Final answer:
The x- and y-components of the given vector are calculated using trigonometric functions, resulting in -5.16 km along the x-axis and 7.37 km along the y-axis, considering the direction of the vector relative to the axes.
Explanation:
The question asks to find the x- and y-components of the vector d⟷ = (9.0 km, 35° left of +y-axis). To solve this, we use trigonometric functions, specifically sine and cosine, because the vector makes an angle with the axis.
Given the vector makes a 35° angle to the left of the +y-axis, this effectively means it is 35° above the -x-axis (or equivalently, 55° from the +x-axis in the second quadrant). We can calculate the components as follows:
y-component: Dy = D*cos(35°) = 9.0 km * cos(35°) = 9.0 km * 0.8191 = 7.37 kmx-component: Dx = -D*sin(35°) = -9.0 km * sin(35°) = -9.0 km * 0.5736 = -5.16 km (negative because it's in the direction of the -x axis)The negative sign in the x-component indicates that the direction is towards the negative x-axis. Therefore, the x- and y-components of the vector are -5.16 km and 7.37 km, respectively.
1. [tex]\(\mathf{d}\): \(d_x = -3.8 \text{ km}, d_y = 8.2 \text{ km}\)[/tex]
2. [tex]\(\mathf{v}\): \(v_x = -4.0 \text{ cm/s}, v_y = 0\)[/tex]
3. [tex]\(\mathf{a}\): \(a_x = 10.3 \text{ m/s}^2, a_y = -14.7 \text{ m/s}^2\)[/tex]
To find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-components of vectors given in terms of magnitude and direction, we need to decompose the vectors using trigonometric functions. Let's go through each problem step by step.
1. Find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-components of the vector [tex]\( \mathf{d} = (9.0 \text{ km}, 25^\circ \text{ left of } +\mathbf{y}\text{-axis}) \)[/tex].
First, understand that "25° left of +[tex]\( y \)[/tex]-axis" means the vector is rotated 25° counterclockwise from the [tex]\( y \)[/tex]-axis.
Decomposition:
- Magnitude, [tex]\( d = 9.0 \text{ km} \)[/tex]
- Angle from the [tex]\( y \)-axis, \( \theta = 25^\circ \)[/tex]
To find the components:
- [tex]\( d_x = d \sin(\theta) \)[/tex]
- [tex]\( d_y = d \cos(\theta) \)[/tex]
However, since the angle is counterclockwise from the [tex]\( y \)-axis[/tex] and to the left, the [tex]\( x \)[/tex]-component is negative.
Therefore:
- [tex]\( d_x = -9.0 \sin(25^\circ) \)[/tex]
- [tex]\( d_y = 9.0 \cos(25^\circ) \)[/tex]
Calculating these:
- [tex]\( d_x \approx -9.0 \times 0.4226 \approx -3.8 \text{ km} \)[/tex]
- [tex]\( d_y \approx 9.0 \times 0.9063 \approx 8.2 \text{ km} \)[/tex]
So, the components are:
- [tex]\( d_x \approx -3.8 \text{ km} \)[/tex]
- [tex]\( d_y \approx 8.2 \text{ km} \)[/tex]
2. Find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-components of the vector [tex]\( \mathf{v} = (4.0 \text{ cm/s}, -x \text{-direction}) \).[/tex]
Since the vector is given in the [tex]\(-x\)[/tex]-direction, it means the entire magnitude is in the [tex]\( x \)[/tex]-direction and negative.
Decomposition:
- Magnitude, [tex]\( v = 4.0 \text{ cm/s} \)[/tex]
- Direction: [tex]\(-x\)[/tex]
Therefore:
- [tex]\( v_x = -4.0 \text{ cm/s} \)[/tex]
- [tex]\( v_y = 0 \text{ cm/s} \)[/tex]
So, the components are:
- [tex]\( v_x = -4.0 \text{ cm/s} \)[/tex]
- [tex]\( v_y = 0 \text{ cm/s} \)[/tex]
3. Find the [tex]\( x \)[/tex]- and [tex]\( y \)[/tex]-components of the vector [tex]\( \mathbf{a} = (18 \text{ m/s}^2, 35^\circ \text{ left of } -y \text{-axis}) \)[/tex].
"35° left of -[tex]\( y \)[/tex]-axis" means the vector is rotated 35° counterclockwise from the negative [tex]\( y \)[/tex]-axis.
