Precalc: Vector word problem:
A channel flows from north to south at a rate of 15 mph.
A sailboat heads at an angle of 30 degrees upstream at 40 mph.
A strong wind blows towards northwest at 30 mph.
a. Write each vector in component form.
Answers
I can make this much from that, check the picture below.
A vector can be defined as an element of a vector space, which represents an object that has both magnitude and direction.
Assuming that; [tex]\left \{ {{x=rcos \theta} \atop {y=rsin \theta}} \right.[/tex]
For the channel:
Rate, r = 15 mph.The vector representing the flow from North to South is given by:
[tex]15 cos (\frac{3\pi}{2} ), \; 15 sin (\frac{3\pi}{2} )\\\\15(0),\; 15(-1)[/tex]
Vector = 0, -15
For the sailboat:
Rate, r = 40 mph.Angle = 30 degrees upstream.The vector representing the sailboat is given by:
[tex]40 cos (30 ), \; 40 sin (30)\\\\30(\frac{\sqrt{3} }{2} ),\; 30(\frac{1 }{2} )\\\\15\sqrt{3} ,\;15[/tex]
Vector = [tex]15\sqrt{3} ,\;15[/tex]
For the strong wind:
Rate, r = 30 mph.The vector representing the strong wind blowing towards northwest is given by:
[tex]30 cos (\frac{3\pi}{4} ), \; 30 sin (\frac{3\pi}{4} )\\\\30(\frac{-\sqrt{2} }{2} ),\; 30(\frac{\sqrt{2} }{2} )\\\\-15\sqrt{2},\;15\sqrt{2}[/tex]
Vector = [tex]-15\sqrt{2},\;15\sqrt{2}[/tex]
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Related Questions
URGENT!!!
Kay was walking along the bridge above the river. She accidentally lost her grip on her camera when she was trying to get it out to photograph massive bird nests in the trees. The height of the camera above the water can be modeled by the equation
y = -16t^2 + 400, where t represents the seconds after the camera is dropped. How long will it take (in seconds) before Kay sees the splash from the camera hitting the water.
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The given equation in this case is:
y = -16 t^2 + 400
We can see that the slope of the equation is negative (-16) which means that the position of Kay is on the positive y-axis , while the bottom of the river is in the origin. This is supported by taking t = 0, the initial y becomes y = 400. Therefore the position of Kay now is at (0, 400).
So to find for the time when the camera hits the water, then y must be zero (t, 0) so we have to find for t in this case.
When y = 0
0 = -16 t^2 + 400
16 t^2 = 400
t^2 = 25
t = 5 s
Therefore it takes 5 seconds for the camera to hit the water and for Kay to see a splash.
Answer:
5 seconds
Step-by-step explanation:
Kay was walking along the bridge above the river. She accidentally lost her grip on her camera when she was trying to get it out to photograph massive bird nests in the trees. The height of the camera above the water can be modeled by the equation
y = -16t^2 + 400, where t represents the seconds after the camera is dropped.
When splash from the hits the water then the height becomes 0
height is y so we replace y with 0 and solve for t
0 = -16t^2 + 400
Subtract 400 on both sides
-400 = -16t^2
Divide by 16 on both sides
25= t^2
take square root on both sides
t= 5
It takes 5 seconds
there are 24 people using the gym, tha ratio of men to women is 2;1, hoiw many men are using the gym
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2 men for every 1 woman
24/3 =8
8*2 = 16
there are 16 men and 8 women
There are 60 minutes in an hour. The total number of minutes is a function of the number of hours, as shown in the table. Does this situation represent a linear or non-linear function, and why?
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Answer:
It represents a linear function because there is a constant rate of change.
Step-by-step explanation:
Since, if the rate of change for y ( output value ) with respect to x ( input value ) remains constant for a function then it is called linear function.
Here, number of hours represents input value and minutes represents the output value,
Now, By the given table,
The function is passing through the points (1,60), (2,120), (3,180), (4,240) and (5,300),
Also,
[tex]\frac{120-60}{2-1}=\frac{180-120}{3-2}=\frac{240-180}{4-3}=\frac{300-240}{5-4}=60[/tex]
⇒ The rate of change output value with respect to input value remains constant,
Hence, It represents a linear function because there is a constant rate of change.
Jeanie has $500 to spend on dance lessons there's $50 one time registration fee and the lessons cost 35 each how many lessons can Jeanie take
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450 = 35x
450/35 = x
12.85 = x
12 lessons
There are 6 students in a small class. To make a team, the names of 2 of them will be drawn from a hat. How many different teams of 2 students are possible?
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Using the combinatorial formula
a.Find the perimeter of the triangle.
b.Find the area of the triangle.
c. If tile cost $6 per square centimeter, how much will it cost to tile the triangle?
