Question

The volume of a tetrahedron DABC is 1/6. Also, angle(ACB)=45° and AD + BC + (AC/√2) = 3. Then....
(A)CD=√3. (B)AC=√2. (C)AD=1. (D)Angle ABC =90°​

Answer 1

Answer:

I don't know

Answer 2

Answer:

The correct answer is  (A) CD=√3.

Explanation:

From the above question,

They have given :

The volume of a tetrahedron DABC is 1/6.

Also, angle(ACB)=45° and AD + BC + (AC/√2) = 3.

The formula for the volume of a tetrahedron is (1/3)Bh, where B is the area of the base and h is the height. Since the volume of the tetrahedron DABC is 1/6, we can write:

(1/3)Bh = 1/6

Since angle ACB is 45°, we can consider triangle ACB as a right triangle, with AC being the hypotenuse. Therefore, we can write:

AC = BC / √2

Since AD + BC + (AC / √2) = 3, we can substitute the expression for AC in terms of BC:

AD + BC + (BC / √2) = 3

Expanding the right side:

AD + BC + BC / √2 = 3

Multiplying both sides by √2:

AD√2 + BC√2 + BC = 3√2

Since the height h of the tetrahedron is equal to AD, we can substitute AD = h:

h√2 + BC√2 + BC = 3√2

Expanding the area of the base B:

h√2 + 2BC√2 = 3√2

Dividing both sides by √2:

h + 2BC = 3

Therefore, h = 3 - 2BC. Substituting h back into the equation for the volume:

(1/3)Bh = 1/6

(1/3)(B)(3 - 2BC) = 1/6

Expanding the area B:

(1/3)(BC√2)(3 - 2BC) = 1/6

Multiplying both sides by 3:

BC√2(3 - 2BC) = 2

Expanding the right side:

BC√2(3) - BC√2(2BC) = 2

Simplifying the left side:

3BC√2 - 2BC^2√2 = 2

Dividing both sides by √2:

3BC - 2BC^2 = √2

Solving for BC:

BC = √2

Finally, since AC = BC / √2, we can substitute BC = √2:

Answer:

     (A) CD=√3. This is correct because the volume of a tetrahedron is equal to one third of the base area multiplied by the height, and since the base area is 3 and the height is √2, the volume is 1/6.

(B)AC=√2 is correct because the angle (ACB) is 45°, which forms two right triangles, each with sides AC and √2, so AC=√2.

(C)AD=1 is correct because AD + BC + (AC/√2) = 3, so AD must equal 1 for the equation to be true.

(D)Angle ABC =90° is incorrect, because the angle (ACB) is 45°, not 90°.

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