734 Chapter 12: Vector-Valued Functions
12.5 Arc Length and Curvature
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the length of the plane curve given below.
a. 20
b. 9
c. 17
d. 19
e. 18
____ 2. A baseball is hit 5 feet above the ground at 140 feet per second and at an angle of
with respect to the ground. Find the vector-valued function for the path of the baseball.
a.
b.
c.
d.
e.
____ 3. A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of
with respect to the ground. Find the maximum height. Round your answer to the nearest integer.
a. 66 feet
b. 65 feet
c. 53 feet
d. 56 feet
e. 61 feet
____ 4. A baseball is hit 3 feet above the ground at 80 feet per second and at an angle of
with respect to the ground. Find the range. Round your answer to one decimal place.
a. 203 feet
b. 259.5 feet
c. 275.7 feet
d. 386.6 feet
e. 215.6 feet
12.5 Arc Length and Curvature 735
____ 5. A baseball is hit 5 feet above the ground at 95 feet per second and at an angle of
with respect to the ground. Find the arc length of the trajectory. Round your answer to one
decimal place.
a. 270.8 feet
b. 298 feet
c. 293.7 feet
d. 239.1 feet
e. 330.7 feet
____ 6. Find the arc length for over the interval . Round your
answer to two decimal places.
a. 67.63
b. 73.37
c. 79.40
d. 49.75
e. 62.97
____ 7. Find the length of the space curve given below.
a.
b.
c.
d.
e.
____ 8. Find the curvature K of the curve given below.
a.
b.
c.
736 Chapter 12: Vector-Valued Functions
d.
e.
____ 9. Find the curvature K of the curve at the point . Round your
answer to three decimal places.
a. 0.883
b. 0.994
c. 0.019
d. 0.012
e. 0.017
____ 10. Find the curvature K of the curve at the point .
Round your answer to three decimal places.
a. 7.286
b. 9.952
c. 0.021
d. 0.014
e. 0.009
____ 11. Find the curvature of the plane curve at . Round your answer to
three decimal places.
a. 0.498
b. 0.332
c. 0.015
d. 0.005
e. 0.003
____ 12. Find the radius of curvature of the plane curve at . Round your
answer to three decimal places.
a. 24.694
b. 85.333
c. 28.133
d. 34.510
e. 37.510
12.5 Arc Length and Curvature 737
____ 13. Find the point on the curve at which the curvature K is a maximum.
a.
b.
c.
d.
e.
____ 14. Find the point on the curve given below at which the curvature K is zero.
a.
b.
c.
d.
e.
____ 15. The smaller the curvature in a bend of a road, the faster a car can travel. Assume that
the maximum speed around a turn is inversely proportional to the square root of the curvature. A car
moving on the path (x and y are measured in miles) can safely go 25 miles per hour at
. How fast can it go at ? Round your answer to two decimal places.
a. 259.31 mi/h
b. 149.71 mi/h
c. 119.77 mi/h
d. 75.30 mi/h
e. 299.42 mi/h
738 Chapter 12: Vector-Valued Functions
____ 16. A 5700-pound vehicle is driven at a speed of 25 miles per hour on a circular
interchange of radius 90 feet. To keep the vehicle from skidding off course, what frictional force must
the road surface exert on the tires? Round your answer to one decimal place.
a. 2,670.2 lb
b. 3,337.8 lb
c. 2,993.5 lb
d. 2,660.9 lb
e. 3,503.9 lb
12.5 Arc Length and Curvature 739
12.5 Arc Length and Curvature
Answer Section