Decomposition:
- Magnitude, [tex]\( a = 18 \text{ m/s}^2 \)[/tex]
- Angle from the [tex]\(-y \)[/tex]-axis, [tex]\( \theta = 35^\circ \)[/tex]
To find the components:
- [tex]\( a_x = a \sin(\theta) \)[/tex]
- [tex]\( a_y = -a \cos(\theta) \)[/tex]
However, since the angle is counterclockwise from the [tex]\(-y \)[/tex]-axis and to the left, the [tex]\( x \)[/tex]-component is positive.
Therefore:
- [tex]\( a_x = 18 \sin(35^\circ) \)[/tex]
- [tex]\( a_y = -18 \cos(35^\circ) \)[/tex]
Calculating these:
- [tex]\( a_x \approx 18 \times 0.5736 \approx 10.3 \text{ m/s}^2 \)[/tex]
- [tex]\( a_y \approx -18 \times 0.8192 \approx -14.7 \text{ m/s}^2 \)[/tex]
So, the components are:
- [tex]\( a_x \approx 10.3 \text{ m/s}^2 \)[/tex]
- [tex]\( a_y \approx -14.7 \text{ m/s}^2 \)[/tex]
The correct question is:
1. Find the [tex]$x$[/tex] - and [tex]$y$[/tex]-components of the vector [tex]$d \boxtimes=(9.0 \mathrm{~km}, 25 \boxtimes$[/tex] left of [tex]$+\mathrm{y}$[/tex]-axis).
2. Find the [tex]$\mathrm{x}$[/tex] - and [tex]$\mathrm{y}$[/tex]-components of the vector [tex]$\mathrm{v} \boxtimes=(4.0 \mathrm{~cm} / \mathrm{s},-\mathrm{x}$[/tex]-direction [tex]$)$[/tex].
3. Find the [tex]$x$[/tex] - and [tex]$y$[/tex]-components of the vector [tex]$a \boxtimes=(18 \mathrm{~m} / \mathrm{s} 2,35 \boxtimes$[/tex] left of [tex]$-y$[/tex]-axis [tex]$)$[/tex].
Evaluate f(x) when x=9
The initial temperature of a cup of tea is 200ºF. The surrounding temperature is 70ºF, and the value of the constant k is 0.6.
Applying Newton's cooling model, the temperature of the tea after 2 hours will be ___
ºF. round to the nearest integer.
Newton's Law of Cooling states that the change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature over time.
Therefore when expressed mathematically, this is equivalent to:
dT = - k (T – Ts) dt
dT / (T – Ts) = - k dt
Integrating:
ln [(T2– Ts) / (T1– Ts)] = - k (t2 – t1)
Before we plug in the values, let us first convert the temperatures into absolute values R (rankine) by adding 460.
R = ˚F + 460
T1 = 200 + 460 = 660 R
Ts = 70 + 460 = 530 R
ln [(T2– 530) / (660 – 530)] = - 0.6 (2 - 0)
T2 = 569.16 R
T2 = 109 ºF
Answer: After 2 hours, it will be 109 ºF
Answer:
109 degrees
Step-by-step explanation:
The measure of an inscribed angle is 110°. What is the measure of the intercepted arc? 55° 110° 220°
the measurement of an inscribed angle is half of the measure of the intercepted arc.
the inscribed angle is 110 degrees, so we multiply 110 by 2.
110*2 = 220 degrees
The measure of the intercepted arc is 220 degrees
220° is the measure of the intercepted arc
The variable Z is inversely proportional to X. When X is 6, Z has the value 2. What is the value of z . When x = 13
Round to at least the thousandths place if needed.
In the diagram below, what is the approximate length of the minor arc ?
arc length = given angle/360 x 2 x PI x radius
120/450 x 2 x 3.14 x 23 = 48.17
round to 48 cm
Answer:
Option A is correct
the approximate length of the minor arc is, 48 cm
Step-by-step explanation:
Length of an arc is given by:
[tex]l = r \theta[/tex] .....[1]
here r is the radius of the circle from the center and [tex]\theta[/tex] is the angle in radian.
From the given figure, we have;
r = 23 cm
Use conversion:
1 degree = 0.0174533 radian
then
120 degree = 2.094396
[tex]\theta = 2.094396[/tex] radian.
