Austin Cholley Zhu MAT 267 ONLINE A Spring 2021
Assignment Midterm 1 due 01/29/2021 at 08:42pm MST
Problem 1.
1. (1 point)
Consider the plane which passes through the three points:
(−6,5,9),(−7,6,4), and (−7,7,6).
Find the vector normal to this plane which has the form:
( -7, , )
Answer(s) submitted:
•2
•1
(score 0.5)
Correct Answers:
•-2
•1
Problem 2.
2. (1 point)
Find the point at which the line h4,−4,1i+th3,−5,−4iinter-
sects the plane −4x+4y−3z=−195.
( , , )
Answer(s) submitted:
•7
•-9
•-3
(incorrect)
Correct Answers:
•28
•-44
•-31
Problem 3.
3. (1 point)
Find the distance from the point (5, 3, 3) to the plane x−y−3z=
5.
Answer(s) submitted:
•-((12)/(sqrt(35)))
(incorrect)
Correct Answers:
•3.61813613493316
Problem 4.
4. (1 point)
Find the area of the triangle with vertices:
Q(-2,-2,2), R(-2,0,-1), S(1,1,-1).
Answer(s) submitted:
•(1/2)sqrt(126)
(correct)
Correct Answers:
•5.61248608016091
Problem 5.
5. (1 point) Answer the following questions about the sphere
whose equation is given by
x2+y2+z2−10x+4y=−4.
1. Find the radius of the sphere.
Radius: r=
2. Find the center of the sphere. Write the center as a
point (a,b,c)where a,b, and care numbers.
Center:
Answer(s) submitted:
•5
•-5,2,1
(score 0.5)
Correct Answers:
•5
•(5,-2,0)
Problem 6.
6. (1 point)
Find the equation of the sphere centered at (−7,−6,−3)
with radius 3. Normalize your equations so that the coefficient
of x2is 1.
= 0.
Give an equation which describes the intersection of this
sphere with the plane z=−2.
= 0.
Answer(s) submitted:
•(x+7)ˆ(2)+(y+6)ˆ(2)+(z+3)ˆ(2)-9
•(x+7)ˆ(2)+(y+6)ˆ(2)-2
(score 0.5)
Correct Answers:
•(x - -7)**2 + (y - -6)**2 + (z - -3)**2 - 3**2
•(x - -7)**2 + (y - -6)**2 + 1 - 3**2
1
Problem 7.
7. (1 point)
For the given position vectors r(t)compute the unit tangent
vector T(t)for the given value of t.
A) Let r(t) = (cos3t,sin3t).
Then T(π
4)= ( , )
B) Let r(t) = (t2,t3).
Then T(2)= ( , )
C) Let r(t) = e3ti+e−2tj+tk.
Then T(5)=i+j+k.
Answer(s) submitted:
•-1
•0
•1
•1
•(((eˆ(3(3))))/(|eˆ(3(3))|))
•-((1eˆ(-3))/(sqrt(9eˆ(18)+1eˆ(-6)+1)))
•(1/(sqrt(9eˆ(18)+1eˆ(-6)+1)))
(score 0.142857142857143)
Correct Answers:
•-0.707106781186548
•-0.707106781186547
•0.316227766016838
•0.948683298050514
•0.999999999999995
•-9.2586292433093E-12
•1.01967440167275E-07
Problem 8.
8. (1 point)
Find an equation of the plane parallel to the plane 8x+5y−
10z=−2
which passes through the point (0, 0, -7).
Give your answer in the form ax +by +cz =d(with a=8).
a =
b =
c =
d =
Answer(s) submitted:
•8
•5
•70
•-2
(score 0.5)
Correct Answers:
•8
•5
•-10
•70
Problem 9.
9. (1 point)
Compare the the planes below to the plane −3x−y+3z=
−1. Match the letter corresponding to the words parallel, or-
thogonal, or ”neither” which describes the relation of the two
planes.
1. −9x−3y+9z=2
2. −3x+3z=1
3. x−3y=−5
A. orthogonal
B. neither
C. parallel
Answer(s) submitted:
•A
•C
•B
(score 0.33)
Correct Answers:
•C
•B
•A
Problem 10.
10. (1 point)
A projectile is fired from ground level with an initial speed
of 500 m/sec and an angle of elevation of 30 degrees. Use that
the acceleration due to gravity is 9.8 m/sec2.
The range of the projectile is meters.
The maximum height of the projectile is me-
ters.
The speed with which the projectile hits the ground is
m/sec.
Answer(s) submitted:
•22092.4847
•3188.7755
•500
(correct)
Correct Answers:
•22092.4847904194
•3188.77551020408
•500
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