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B. You can find the height by dividing the traingle into 2 right traingles with 5 being the hypotenuse and 3 being one of the legs.
5^2=3^2+x^2
25=9+x^2
16=x^2
x=4
Area= 6*4/2= 12 units^2
C. 12*6=$72
Jan needed to empty 18 gallons of water from a tub into jugs that hold 0.5 gallons of water. How many jugs did Jan need?
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divide 18 by 0.5
18/0.5 = 36
36 jugs are needed
The school choir has 84 members. The ratio of girls to boys in the color is 3: 4. How many members are girls?
show steps !
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To calculate " 3 fourth" of 84:
i) we divide 84 by 4, that is 84/4=21,
ii) we multiply the result of the division by 3: 21*3=63
Answer: 63 members of the choir are girls.
Remark: whenever we want to find a/b or "something":
i) divide "something" by b
ii) multiply the result by a
The parent function, f(x) = 5x, has been vertically compressed by a factor of one-half, shifted to the left three units and up two units.
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The parent function, f(x) = 5x, has been vertically compressed by a factor of one-half, shifted to the left three units and up two units.
Explanation:To vertically compress the parent function f(x) = 5x by a factor of one-half, we multiply the function by the compression factor, which is 1/2. So the compressed function is f(x) = (1/2)(5x) = 2.5x.
To shift the function to the left three units, we subtract the shift amount from the x-variable. So the shifted function is f(x + 3) = 2.5(x + 3) = 2.5x + 7.5.
To shift the function up two units, we add the shift amount to the y-variable. So the final transformed function is f(x + 3) + 2 = 2.5x + 7.5 + 2 = 2.5x + 9.5.
Anyone Know This Answer ASAP ?
Answers
multiply 35 by tan(42)
this equals 31.5141
rounded to 31.51
What is the simplified form of 14x^5y^9 divided by 2xy^3
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7x^5y^9^+^1y^3
In triangle ABC angle c is a right angle find the value of the trig function indicated. find sec A if c=25, a=7
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The value of trigonometric function sec A is,
⇒ sec A = 25/24
What is mean by Triangle?A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.
Given that;
In triangle ABC angle c is a right angle.
And, c = 25, a = 7
Now, We can draw the tringle ABC.
And, Find the value of side b by using Pythagoras theorem as;
⇒ c² = a² + b²
⇒ 25² = 7² + b²
⇒ 625 = 49 + b²
⇒ b² = 625 - 49
⇒ b = 24
We know that;
⇒ sec A = Hypotenuse / adjacent
Hence, We get;
⇒ sec A = Hypotenuse / adjacent
⇒ sec A = 25 / 24
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Write the given system without the use of matrices. d/dt (x y z) =( 1 −1 6 3 −8 1 −2 7 5) (x y z) + (1 2 2)
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[tex]\begin{cases}\dfrac{\mathrm dx}{\mathrm dt}=x-y+6z+1\\\\\dfrac{\mathrm dy}{\mathrm dt}=3x-8y+z+2\\\\\dfrac{\mathrm dz}{\mathrm dt}=-2x+7y+5z+2\end{cases}[/tex]
1. Describe how factoring a quadratic expression ax2 + bx + c, where a ≠ 1, is different from factoring x2 + bx + c.
2. Two students factored 2x2 + 6x – 20. Keiko said that the factorization was (2x – 4)(x + 5). Ray gave the factorization as (x – 2)(2x + 10). Confirm that both of these factorizations are correct. Then explain why they are not complete.
3. Explain the relationship between the factors of a quadratic expression, the roots of the related quadratic equation, and the x-intercepts of the graph of the related function.
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Answer:
1. Dividing the expressions [tex]ax^2+bx+c[/tex] by a is a different step.
2. Yes, both of these factorization are correct. They are not complete because they can be factored further.
3. The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
Step-by-step explanation:
1.
To factorize the quadratic expressions [tex]ax^2+bx+c[/tex] first we divide it by a. Then we factorize it same as [tex]x^2+bx+c[/tex].
It means all the steps of factoring a quadratic expressions [tex]ax^2+bx+c[/tex] and [tex]x^2+bx+c[/tex] are same except the first step, i.e., divide the expressions [tex]ax^2+bx+c[/tex] by a.
2.
The given quadratic expression is
[tex]P(x)=2x^2+6x-20[/tex]
[tex](2x-4)(x+5)=2x(x+5)-4(x+5)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
[tex](x-2)(2x+10)=x(2x+10)-2(2x+10)\Rightarrow 2x^2+10x-4x-20=2x^2+6x-20=P(x)[/tex]
The product of factors is equal to the given expression. It means both of these factorization are correct.
They are not complete because they can be factored further.
[tex](2x-4)(x+5)=2(x-2)(x+5)[/tex]
[tex](x-2)(2x+10)=(x-2)2(x+5)[/tex]
3.
