Find the volume of revolution bounded by the curves y = 4 – x2 , y = x, and x = 0, and is revolved about the vertical axis.
What kind of transformation is illustrated in this figure ?
Find the x- and y-components of the vector d⃗ = (9.0 km , 35 ∘ left of +y-axis).
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].
Four less than a number is greater than -28
Find all complex solutions of 3x^2+3x+4=0.
(If there is more than one solution, separate them with commas.)
The complex solutions to the equation[tex]3x^2+3x+4=0 are x = (-3 + i\sqrt{39})/6 and x = (-3 - i\sqrt{39})/6.[/tex]
Explanation:To find all complex solutions of the quadratic equation [tex]3x^2+3x+4=0[/tex]e the quadratic formula:
[tex]x = \((-b \pm \sqrt{b^2-4ac})/(2a)\).[/tex]
Here, a = 3, b = 3, and c = 4. Plugging these values into the formula, we get:
[tex]x = \((-3 \pm \sqrt{3^2-4 \cdot 3 \cdot 4})/(2 \cdot 3)\).[/tex]
This simplifies to:
[tex]x = \((-3 \pm \sqrt{-39})/6\).[/tex]
Since the discriminant (under the square root sign) is negative, we know that the solutions will be complex. Using i to represent the square root of -1, we can write the solutions as:
[tex]x = \((-3 \pm i\sqrt{39})/6\).[/tex]
So, the complex solutions are [tex]x = (-3 + i\sqrt{39})/6 and x = (-3 - i\sqrt{39})/6.[/tex]
Approximately what percentage of scores falls below the mean in a standard normal distribution
A dress is priced at $21.42. The store is having a 30%-off sale. What was the original price of the dress?
During one year about 163 million adults over 18 years old in the United States spent a total of about 93 billion hours online at home. On average, how many hours per day did each adult spent online at home?
1. How do you write each number in scientific notation?
2 How do you convert the units to hours per day.
Final answer:
To write each number in scientific notation, express it as a product between 1 and 10 and a power of 10. Each adult spent an average of 570.55 hours per day online at home.
Explanation:
To write each number in scientific notation, we need to express it as a product of a number between 1 and 10 and a power of 10.
163 million can be written as 1.63 x 10⁸
93 billion can be written as 9.3 x 10¹⁰
To convert the units to hours per day, we need to divide the total number of hours by the number of adults.
So, each adult spent an average of (93 x 10¹⁰) / (163 x 10⁸) = 570.55 hours per day online at home. Scientific notation simplifies large numbers, facilitating computations and providing a concise representation of quantities in mathematical contexts.
Answer please math isn't my forte
How many three digit numbers can be made from the digits 1,\ldots,9 if repetitions of digits are not allowed?
There are 84 three digit numbers can be made from the digits 1, ..., 9
What is Combination?
A combination is a technique to determines the number of possible arrangements in a collection of items where the order of the selection does not matter.
Given that;
The numbers are,
⇒ 1, 2, ..., 9
Now,
All the three digit numbers can be made from the digits 1, ..., 9 are;
⇒ [tex]^{9} C_{3}[/tex]
⇒ 9! / 3! 6!
⇒ 9 × 8 × 7 / 6
⇒ 84
Thus, There are 84 three digit numbers can be made from the digits
1, ..., 9.
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A quadratic equation has a discriminant of 0. which describes the number and type of solutions of the equation?
Discriminant 0 in quadratic equation means 1 real solution—a repeated root where parabola touches x-axis once.
When the discriminant of a quadratic equation is 0, it means that the quadratic equation has exactly one real solution. This solution is considered a "double root" or "repeated root," meaning that the parabola defined by the quadratic equation touches the x-axis at exactly one point. Mathematically, this occurs when the quadratic equation has two identical roots.
The general form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], and the discriminant, denoted by [tex]\(b^2 - 4ac\),[/tex] helps determine the nature of the roots.
When the discriminant is zero [tex](\(b^2 - 4ac = 0\))[/tex], the quadratic equation has one real root. This happens when the parabola defined by the equation just touches the x-axis at one point. The solution is given by:
[tex]\[x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{{2a}}\][/tex]
The following conditions are:
D < 0 ; there are two non-real or imaginary roots which are complex conjugates
D = 0 ; there is one real root and one imaginary (non-real)
D > 0 ; there are two real distinct roots
Therefore the answer to this question is:
The solution has one real root and one imaginary root.
How to find the area of region composed of rectangles and/or right triangles?
If XYZ measures 75, what is the measure of XWZ ? A. 285 B. 210 C. 75 D. 150
(Its a circle, and its saying that the arc is 775 and wants to know what the rest of the circle is)
The measure of XWZ is 285
what is arc?The arc of a circle is defined as the part or segment of the circumference of a circle.
Given:
<XYZ = 75
As, we know central angle of 360
So, arc(XWZ) + <XYZ = 360
arc (XWZ) = 360 - 75
arc(XWZ) = 285
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Avery can run at 10 uph. The bank of a river is represented by the line 4x + 3y = 12, and Avery is at (7, 5). How much time does Avery need to reach the river?
ANSWER ASAP Find the value of x. A. sqrt 3 B. 3 sqrt 2/2 C. 3 sqrt 2 D. 3 sqrt 3
Answer:
B.
Step-by-step explanation:
If sam walks 650 meters in x minutes then write an algebraic expression which represents the number of minutes it will take sam to walk 1500 meters at the same average rate.
Find the distance between the points (13, 20) and (18, 8).