Substitute these in [1] we have;
[tex]l = 23 \cdot 2.094396 = 48.171108[/tex] cm
Therefore, the approximate length of the minor arc is, 48 cm
Identify the cross section shown.
circle
trapezoid
rectangle
pentagon
The cross section shown in the picture below include the following: B. rectangle.
In Mathematics and Euclidean Geometry, a rectangle refers to a type of quadrilateral or polygon in which its opposite sides are equal and all the angles that are formed are right angles.
Generally speaking, a circle has no edge or sides. A trapezoid is a type of quadrilateral that has one pair of parallel sides and one pair of non-parallel sides. Additionally, a pentagon is a regular polygon that comprises 5 sides.
By critically observing the cross section shown in the picture below, we can logically deduce that it represents a rectangle because its opposite sides are equal.
Which is a counterexample that disproves the conjecture? After completing several multiplication problems, a student concludes that the product of two binomials is always a trinomial.
The sum of 2 numbers is 7. if one number is subtracted from the other, the result is -1. find the numbers
I need help with #18 c-i
Find the angle between the given vectors to the nearest tenth of a degree. u = <-5, 8>, v = <-4, 8>
Answer: 5.4 degrees
Step-by-step explanation:
look up online "how to find the angle between 2 vectors, and it'll show you how"
A gardener has 27 pansies and 36 daisies. He plants an equal number of each type of flower in each row.What is the greatest possible number of pansies in each row?
Can you use the ASA Postulate or the AAS Theorem to prove the triangles congruent?
Answer:
ASA only
Step-by-step explanation:
Given is a picture of two triangles with one side and one angle congruent.
Comparison of these two triangles given
side = side
one angle = one angle (given)
Second angle = second angle (Vertically opposite angles)
Thus we find here that two angles and one corresponding side are congruent.
HEnce we say that these two triangles are congruent by ASA theorem
ASA theorem can be applied here because the equal side is between the two congruent angles.
Instead if the side is not between the congruent angles but corresponding side then we can only use AAS
SO here ASA is correct.
Answer:
ASA only
Step-by-step explanation:
We are given that two triangles in which
An angle of triangle is equal to its corresponding angle of second triangle.
One side of a triangle is equal to one side of other triangle.
ASA postulate: It states that two angles and included side of one triangle are congruent to its corresponding angles and corresponding side of other triangle , then the two triangles are congruent.
AAS postulate: It states that two angles and non- included side of one triangle are congruent to its corresponding two angles and its corresponding side of another triangle, then the triangles are congruent by AAS postulate.
In triangle AOB and COD
[tex]\angle AOB= \angle COD[/tex] (Vertical angles are equal )
[tex]\angle ABO=\angle CDO[/tex] ( Given )
[tex]OB=OD[/tex] (Given )
[tex]\triangle AOB\cong \triangle COD[/tex] ( ASA Postulate )
Answer: ASA only
When a line has an undefined slope what will any two points on the line have in common
HELP! Type the correct answer in each box. Use numerals instead of words, if necessary, use / for the fractions bar(s)
The piecewise function is:
[tex]\[ f(x) = \begin{cases} 4 & \text{if } -1 \leq x \leq 1 \\ x - 1 & \text{if } 3 \leq x \leq 5 \end{cases} \][/tex]
To determine the piecewise function represented by the given coordinates, let's examine the points and their corresponding intervals.
The coordinates are (-1, 4), (1, 4), (3, 2), and (5, 4).
The points (-1, 4) and (1, 4) have the same y-coordinate, indicating that on the interval [tex]\(-1 \leq x \leq 1\)[/tex], the function has a constant value of 4. Therefore, the piecewise function for this interval is f(x) = 4 for [tex]\(-1 \leq x \leq 1\).[/tex]
The points (3, 2) and (5, 4) indicate that on the interval [tex]\(3 \leq x \leq 5\)[/tex], the function is a line passing through these two points. We can find the slope (m) and y-intercept (b) for this line.
[tex]\[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{4 - 2}{5 - 3} = \frac{2}{2} = 1 \][/tex]
Using the point (3, 2), we can find the y-intercept:
2 = 1(3) + b
b = -1
Therefore, the equation for the line on the interval [tex]\(3 \leq x \leq 5\) is \(f(x) = x - 1\) for \(3 \leq x \leq 5\).[/tex]
Putting it all together, the piecewise function is:
[tex]\[ f(x) = \begin{cases} 4 & \text{if } -1 \leq x \leq 1 \\ x - 1 & \text{if } 3 \leq x \leq 5 \end{cases} \][/tex]
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Using the graph, find this information about the child’s movement in the vertical (y) direction:
1. the best trigonometric function to start with
2. the amplitude
3. the period
4. any horizontal displacement of the graph.
