Name: ________________________ Class: ___________________ Date: __________
ID: A
Calc III test 1 practice
1. a. Find an equation of the sphere that passes
2. Find parametric equations for the line through
through the point (6, − 6, 3) and has center
(−3, 6, 3).
(−4, 3, 3) and (4, 5, 6).
b. Find the curve in which this sphere intersects
the xy-plane.
3. Find parametric equations for the line through
(− 6, 4, 7) and parallel to the vector 7, 5,−7 .
4. Find the point of intersection.
L1:
y − 58 z − 23
y − 26
x − 17
x − 49
=
=
, L2 :
=
= z − 15
3
8
2
7
4
6. Find the length of each side of the triangle ABC
and determine whether the triangle is an isosceles
triangle, a right triangle, both, or neither.
5. Find the center and radius of the sphere.
x 2 − 6x − 36 + y 2 + 24y + z 2 − 18z = 0
A (–1, 0, 1), B (1, 1, –1), C (1, 1, 1)
7. Find the distance from (−10, 2, − 8) to the
xy-planes.
8. The tension T at each end of the chain has magnitude 27 N and makes an angle ∠a = 27° with the
horizontal. What is the weight of the chain? Round the result to the nearest hundredth.
1
Name: ________________________
ID: A
9. Find an equation of the sphere with center (6, − 3, 6) that touches the xy-plane.
10. Two forces F 1 and F 2 with magnitudes 8 lb and 12 lb act on an object at a point P as shown in the figure.
Find the magnitude of the resultant force F acting at P. Round the result to the nearest tenth.
15. Let v = 7j and let u be a vector with length 5
11. Find the unit vectors that are parallel to the
tangent line to the curve y = 2x at the point
(2, 8).
2
that starts at the origin and rotates in the xy plane. Find the maximum value of the length of
the vector |u × v| .
12. Find the values of x such that the vectors
16. Find a nonzero vector orthogonal to the plane
2x, x, 3 and 6, x, 9 are orthogonal.
through the points P, Q, and R.
P (1, 0, 0), Q (7, 8, 0), R (0, 8, 1)
13. Find the work done by a force F = i + 5j+ 7k
that moves an object from the point (3, 0, 5) to
the point (7, 5, 10) along a straight line. The
distance is measured in meters and the force in
newtons.
17. Find | u × v| correct to three decimal places
where | u| = 18, | v| = 3, ∠θ = 80 ° .
18. Which of the given lines is parallel to the line
14. Calculate the angle between a and b (correct to
x = 8 + t, y = t, z = −7 − 10t?
the nearest degree).
a = 9i − 8j, b = −7j + k
2
ID: A
Calc III test 1 practice
Answer Section
1. ANS: a. (x + 3 ) + ÊÁË y − 6 ˆ˜¯ 2 + ( z − 3) 2 = 225
2
2
b. (x + 3 ) + ÁÊË y − 6 ˜ˆ¯ 2 = 225
PTS: 1
2. ANS: x = − 4 + 8t, y = 3 + 2t, z = 3 + 3t
PTS: 1
3. ANS: x = − 6 + 7t, y = 4 + 5t, z = 7 − 7t
PTS: 1
4. ANS: (−7, − 6, 7)
PTS: 1
5. ANS:
none of these
PTS: 1
6. ANS:
3, 2, 5 , right
PTS: 1
7. ANS:
8
PTS: 1
8. ANS: 51.64
PTS: 1
9. ANS:
(x − 6 ) 2 + ÊÁË y + 3 ˆ˜¯ 2 + ( z − 6) 2 = 36
PTS: 1
10. ANS:
11.7 lb
PTS: 1
11. ANS:
± ÊÁË i + 8j ˆ˜¯
65
PTS: 1
1
ID: A
12. ANS:
x = −3, x = −9
PTS: 1
13. ANS:
64 J
PTS: 1
14. ANS:
49°
PTS: 1
15. ANS:
|u × v| = 35
PTS: 1
16. ANS:
8i − 6j + 56k
PTS: 1
17. ANS:
53.179
PTS: 1
18. ANS:
r = −6, − 10, 0 + t 2, 2, − 20
PTS: 1
2