Answer: 13 units
Step-by-step explanation:
The distance formula to calculate distance between two points (a,b) and (c,d) is given by :-
[tex]d=\sqrt{(d-b)^2+(c-a)^2}[/tex]
Given points : (13, 20) and (18, 8)
Now, the distance between the points (13, 20) and (18, 8)is given by ;-
[tex]d=\sqrt{(8-20)^2+(18-13)^2}\\\\\Rightarrow\ d=\sqrt{(-12)^2+(5)^2}\\\\\Rightarrow\ d=\sqrt{144+25}\\\\\Rightarrow\ d=\sqrt{169}\\\\\Rightarrow\ d=13\text{units}[/tex]
Hence, the distance between the points (13, 20) and (18, 8) is 13 units.
Rationalize the denominator
Answer:
[tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex]
Step-by-step explanation:
Hello!
To rationalize the denominator, we have to remove any root operations from the denominator.
We can do that by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate simply means the same terms with different operations.
Rationalize[tex]\frac{6 - \sqrt10}{10 + \sqrt3}[/tex][tex]\frac{6 - \sqrt10}{10 + \sqrt3} * \frac{10 - \sqrt3}{10 - \sqrt3}[/tex][tex]\frac{(6 - \sqrt10)(10 - \sqrt3)}{100 - 3}[/tex][tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex]The answer is [tex]\frac{60 - 10\sqrt10-6\sqrt3+\sqrt30}{97}[/tex].
If a radius of a circle bisects a chord which is not a diameter, then ______
Answer:
the radius is perpendicular to the chord.
Step-by-step explanation:
The geometry is drawn in the image shown below in which AB is the chorh and O is the centre of the circle. Om is the radius which bisects the chord.(Given) So, AN = NB
From the image, considering ΔAON and ΔBON,
AO = BO (radius of circle)
AN = NB (given)
ON = ON (common)
So,
ΔAON ≅ ΔBON
Hence, ∠ANO = ∠BNO
Also, ∠ANO + ∠BNO = 180° (Linear Pair)
So,
∠ANO = ∠BNO = 90°
Hence, it is perpendicular to the chord.
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 24, leg BC has length 32, and hypotenuse AB has length 40.
In this question , it is given that
A right triangle ABC is shown. Leg AC has length 24, leg BC has length 32, and hypotenuse AB has length 40.
And we have to find the values of sin A and cos A .
[tex]sin A = \frac{opposite}{hypotenuse} = \frac{32}{40}[/tex]
[tex]sin A = \frac{4}{5}[/tex]
And
[tex]cos A = \frac{adjacent}{hypotenuse} = \frac{24}{40}[/tex]
[tex]cos A = \frac{3}{5}[/tex]
And these are the required values of sin A and cos A .
Find the probability of at least 2 girls in births. Assume that male and female births are equally likely and that the births are independent events. Round to three decimal places.
Need help with this please
If ON = 9x – 4, LM = 8x + 7, NM = x – 3, and OL = 4y – 8, find the values of x and y for which LMNO must be a parallelogram.
Find all complex solutions of 4x^2-5x+2=0.
(If there is more than one solution, separate them with commas.)
A cone has a volume of about 28 cubic inches. Which are possible dimensions for the cone?
a) radius 6 inches, height 3 inches
b) diameter 6 inches, height 3 inches
c) diameter 4 inches, height 6 inches
d) diameter 6 inches, height 6 inches
Now you can probe those options to see which leads to an approximate volume of 28 cubic inches.
a) radius 6 in, height 3 in
=> V = (1/3)*3.14*(6in)^2 * 3in = 113.04 in^3 => not possible
The area of a triangle is 92 cm2. the base of the triangle is 8 cm. what is the height of the triangle?
a. 11.5 cm
b. 23 cm
c. 4 cm
d. 46 cm
Answer: b. 23 cm
Step-by-step explanation:
We know that area of a triangle is given by :-
[tex]\text{Area}=\dfrac{1}{2}bh[/tex], where b is the base of triangle and h is the height of the triangle.
Given : The area of a triangle is [tex]92\ cm^2[/tex]
The base of the triangle = 8 cm
Let h be the height of the triangle , then we have
[tex]92=\dfrac{1}{2}(8)h\\\\\Rightarrow\ h=\dfrac{92}{4}\\\\\Rightarrow\ h = 23[/tex]
Hence, the height of the triangle = 23 cm
The following table shows information collected from a survey of students regarding their grade level and transportation they use to arrive at school. What is the probability that a randomly selected eighth grader takes the bus?
Write 11•47 using the distributive property. Then simplify.
Solve the following system by graphing.
x - y = 4
x + y = 2
What is the solution of the system?
(3, -1)
(3, 1)
(-1, 3)
The solution for the system of equation x - y = 4 and x + y = 2 is (3, -1).
What is an equation ?An equation is a combination of different variables, in which two mathematical expressions are equal to each other.
The given pair of equations,
x - y = 4 (1)
And x + y = 2 (2)
To find the solution of the equations,
Add both the equations,
x - y + x + y = 4 + 2
2x = 6
x = 3,
Substitute the value of x = 3 in equation (1),
3 - y = 4
y = -1
The values of x and y are 3 and -1 respectively.
Hence, option (A) is correct.
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What value of x makes the denominator of the function equal zero? y= 6/4x-40
The value of x makes the denominator of the function equal zero is 10
what is an equation?An equation is a mathematical expression that contains an equals symbol. Equations often contain algebra.
Given that:
y = [tex]\frac{6}{4x-40}[/tex]
Now the denominator is: 4x - 40
So, 4x =40
x =40/4
x =10
So, make the denominator 0, put x= 10.
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