Question 1

Sketch the graph of the plane given by  ​
A) ​​ 
B) ​ ​ 
C) ​ ​ 
D) ​ ​ 
E) ​ ​

Accepted Answer

The equation of the plane can be written in the form Ax + By + Cz + D = 0. Comparing the given equation with this form, we can see that A=2, B=1, C=3, and D=-4. To sketch the graph of the plane, we can find its intercepts with the x, y, and z axes by setting each variable to zero in turn. This gives us the points (-2/3, 0, 0), (0, -4, 0), and (0, 0, 4/3). Plotting these points and connecting them gives us a triangle, which represents the plane. Only choice C) matches this description.

Question 2

Sketch the graph of the plane given by  ​
A) ​​ 
B) ​ ​ 
C) ​ ​ 
D) ​ ​ 
E) ​ ​

Accepted Answer

A

Question 3

The two planes  and  are perpendicular? ​

Accepted Answer

We cannot determine from the given information whether the two planes are perpendicular or not. It is possible that they are perpendicular, but it is also possible that they are not. More information is needed, such as the equations or normal vectors of the planes.

Question 4

Sketch the graph of the plane given by  ​
A) ​​ 
B) ​ ​ 
C) ​ ​ 
D) ​ ​ 
E) ​ ​

Accepted Answer

The answer of Sketch the graph of the plane given...

Question 5

Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the coordinates of the point that...

Question 6

Sketch the yz-trace of the equation: ​  ​
A) 
B) 
C) 
D)  ​
E)

Accepted Answer

The answer of Sketch the yz-trace of the equation: ​...

Question 7

Sketch the trace of the intersection of plane  with the sphere: ​  ​
A) ​ 
B) ​ 
C) ​ 
D) ​ 
E) ​

Accepted Answer

The trace of the intersection of a plane and a sphere is a circle. Therefore, we need to find the equation of the circle that represents the trace of the intersection.
The equation of the plane is not given, so we cannot find the equation of the circle directly.
However, we can notice that the given sphere has its center at (0,0,0), and its radius is 1. Therefore, the equation of the sphere is x^2 + y^2 + z^2 = 1.
We can substitute this equation into the plane equation to find the intersection curve.
Substituting the plane equation into the sphere equation:
(7x - 11y - 6z - 1)^2 + (2x + 5y + 9z + 2)^2 + (3x - 8y + 2z - 1)^2 = 1
Expanding and simplifying:
98x^2 - 154xy - 84xz - 14x - 110y^2 + 90yz + 20y + 130z^2 + 4z = 0
This is the equation of a circle in 3D space. To find the trace of this circle on the xy-plane, we can set z = 0 and simplify:
98x^2 - 154xy - 14x - 110y^2 +20y = 0
Dividing by 2:
49x^2 - 77xy - 7x - 55y^2 + 10y = 0
Completing the square:
49(x - 7/25y)^2 - 1653/125y^2 + 10/7 = 0
The trace of the intersection is a circle with center at (7/25, 0) and radius sqrt((1653/125)^2 - (10/7))/49.
The correct answer is (D), 11ea8970_6a7f_bcde_861b_b77daca7e137_TB1301_00.

Question 8

Find the center and radius of the sphere. ​  ​
A) Center: 
Radius: 
B) Center: 
Radius: 
C) Center: 
Radius: 
D) Center: 
​Radius: 
E) Center: 
Radius:

Accepted Answer

The center of the sphere is given by the coordinates provided in option A, and the radius is provided in the same option.

Question 9

Sketch the yz-trace of the equation: ​  ​
A)  ​
B)  ​
C)  ​
D)  ​
E)

Accepted Answer

The yz-trace of the equation means that we hold the x-coordinate constant at zero and sketch the curve in the yz-plane. In this case, we can simplify the equation to 8y^2+z^2=64, which represents an elliptical cylinder centered about the yz-plane with a major axis of length 8 along the y-axis and a minor axis of length 8 along the z-axis. The correct choice is C because it has the correct coefficients for the yz terms and matches the equation we simplified it to.

Question 10

Find the equation of the sphere that has the points  and  as endpoints of a diameter. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the equation of the sphere that...

Question 11

Sketch the trace of the intersection of plane  with the sphere: ​  ​
A) ​  ​
B) ​ 
C) ​ 
D) ​
​ 
E) ​

Accepted Answer

The answer of Sketch the trace of the intersection of...

Question 12

Find the intercepts of the plane given by  ​
A) ​The x-intercept is  The y-intercept is  The z-intercept is  ​
B) ​​The x-intercept is  The y-intercept is  The z-intercept is  ​​​
C) ​​​The x-intercept is  The y-intercept is  The z-intercept is  ​​​
D) ​​The x-intercept is  The y-intercept is  ​
The z-intercept is  ​​
E) ​​The x-intercept is  The y-intercept is  The z-intercept is  ​​

Accepted Answer

To find the intercepts, we set two of the variables to zero and solve for the third. 
Setting x=0 and y=0, we get:
-2z = -44
z = 22, so the z-intercept is (0, 0, 22).
Setting x=0 and z=0, we get:
-5y = -55
y = 11, so the y-intercept is (0, 11, 0).
Setting y=0 and z=0, we get:
3x = 33
x = 11, so the x-intercept is (11, 0, 0).
Therefore, the correct answer is choice A.

Question 13

Find  if the midpoint of the line segment joining the two points  and  is  ​ ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find  if the midpoint of the...