If the factored form of a quadratic expression is defined as
[tex](x-a)(x-b)[/tex]
Then the related quadratic equation is
[tex](x-a)(x-b)=0[/tex]
[tex]x=a,b[/tex]
The roots of the quadratic equation are a and b.
The related function is
[tex]f(x)=(x-a)(x-b)[/tex]
The x-intercepts of the function are a and b because at x=a and x=b the value of function is 0.
The roots of the related quadratic equation are the x-intercepts of of the related function and factors of the expression are difference of x and roots of the related quadratic equation.
It should be noted that in order to factorize the quadratic expression, one will have to divide it by a.
FactorizationThe factorization of the quadratic expression ax² + bx + c is different from factoring x² + bx + c as one has to first divide it by a.
Secondly, the factorization by the students isn't complete because they can be factored further.
Lastly, the relationship between the factors of a quadratic expression, the roots of the related quadratic equation is that the roots are the x-intercept of the related function.
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12 pounds of apples. Each pounds costs $3. If she gives the cashier two $20 bills, how much change should she receive.
Answers
36
2 20 dollar bills = 40
4 dollars in change back
What percentage of muscle weight would you have if 60 pounds out of 140 pounds is muscle weight? If muscle weight is 90 pounds, how much does the person weigh?
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solve for "x".
The area of a rectangle is 35 feet^2, and the length of the rectangle is 8 feet less than three times the width. Find the dimensions of the rectangle
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L=3W-8 (feet)
The formula for the area of this rect. is now A = W(3W-8) = 35 ft^2
3W^2 - 8W - 35 = 0
There are various ways in which we could solve this equation for W.
I have chosen to "factor by grouping."
3W^2 - 8W -35 = 0
3W^2 - 15W + 7W - 35 = 0
3W(W-5) + 7(W-5) = 0
Then (3W+7)(W-5)=0, and W=-7/3 or W=5. Since W denotes width, and width cannot be negative, omit W=-7/3. Thus, W=width=5 feet.
Next, L=3W-8=3(5)-8=15-8=7 feet.
Check: Does (5 feet)(7 feet) come out to 35 square feet? YES.
The dimensions of this rectangle are 5 by 7 feet.
The width of the rectangle is 5 feet, and the length is 7 feet.
To find the dimensions of the rectangle, set up the equation (3W - 8)W = 35, where W is the width and solve for W. You'll find that the width is 5 feet, and the length, which is 8 less than three times the width, is therefore 7 feet.
To find the dimensions of the rectangle with an area of 35 feet2 where the length is 8 feet less than three times the width, let's denote the width as W and the length as L. The problem gives us two equations:
Area: L imes W = 35Length Relationship: L = 3W - 8Now, replace L in the first equation with 3W - 8, so we have:
(3W - 8)W = 35
Expanding the equation, we get:
3W2 - 8W = 35
Solve this quadratic equation to find the width W, and then substitute W back into L = 3W - 8 to find the length L. Let's solve the quadratic equation:
3W2 - 8W - 35 = 0
By factoring or using the quadratic formula, we can find that W=5 and L=7 are the solutions that makes sense for the dimensions of the rectangle.
So, the width of the rectangle is 5 feet, and the length is 7 feet.
4% of what number is 6
Answers
4% = 0.04
divide 6 by 0.04 = 150
6 is 4% of 150
double check 150*0.04 = 6
The waitress and hostess at a restaurant share the tips at the end of every shift. Between the two of them, they earn an average of $75 in tips. The waitress take $55 and the hostess takes $20. If they continue to earn money at this rate, how much will the waitress receive if they earn $300 at the end of their next shift?
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thy get 55/75 and you want to know x/300
55/75 x X/300
cross multiply 55 x 300 = 16500
divide that by 75
16500/75 = 220
x = 220
they will get $220
11:4...added = 15
waitress : 11/15(300) = 220 <==
hostess : 4/15(300) = 80
Prove that if a and b are two matrices such that a+b and a*b are both defined, then a and b are square matrices
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six times the difference between a number and 4 equals the product of the number and -2. find the number.
I just want maybe a reteach for this.
*Solving problems with unknown numbers*
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6a - 24 = a - 2
-a -a
5a - 24 = -2
+ 24 +24
5a = 22
/5 /5
a = 22/5
For which function is f(x) not equal to f-1(x)?
A- f(x)=2-x
B-f(x)=2/x
C=f(x)=-x
D=f(x)=2x
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If f(x) or y = 2x, the inverse is x = 2y or y=x/2 which is not f(x).
