ 2 + Y ^ 2 – X ^ 2 – Z ^ 2 = 13 – 5.

Y ^ 2 – Z ^ 2 = 8.

Add equation 3 to the last equation.

Y ^ 2 – Z ^ 2 + Z ^ 2 + Y ^ 2 = 8 + 10.

2 * Y ^ 2 = 18.

Y ^ 2 = 18/2 = 9.

Y = 3 cm.

We put this value in equations 1 and 3.

X ^ 2 + 3 ^ 2 = 13.

X ^ 2 = 4.

X = 2 cm.

Z2 + 3 ^ 2 = 10.

Z2 = 1.

Z = 1 cm.

Let’s define the volume of the parallelepiped.

V = X * Y * Z = 2 * 3 * 1 = 6 cm3.

Answer: V = 6 cm3.



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