7.The equation of a helix is x=2 sin 2t, y=2 cos 2t, z=3t. a) Find the arc length s from
arbitrary point (2 sin 2t, 2 cos 2t, 3t) on the helix. b) Compute the arc length from (0,2,0) to (0,-2,3π/2) c) Compute the vectors T, N and B at (0,-2,3π/2) d) Compute the curvature at (0,-2,3π/2) e) Find the angle between T and the z-axis at (0,-2,3π/2) to the nearest tenth of a degree.
8.The equation of a helix is x=2 sin 2t, y=2 cos 2t, z=3t.
a) Find the arc length s
an arbitrary point (2 sin 2t, 2 cos 2t, 3t) on the helix.
b) Compute the arc length from (0,2,0) to (0,-2,3π/2)
c) Compute the vectors T, N and B at (0,-2,3π/2)
d) Compute the curvature at (0,-2,3π/2)
e) Find the angle between T and the z-axis at (0,-2,3π/2) to the nearest tenth of a degree.
20.The Sydney Harbour Bridge roadway is 504m long. At a distance of 108.5m from each pylon, there is a vertical
vertical strut extending from the lower arch to the roadway (as shown in the image). Here the lower arch is 80m above sea level and the upper arch is 49m above the roadway. At the vertices, the lower arch is 118m above sea level and the upper arch is 73m above the roadway.
Find the quadratic equations which describe the parabolas of the lower arch in:
vertex form, y=a(x-h)^2+k;
intercept form, y=a(x-α)(x-β)
general form, y=ax^2+bx+c
The lower arch intersects the roadway 181.5m from the vertex. Calculate how much higher is the upper arch than the lower at the middle of the bridge?
Using technology, determine the total length of all 19 pairs of equally-spaced, vertical struts between the lower arch and the roadway.
22.You are sleeping on an air mattress, which forms a rectangular "box" shape. The length and width don't change, but
't change, but throughout the night the height of the mattress is decreasing. The rate at which it decreases varies over time: at the beginning it is just shrinking a tiny bit, but after a while it starts shrinking faster. If the height of the of the mattress follows the formula h=18-0.2x^2, where h is the height in inches and x is the time in hours, and the length of the mattress is always 74 inches, and the width of the air mattress is always 54 inches, please find the rate at which the VOLUME of the air mattress is changing after 2 hours. V ' =
cubic inches per hour after 2 hours. (Round to one decimal place, but take a picture of your work and send it to me if you're worried your answer might be slightly off.)
26.A uniform beam of length L
and mass m shown in Figure
P12.16 is inclined at an angle
u to the horizontal. Its
izontal. Its upper
end is connected to a wall by
a rope, and its lower end rests
on a rough, horizontal sur-
face. The coefficient of static
friction between the beam
and surface is ms. Assume
the angle u is such that the static friction force is at its
maximum value. (a) Draw a force diagram for the beam.
(b) Using the condition of rotational equilibrium,
find an expression for the tension T in the rope in
terms of m, g, and u. (c) Using the condition of trans-
lational equilibrium, find a second expression for T in
terms of ms, m, and g. (d) Using the results from parts
(a) through (c), obtain an expression for ms
vertical component of this force. Now solve the same
problem from the force diagram from part (a) by com-
puting torques around the junction between the cable
and the beam at the right-hand end of the beam. Find
(e) the vertical component of the force exerted by the
pole on the beam, (f) the tension in the cable, and
(g) the horizontal component of the force exerted
by the pole on the beam. (h) Compare the solution
to parts (b) through (d) with the solution to parts
(e) through (g). Is either solution more accurate?
19. Sir Lost-a-Lot dons his armor and sets out from the
castle on his trusty steed (Fig. P12.19). Usually, the
drawbridge is lowered to a horizontal position so that
the end of the bridge rests on the stone ledge. Unfor-
squire didn’t lower the draw-
ing only the angle u. (e) What happens if the ladder
is lifted upward and its base is placed back on the
ground slightly to the left of its position in Figure
27.We analyzed a program which was supposedly used for cracking some cryptographic algorithms.During the investigations we determined that the program
e investigations we determined that the program input size can be varied in a wide range and for N-bit input the program result is also always N – Bit long. Additionally we found that the program working time depends significantly on input length N, especially when N is greather than 10- 15 . Our test also revealed, that the program working time depends only on input length ,not the input itself.
During our tests, we fixed the following working times (with an accuracy of one hundredth of a second)
N = 2 – 16.38 seconds
N = 5 – 16.38 seconds
N = 10 – 16.44 seconds
N = 15 – 18.39 seconds
N = 20 – 1 minute 4.22 seconds
We also planned to test the program for N = 25 and N = 30, but for both cases the program didn’t finish within half and hour was forced to terminate it. Finally, we decided for N = 30 to not terminate the program,but to wait a little bit longer.The result was 18 hours 16 minutes 14.62 seconds . We repeated the test for N = 30 and it gave us exactly the same result, more than 18 hours.
a) Find the program working times for the following three cases N = 25, N = 40 and N = 50.
b) Explain your result and solution process.
28.1, A ship is conned around a lighthouse on a submerged reef at a constant distance of 10 nautical miles.
les. Draw a neat diagram and find the distance between the instants when the lighthouse 040° T and 120° T.
HINT: This is NOT a problem to be solved using cosine rule. Since the distance is constant, the ship will travel along the arc. You should therefore use radian measure to workout the distance along the arc. Also remember that all bearings are from seawards .
2, The length of a shadow of a vertical flag mast 2.5 meters high is 5.2 meters (shadow's length). Find the sun's altitude.
3, From a ship, a light on the edge of a breakwater bore 045° T . The ship was steering a course of 060° T. After travelling for 30 minutes, the same light bore 280° T and distance off 2.0 nautical miles. Find the speed of the vessel.