Answer:
[tex]f(x)= 2x[/tex]
Step-by-step explanation:
For which function is f(x) not equal to f^-1(x)
LEts check with each function, we find out inverse for each option
[tex]f(x)= 2-x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= 2-x[/tex]
[tex]x= 2-y[/tex], Add y on both sides
[tex]x+y= 2[/tex], Subtract x from both sides
[tex]y= 2-x[/tex], Inverse is equal to f(x)
[tex]-f(x)= \frac{2}{x}[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]-y= \frac{2}{x}[/tex]
[tex]-x= \frac{2}{y}[/tex], cross multiply it
[tex]-xy = 2[/tex], divide by x on both sides
[tex]-y= \frac{2}{x}[/tex], Inverse is equal to f(x)
[tex]f(x)= -x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= -x[/tex]
[tex]x=-y[/tex], Add y on both sides
[tex]x+y= 0[/tex], Subtract x from both sides
[tex]y=-x[/tex], Inverse is equal to f(x)
[tex]f(x)= 2x[/tex]
Replace f(x) by y, then switch the variables and solve for y
[tex]y= 2x[/tex]
[tex]x= 2y[/tex], divide by 2 on both sides
[tex]\frac{x}{2}=y[/tex], Inverse is not equal to f(x)
How to find two fractions between 1/2 and 1/3
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the common denominator must be a multiple of both 2 and 3, for example 6, 12, 24 etc...
let the common denominator be 24
(1/2)(12/12)=12/24
(1/3)(8/8)=8/24
2. so , 9/24, 10/24, 11/24 are all fractions between 1/2 and 1/3
Evaluate the line integral for x^2yds where c is the top hal fo the circle x62 _y^2 = 9
Answers
[tex]\mathbf r(t)=\langle x(t),y(t)\rangle=\langle3\cos t,3\sin t\rangle[/tex]
where [tex]0\le t\le\pi[/tex]. Then the line integral is
[tex]\displaystyle\int_Cx^2y\,\mathrm dS=\int_{t=0}^{t=\pi}x(t)^2y(t)\left\|\frac{\mathrm d\mathbf r(t)}{\mathrm dt}\right\|\,\mathrm dt[/tex]
[tex]=\displaystyle\int_{t=0}^{t=\pi}(3\cos t)^2(3\sin t)\sqrt{(-3\sin t)^2+(3\cos t)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle3^4\int_{t=0}^{t=\pi}\cos^2t\sin t\,\mathrm dt[/tex]
Take [tex]u=\cos t[/tex], then
[tex]=\displaystyle-3^4\int_{u=1}^{u=-1}u^2\,\mathrm du[/tex]
[tex]=\displaystyle3^4\int_{u=-1}^{u=1}u^2\,\mathrm du[/tex]
[tex]=\displaystyle2\times3^4\int_{u=0}^{u=1}u^2\,\mathrm du[/tex]
[tex]=54[/tex]
What is the graph of the system? y ≤ -x – 1 y ≥ 2x + 4
Answers
See the attached graph
Step-by-step explanation:The comparison operator includes "or equal to" in each case, so the graph of the boundary line is solid. Each bounary line can be graphed in the usual way. Considering the y-intercept and slope is often easiest when the equations are presented in this form:
y = -x -1y = 2x +4Since the solution space includes y-values less than those in the first line, the graph is shaded below that line.
Since the solution space includes y-values greater than those in the second line, the graph is shaded above that line.
Where the graph is doubly-shaded is the solution space for the system of equations.
What is the sum of two and two
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the sum of three number 20. the sum of twice the first number, and 5 times the third number is 71. the difference between 4 times the first number and the scond number is 27. find the three numbers. Helllp
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b = second number
c = third number
a + b + c = 20 [1]
2a + 5c = 71 [2]
4a - b = 27 [3]
Rearrange [2] and [3] to express in terms of a:
c = (71 - 2a)/5 [2]
b = 4a - 27 [3]
Sub' these into [1], rearrange and solve:
a + (4a - 27) + ((71 - 2a)/5) = 20
5a - 27 + (71 - 2a)/5 = 20
5(5a - 27) + 71 - 2a = 100
25a - 135 + 71 - 2a = 100
23a - 64 = 100
a = 164/23
Sub a-value into rearranged [2] and [3]:
c = (71 - 2(164/23))/5
= (1305/23)/5
= 261/23
b = 4(164/23) - 27
= 656/23 - 27
= 35/23
Polk elementary school has three classes for each grade level kindergarten through fifth grade if there is an average of 27 students per class about how many students attend Polk school
Answers
If there are approximately 27 children in each class, than 27*18 = ? your answer.
There are 147 peaple attending a diner party if each table can seat 7 peaple how many tables are needed for the dinner party?
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Solve for "W"
-1 > 2W + 7
Answers
-1>2W+7
-8>2w
w = -8/2 = -4
w>-4
-8 > 2w
-8/2 > 2w/2
-4 > w
hope this helps