1. A boat sails at a constant speed in a straight line. its position at time S is (30S- 300,10S +500). in the water there are two buoys, A and B. at positions A- (7100,2800) and B - (125700,42500).

a) write an expression in terms of S that is the square of the distance between the boat and buoy A at time S, simplify the answer.

b) use answer a and completing the square to find the closest distance as the boat passes buoy A (answer to nearest ten meters). what time did the closest distance occur?

c) show that the boat passes over buoy b and tell us in minutes how long it takes to get to buoy b. assuming the boat starts at S=0.

d) what is the speed of the boat in metres per second (to 1 decimal place) and it direction as a bearing (to nearest degree).

2. on a second boat trip the boat starts in the same position. (-300,500) and it heading N70^oE. its speed is 30 ms ^-1 but it sails in a current of 12 ms^-1 from direction S10^o E.

Va is velocity of boat in still water.
Vw is velocity of the current
V resultant velocity of the boat.

a) draw a diagram illustrating these three velocities as a triangle, what is the angle between Va and Vw.

b) use the triangle and trigonometry to calculate the overall speed of the boat in m/s^-1 to 1dp. and its direction (bearing) correct to nearest degree.

(without using component form for vectors)

Solution Preview

Please see the attachment.

Problem #1
(a) Let be the distance between the boat and ...

Solution Summary

This solution is comprised of a detailed explanation to write an expression in terms of S that is the square of the distance between the boat and buoy A at time S, simplify the answer.

Please evaluate the following expressions:
92-7(8)
6(32-17)
6(32)-17
Please use the function f(x) = -3x+5 when f (2), (-1) ,(.5) , (-3) and plot the following on Cartesiancoordinates:
(2,3)
(4,-1)
(-.5,3.5)
(-1.5,-4)
(-5,3)

1. Rotate the coordinate axes x and y by the appropriate acute angle theta, so as to eliminate the cross product term (xy) from the equation x^2 + xy + y^2 - 6 = 0. State the value of theta, give the transformed equation in terms of the new rotated coordinates u and v, and identify the figure it represents.
2. Find cartesian

Please assist with the attached problem.
(a) Calculate the Laplacian of function u(x,y,z) = x^3 - 3xy^2 + z^2 in 3D Cartesiancoordinates.
(b) Convert the formula for u into formula for u involving cylindrical polar coordinates. Then compute the Laplacian using the cylindrical polar form. Show that your answer here is the same

1. Find the equation of the tangent line in Cartesiancoordinates of the curve given in polor coordinates by
r = 3 - 2 cos Ø, at Ø= (π / 3)
2.Test for convergence or divergence, absolute or conditional. If the series converges and it is possible to find the sum, then do so.
a) ∑[∞/n=1] (3/ 2^n)

A particle moves along a parametrized curve given by
x(lamda)=cos(lamda), y(lamda)=sin(lamda), z(lamda)=lamda
Express the path of the curve in the spherical polar coordinates {r, theta, pheta}
where x = rsin(theta)cos(pheta)
y=rsin(theta)sin(pheta)
z=rcos(theta)
so that the metric is
ds^2=dr^2+(r^2)d(theta)^2+(r^2)sin

(See attached file for full problem description)
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Given a point P with spherical coordinates (4, pi/6, pi/4). Find the xyz coordinates and cylindrical coordinates for P.
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