1) Inside the graph of r = 1 + cos 3Ό
2) Inside the circle r = -6 cos Ό and outside the circle r = 3
In: Math
Determine whether or not W is a subspace of R3 where W consists of all vectors (a,b,c) in R3 such that
1. a =3b
2. a <= b <= c
3. ab=0
4. a+b+c=0
5. b=a2
6. a=2b=3c
In: Math
Estimate the area under the graph of f(x)=1/(x+4) over the interval [-1,2] using five approximating rectangles and right endpoints.
Rn=
Repeat the approximation using left endpoints.
Ln=
In: Math
Determine the first four terms of the Maclaurin series for sin(6)x and cos(7)x:
b) By replacing x by (6)x and (7)x, respectively, in the power series for sin x and cos x
In: Math
1) Aris uses a boat to cross from one side to the other side of a river of width w = 32 m. The river flows at VC = 8 m/min. and he can row at a speed of VB =30 m/min. in still water. He wants to land at a dock which is L=10 m downstream from a point directly opposite his starting point. a) In what direction (i.e. angle relative to the bank), should he aim his boat upstream? b) How long will it take for him to make this crossing?
2) The speed of an aircraft in still air (air speed) is 600 km/h. The wind velocity is 120 km/h blowing from the east. The aircraft is steered in the direction [E 30° S] that is a bearing of 120°. Find the ground velocity vector of the aircraft. (Draw an appropriate vector diagram.)
3) An airplane whose airspeed is 610 km / h is supposed to fly in a straight path 35.0 o North of East. But a steady 85 km / h wind is blowing from the North. In what direction should the plane head?
In: Math
There are an infinite number of vector and parametric equations for a given plane. Why is the scalar equation of a given plane unique?
In: Math
Q(1): An object moving vertically is at the given heights at the specified times. Find the position equation
s = 1/2at2 + v0t + s0
for the object At
t = 1 second, s = 141 feet
t = 2 seconds, s = 93 feet
t = 3 seconds, s = 13 feet
Q(2):
Find the equation
y = ax2 + bx + c
of the parabola that passes through the points. To verify your result, use a graphing utility to plot the points and graph the parabola.
(0, 3), (1, 4), (2, 3)
Q(3):
A natural history museum borrows $2,000,000 at simple annual interest to purchase new exhibits. Some of the money is borrowed at 5%, some at 6.5%, and some at 7.5%. Use a system of linear equations to determine how much is borrowed at each rate given that the total annual interest is $129,750 and the amount borrowed at 6.5% is four times the amount borrowed at 7.5%. Solve the system of linear equations using matrices.
In: Math
A certain element has a half life of
2.52.5
billion years.
a. You find a rock containing a mixture of the element and lead. You determine that
55?%
of the original element? remains; the other
9595?%
decayed into lead. How old is the?rock?
b. Analysis of another rock shows that it contains
6060?%
of its original? element; the other
4040?%
decayed into lead. How old is the? rock?
In: Math
| Find the family of curves orthogonal to: y4 = k(x − 2)3. |
In: Math
Solve y′′ + 4y′ + 3y = 15e^2t given y(0) = −7,y′(0) = 16 by the method of Laplace transforms.
In: Math
Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
a. f(x, y) = 2x3 + xy2 + 5x2 + y2 + 5
b. f(x, y) = x4 + y4 − 4xy + 3
c. f(x, y) = x3 − 48xy + 64y3
In: Math
If Dave is standing next to a silo of cross-sectional radius
r = 9
feet at the indicated position, his vision will be partially obstructed. Find the portion of the y-axis that Dave cannot see. (Hint: Let a be the x-coordinate of the point where line of sight #1 is tangent to the silo; compute the slope of the line using two points (the tangent point and (12, 0)). On the other hand, compute the slope of line of sight #1 by noting it is perpendicular to a radial line through the tangency point. Set these two calculations of the slope equal and solve for a. Enter your answer using interval notation. Round your answer to three decimal places.)
In: Math
Solve the given equation. 7 sin2(θ) − 36 sin(θ) + 5 = 0
In: Math
I planned on putting some math to work when I visited Golden Gate Park along with my new kite. I made sure my kite was always 100ft above the ground and I was able to calculate that the kite was moving horizontally at a speed of 9ft/s. Then I proceeded to calculate the rate at which the angle between the horizontal and the string was decreasing when I had let out about 250 ft of string. Can you help me determine what that rate was?
In: Math
The number of customers in a local dive shop depends on the amount of money spent on advertising. If the shop spends nothing on advertising, there will be 105 customers/day. If the shop spends $100, there will be 170 customers/day. As the amount spent on advertising increases, the number of customers/day increases and approaches (but never exceeds) 300 customers/day.
(a) Find a linear to linear rational function y = f(x) that calculates the number y of customers/day if $x is spent on advertising.
(b) How much must the shop spend on advertising to have 200 customers/day?
(c) Sketch the graph of the function y = f(x) on the domain 0 ≤ x ≤ 5000.
(d) Find the rule, domain and range for the inverse function from part (c).
Domain:
Range:
Explain in words what the inverse function calculates.
In: Math