5. any vertical displacement of the graph.
Type your response here:
What is the equation of the line?
Evaluate. 5.4 - 1.3
Determine the common ratio and find the next three terms of the geometric sequence.
3/4,3/10,3/25,...
The next three terms of the sequence are 6/125, 12/625 and 24/3125
Geometric sequenceThe nth term of a geometric sequence is given as:
Tn = ar^n-1
Given the geometric sequence
3/4,3/10,3/25,...
Find the common ratio
r = 3/10 * 4/3
r = 2/5
Find the 4th, 5th and 6th terms
T4 = (3/4)(2/5)^3
T4 = 3/4 * 8/125
T4 = 6/125
T5 = (3/4)(2/5)^4
T5 = 3/4 * 16/625
T5 = 12/625
T6 = (3/4)(2/5)^5
T6 = 3/4 * 32/3125
T6 = 24/3125
Hence the next three terms of the sequence are 6/125, 12/625 and 24/3125
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Can someone help me
2.89-2.84 = 0.05 cent increase
0.05/2.84 = 0.0176
0.0176 = 1.8% increase
Find the exact values of sin A and cos A. Write fractions in lowest terms. A right triangle ABC is shown. Leg AC has length 15, leg BC has length 20, and hypotenuse AB has length 25.
In the right triangle with sides 15, 20, and 25, the exact values of sin A and cos A are 4/5 and 3/5, respectively.
In the right triangle ABC, we can use the given side lengths to find the trigonometric ratios for angle A. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (AB) is equal to the sum of the squares of the lengths of the legs (AC and BC).
Let's denote the angle A as angle A:
Find sin A:
sin A = (opposite / hypotenuse) = BC / AB = 20 / 25 = 4/5.
Find cos A:
cos A = (adjacent / hypotenuse) = AC / AB = 15 / 25 = 3/5.
So, the exact values are sin A = 4/5 and cos A = 3/5.
In summary, in the right triangle ABC with side lengths AC = 15, BC = 20, and AB = 25, the exact values of sin A and cos A are 4/5 and 3/5, respectively.
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Which equation is not equivalent to the formula m = ca?
A: [tex]c = \frac{m}{a} [/tex]
B: [tex]a = \frac{c}{m} [/tex]
C: [tex]a = \frac{m}{c} [/tex]
D: m = ac
Answer:m=ca i remember my teacher telling me how to do this.
Step-by-step explanation:
Two containers, one for 3 liters and one for 5 liters, how would you measure 4 liters
Quadrilateral ABCD is a parallelogram. ,  bisects , and . What is the best name for quadrilateral ABCD?
A.
rectangle
C.
isosceles trapezoid
B.
rhombus
A surface on which all points are at the same potential is referred to as
Use Gauss-Jordan elimination to solve the following linear system: 5x – 2y = –2 –x + 4y = 4 A. (–6,–5) B. (2,0) C. (0,1) D. (–1,0)
The answer is C.(0,1)
5x - 2y = -2 -x + 4y = 4
5(0) - 2(1) = -2 -(0) + 4(1) = 4
If four times a number plus 3 is 11, what is the number
Solve cos x +sqrt2 = -cos x for x over the interval (0,2pi) .
Final answer:
The equation cos x + sqrt(2) = -cos x is solved by combining like terms and isolating cos x, resulting in the solutions x = 3π/4 and 5π/4 over the interval (0,2π).
Explanation:
To solve the equation cos x + sqrt(2) = -cos x for x over the interval (0,2π), first combine like terms by adding cos x to both sides of the equation, resulting in:
2 cos x + sqrt(2) = 0
Isolate cos x by subtracting sqrt(2) from both sides:
2 cos x = -sqrt(2)
Divide both sides by 2 to solve for cos x:
cos x = -sqrt(2)/2
The value of cos x = -sqrt(2)/2 corresponds to angles in the second and third quadrants. So, the solutions for x in the interval (0,2π) are:
x = 3π/4, 5π/4
Find the greatest common factor of the following monomials 28g5h2 12g6h5
Also explain step by step how to solve these problems please
A line passes through the point (-8,3) and has a slope of 5/4.
Write an equation in slope-intercept form for this line.
3. Simplify log3 20 - log3 5