Question

x²+y²+z²=4
z=√(x²+y²)
z = x²+y²
z² = x²+y²
z²+4 = x²+y²
z² = x²+y²+4
z = x²−y²
4 = x²y² (shapes a-h with 3d figures: sphere, cone, hyperboloid, paraboloid, etc.)

Explanation:

To solve the problem of matching the equations to the corresponding 3D shapes, we analyze each equation and shape:

1. Equation \( x^2 + y^2 + z^2 = 4 \)

• This is the equation of a sphere with radius \( r = 2 \) (since the standard sphere equation is \( x^2 + y^2 + z^2 = r^2 \)).
• Shape A is a sphere. So this equation matches shape A.

2. Equation \( z = \sqrt{x^2 + y^2} \)

• This is a cone opening upwards (since \( z \geq 0 \) and the cross - section in any plane containing the \( z \) - axis is a straight line with slope 1).
• Shape B is a cone with the vertex at the origin and opening upwards. So this equation matches shape B.

3. Equation \( z = x^2 + y^2 \)

• This is a paraboloid opening upwards (the cross - section in the \( x - z \) or \( y - z \) plane is a parabola \( z=x^2 \) or \( z = y^2 \), and in the \( xy \) - plane (for a fixed \( z \)) it is a circle \( x^2 + y^2=z \)).
• Shape D is a paraboloid opening upwards. So this equation matches shape D.

4. Equation \( z^2=x^2 + y^2 \)

• This can be rewritten as \( z=\pm\sqrt{x^2 + y^2} \), which represents a double - napped cone (two cones opening in opposite directions along the \( z \) - axis).
• Shape C is a double - napped cone. So this equation matches shape C.

5. Equation \( z^2 + 4=x^2 + y^2 \) or \( x^2 + y^2-z^2 = 4 \)

• This is a hyperboloid of one sheet (the standard form of a hyperboloid of one sheet is \( \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 \), here \( a = b = 2,c = 2 \)).
• Shape F is a hyperboloid of one sheet. So this equation matches shape F.

6. Equation \( z^2=x^2 + y^2+4 \) or \( x^2 + y^2 - z^2=- 4 \)

• This is a hyperboloid of two sheets (the standard form of a hyperboloid of two sheets is \( \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1 \), here \( a = b = 2,c = 2 \)).
• Shape E is a hyperboloid of two sheets. So this equation matches shape E.

7. Equation \( z=x^2 - y^2 \)

• This is a hyperbolic paraboloid (the cross - section in the \( x - z \) plane (\( y = 0 \)) is \( z=x^2 \) (a parabola opening upwards), and in the \( y - z \) plane (\( x = 0 \)) is \( z=-y^2 \) (a parabola opening downwards), and in the \( xy \) - plane (\( z = 0 \)) is \( x^2-y^2 = 0\) or \( y=\pm x \) (a pair of straight lines)).
• Shape G is a hyperbolic paraboloid. So this equation matches shape G.

8. Equation \( 4=x^2y^2 \) (assuming it is \( x^2 + y^2=4 \) in the \( z \) - direction, representing a cylinder)

• The equation \( x^2 + y^2 = 4 \) represents a circular cylinder along the \( z \) - axis (since for any \( z \), the cross - section in the \( xy \) - plane is a circle with radius 2).
• Shape H is a cylinder. So this equation matches shape H.

Final Matching:

• \( x^2 + y^2 + z^2 = 4 \): A
• \( z=\sqrt{x^2 + y^2} \): B
• \( z = x^2 + y^2 \): D
• \( z^2=x^2 + y^2 \): C
• \( z^2 + 4=x^2 + y^2 \): F
• \( z^2=x^2 + y^2+4 \): E
• \( z=x^2 - y^2 \): G
• \( x^2 + y^2 = 4 \) (interpreted from \( 4=x^2y^2 \) error, likely \( x^2 + y^2 = 4 \)): H

Answer:

To solve the problem of matching the equations to the corresponding 3D shapes, we analyze each equation and shape:

1. Equation \( x^2 + y^2 + z^2 = 4 \)

• This is the equation of a sphere with radius \( r = 2 \) (since the standard sphere equation is \( x^2 + y^2 + z^2 = r^2 \)).
• Shape A is a sphere. So this equation matches shape A.

2. Equation \( z = \sqrt{x^2 + y^2} \)

• This is a cone opening upwards (since \( z \geq 0 \) and the cross - section in any plane containing the \( z \) - axis is a straight line with slope 1).
• Shape B is a cone with the vertex at the origin and opening upwards. So this equation matches shape B.

3. Equation \( z = x^2 + y^2 \)

• This is a paraboloid opening upwards (the cross - section in the \( x - z \) or \( y - z \) plane is a parabola \( z=x^2 \) or \( z = y^2 \), and in the \( xy \) - plane (for a fixed \( z \)) it is a circle \( x^2 + y^2=z \)).
• Shape D is a paraboloid opening upwards. So this equation matches shape D.

4. Equation \( z^2=x^2 + y^2 \)

• This can be rewritten as \( z=\pm\sqrt{x^2 + y^2} \), which represents a double - napped cone (two cones opening in opposite directions along the \( z \) - axis).
• Shape C is a double - napped cone. So this equation matches shape C.

5. Equation \( z^2 + 4=x^2 + y^2 \) or \( x^2 + y^2-z^2 = 4 \)

• This is a hyperboloid of one sheet (the standard form of a hyperboloid of one sheet is \( \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1 \), here \( a = b = 2,c = 2 \)).
• Shape F is a hyperboloid of one sheet. So this equation matches shape F.

6. Equation \( z^2=x^2 + y^2+4 \) or \( x^2 + y^2 - z^2=- 4 \)

• This is a hyperboloid of two sheets (the standard form of a hyperboloid of two sheets is \( \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1 \), here \( a = b = 2,c = 2 \)).
• Shape E is a hyperboloid of two sheets. So this equation matches shape E.

7. Equation \( z=x^2 - y^2 \)

• This is a hyperbolic paraboloid (the cross - section in the \( x - z \) plane (\( y = 0 \)) is \( z=x^2 \) (a parabola opening upwards), and in the \( y - z \) plane (\( x = 0 \)) is \( z=-y^2 \) (a parabola opening downwards), and in the \( xy \) - plane (\( z = 0 \)) is \( x^2-y^2 = 0\) or \( y=\pm x \) (a pair of straight lines)).
• Shape G is a hyperbolic paraboloid. So this equation matches shape G.

8. Equation \( 4=x^2y^2 \) (assuming it is \( x^2 + y^2=4 \) in the \( z \) - direction, representing a cylinder)

• The equation \( x^2 + y^2 = 4 \) represents a circular cylinder along the \( z \) - axis (since for any \( z \), the cross - section in the \( xy \) - plane is a circle with radius 2).
• Shape H is a cylinder. So this equation matches shape H.

Final Matching:

• \( x^2 + y^2 + z^2 = 4 \): A
• \( z=\sqrt{x^2 + y^2} \): B
• \( z = x^2 + y^2 \): D
• \( z^2=x^2 + y^2 \): C
• \( z^2 + 4=x^2 + y^2 \): F
• \( z^2=x^2 + y^2+4 \): E
• \( z=x^2 - y^2 \): G
• \( x^2 + y^2 = 4 \) (interpreted from \( 4=x^2y^2 \) error, likely \( x^2 + y^2 = 4 \)): H