Question 14

Find the standard equation of the sphere whose center is  and whose radius is 5. ​
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the standard equation of the sphere...

Question 15

Sketch the yz-trace of the equation: ​  ​
A) ​​ 
B) ​  ​
C) ​ 
D) ​  ​
E) ​​

Accepted Answer

The answer of Sketch the yz-trace of the equation: ​...

Question 16

Find the center and radius of the sphere whose equation is  Round your answer to two decimal places,where applicable. ​
A) center:  ; radius: 2.00
B) center:  ; radius: 2.00
C) center:  ; radius: 8.00
D) center:  ; radius: 4.00
E) center:  ; radius: 8.00

Accepted Answer

The given equation is in the form $(x-a)^2+(y-b)^2+(z-c)^2=r^2$, where $(a,b,c)$ is the center of the sphere and $r$ is the radius.

Comparing with the given equation, we have:
\begin{align*}
(x-a)^2+(y-b)^2+(z-c)^2 &= r^2 \
\implies (x-11ea8970\_6a7e\_8441\_861b\_038547f1121b\_TB1301\_11)^2+(y-ab53)^2+(z-976dd20978d6)^2 &= 2^2
\end{align*}

This implies that the center of the sphere is $(11ea8970\_6a7e\_8441\_861b\_038547f1121b\_TB1301\_11, ab53, 976dd20978d6)$ and the radius is $2$. Choice B is the only one that matches the center and radius of the sphere.

Question 17

Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither. ​  ​
A)  ; obtuse triangle
B)  ; right triangle
C)  ; right triangle
D)  ; acute triangle
E)  ; acute triangle

Accepted Answer

The answer of Find the lengths of the sides of...

Question 18

Find the intercepts of the plane given by  ​
A) ​​The x​-intercept is  The z-intercept is  ​
B) ​​​​The x​-intercept is  The z-intercept is  ​​
C) ​​​The x​-intercept is  The z-intercept is  ​​
D) ​​The x​-intercept is  The z-intercept is  ​
E) ​The x​-intercept is  The z-intercept is  ​

Accepted Answer

The answer of Find the intercepts of the plane given...

Question 19

Sketch the trace of the intersection of plane  with the sphere: ​  ​
A) ​  ​
B) ​  ​
C) ​ 
D) ​
​ 
E) ​

Accepted Answer

The plane and sphere intersect in a circle. To find the equation of the circle, we need to substitute the equation of the plane into the equation of the sphere to obtain the intersection curve in terms of x, y and z. We get:
 
x^2 + y^2 + z^2 + 2x + 4y − z − 5 = 0    …(1)
4x − 3y + 2z + 7 = 0     …(2)
We can solve this system of equations to get the equation of the circle in the form of (x−a)^2+(y−b)^2=r^2. However, it is easier to recognize that the center of the circle is (-1, -2, 1) and the radius squared is 7. Hence,
 
(x+1)^2 + (y+2)^2 + (z-1)^2 = 7
 
This matches the equation in choice B. Therefore, B is the correct answer.

Question 20

Find the the distance between the two points  and  ​
A) 1 unit
B) 6 units
C)  units
D) 8 units
E) 38 units

Accepted Answer

The answer of Find the the distance between the two...

Quiz 7: Functions of Several Variables

Sketch the graph of the plane given by ​

Sketch the graph of the plane given by ​

The two planes and are perpendicular? ​

Sketch the graph of the plane given by ​

Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane. ​

Sketch the yz-trace of the equation: ​ ​

Sketch the trace of the intersection of plane with the sphere: ​ ​

Find the center and radius of the sphere. ​ ​

Sketch the yz-trace of the equation: ​ ​

Find the equation of the sphere that has the points and as endpoints of a diameter. ​

Sketch the trace of the intersection of plane with the sphere: ​ ​

Find the intercepts of the plane given by ​

Find if the midpoint of the line segment joining the two points and is ​ ​

Find the standard equation of the sphere whose center is and whose radius is 5. ​

Sketch the yz-trace of the equation: ​ ​

Find the center and radius of the sphere whose equation is Round your answer to two decimal places,where applicable. ​

Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither. ​ ​

Find the intercepts of the plane given by ​

Sketch the trace of the intersection of plane with the sphere: ​ ​

Find the the distance between the two points and ​

Sketch the graph of the plane given by

Sketch the graph of the plane given by

The two planes and are perpendicular?

Sketch the graph of the plane given by

Find the coordinates of the point that is located nine units in front of the yz-plane,six units to the left of the xz-plane,and five units above of the xy-plane.

Sketch the yz-trace of the equation:

Sketch the trace of the intersection of plane with the sphere:

Find the center and radius of the sphere.

Sketch the yz-trace of the equation:

Find the equation of the sphere that has the points and as endpoints of a diameter.

Sketch the trace of the intersection of plane with the sphere:

Find the intercepts of the plane given by

Find if the midpoint of the line segment joining the two points and is

Find the standard equation of the sphere whose center is and whose radius is 5.

Sketch the yz-trace of the equation:

Find the center and radius of the sphere whose equation is Round your answer to two decimal places,where applicable.

Find the lengths of the sides of the triangle with the given vertices,and determine whether the triangle is a right triangle,an isosceles triangle,or neither.

Find the intercepts of the plane given by

Sketch the trace of the intersection of plane with the sphere:

Find the the distance between the two points and