a
particle moves along a path in the xy-plane. the path is given by
the parametric equations x(t)=sin(3t) and y(t)=cos(3t), help with
steps A-E
a. Find the velocity b. Find the acceleration. c. Find the speed and simplify your answer completely. d. Find any times at which the particle stops. Thoroughly explain your answer. e. Use calculus to

The expression that represents the amount of ore sold and how much ore can the mine sell after extracting ore for 12 hours is option B: The expression is 2t−14t. The amount of ore is 21 metric tons.

The reasoning for the selection of the expression and amount of ore can the mine sell after extracting ore for 12 hours  is as follows.

1: Determine the amount of coal used for electricity generation in terms of t.

The mine uses 14 tons of coal every hour, so the total amount used for electricity generation is 14t.

2: Determine the total amount of coal extracted in terms of t.

The mine extracts 2 tons of coal every hour, so the total amount extracted is 2t.

3: Calculate the amount of coal sold in terms of t.

To find the amount of coal sold, subtract the amount used for electricity generation from the total amount extracted: 2t - 14t.

4: Determine the amount of coal sold after 12 hours.

Substitute t = 12 into the expression:

2(12) - 14(12) = 24 - 168 = -144.

However, since the mine uses 14 tons of the extracted coal every hour, it cannot sell more coal than it extracts. So, the correct expression should be 2t - 14 (without the t for the amount used for electricity generation).

5: Calculate the amount of coal sold after 12 hours using the corrected expression.

Substitute t = 12 into the expression: 2(12) - 14 = 24 - 14 = 10 metric tons.

The correct expression should be 2t - 14, and the amount of coal the mine can sell after extracting coal for 12 hours is 10 metric tons. Hence, the correct answer is option B.

Note: The question is incomplete. The complete question probably is: A mine extracts 2 metric tons of coal in an hour. The mine uses 14 ton of the extracted coal every hour to generate electricity for the mine and sells the rest. If t is the number of hours spent mining, which expression represents the amount of ore sold? How much ore can the mine sell after extracting ore for 12 hours? A) The expression is 2t−1/4t. The amount of ore is 23 3/4 metric tons. B) The expression is 2t−1/4t. The amount of ore is 21 metric tons. C) The expression is 2t+1/4t. The amount of ore is 24 metric tons. D) The expression is 2t+1/4t. The amount of ore is 24 1/4 metric tons.

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The coordinates of point M come out to be 4.8, 4.4

This case is solved by using the section formula which states that

The coordinate of point P that divides the line segment AB in the ratio of m:n where the coordinate of A is [tex]x_1,y_1[/tex] and the B is [tex]x_2,y_2[/tex] is described as

[tex]\frac{mx_2+nx_1}{m+n}[/tex],[tex]\frac{my_2+ny_1}{m+n}[/tex]

The line to be divided = PQ

Coordinates of P = (6,2)

Coordinates of Q = (3,8)

Ratio = 2:3

Thus, the coordinates of M = [tex]\frac{2*3+3*6}{2+3}[/tex],[tex]\frac{8*2+2*3}{2+3}[/tex]

= 24/5 , 22/5

= 4.8, 4.4

Point M with coordinates (4.8,4.4) lies on PQ and divides the segment so that the ratio of PM-MQ is 2-3

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The correct statement is given as follows:

The function g(t) reveals the market value of the house increases by 3.6% each year.

An exponential function has the definition presented as follows:

[tex]y = ab^x[/tex]

In which the parameters are given as follows:

The parameter b for this problem is given as follows:

b = 1.036.

As the parameter b has an absolute value greater than 1, the function is increasing, with a rate given as follows:

1.036 - 1 = 0.036 = 3.6% a year.

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Answer:

Step-by-step explanation:

Answer:

13 years

Step-by-step explanation:

Intuition for how sons can collectively win after a certain period of time:- After a certain period of time the father's age will increase by that certain period of time (say 5 years) but for the sons (since there are 3 of them) their collective age will increase by three times that of their father (5 for each 1 one them). Therefore there exist a time after which collective increase in sons' age can cover the current gap of 13 years.

The inverse for each relation:

1. {(1,‐2), (2, 3),(3, ‐3),(4, 2)} - {(-2, 1), (3, 2), (-3, 3), (2, 4)}

2. {(4,2),(5,1),(6,0),(7,‐1)} - {(2, 4), (1, 5), (0, 6), (-1, 7)}

3. Inverse equation: y=(-1/8)x+3/8

4. Inverse equation: y=3/2x+15/2

5. Inverse equation: y=2x-20

6. Inverse equation: y=[tex]x^{(1/2)}+3[/tex]

7. Since fog(x) = gof(x) = x, f and g are inverse functions.

8. Since fog(x) = gof(x) = x, f and g are inverse functions.

1. To find the inverse of the relation, we need to swap the positions of x and y for each point and then solve for y.

{(1, -2), (2, 3), (3, -3), (4, 2)}

Inverse: {(-2, 1), (3, 2), (-3, 3), (2, 4)}

2. Again, we swap x and y and solve for y.

{(4, 2), (5, 1), (6, 0), (7, -1)}

Inverse: {(2, 4), (1, 5), (0, 6), (-1, 7)}

3. To find the inverse equation for y=-8x+3, we swap x and y and solve for y.

x=-8y+3

x-3=-8y

y=(x-3)/-8

Inverse equation: y=(-1/8)x+3/8

4. To find the inverse equation for y=2/3x-5, we swap x and y and solve for y.

x=2/3y-5

x+5=2/3y

y=3/2(x+5)

Inverse equation: y=3/2x+15/2

5. To find the inverse equation for y=1/2x+10, we swap x and y and solve for y.

x=1/2y+10

x-10=1/2y

y=2(x-10)

Inverse equation: y=2x-20

6. To find the inverse equation for y=(x-3)², we swap x and y and solve for y.

x=(y-3)²

[tex]x^{(1/2)}=y-3[/tex]

[tex]y=x^{(1/2)}+3[/tex]

Inverse equation: [tex]y=x^{(1/2)}+3[/tex]

7. To verify that f(x)=5x+2 and g(x)=(x-2)/5 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.

fog(x) = f(g(x)) = f((x-2)/5) = 5((x-2)/5) + 2 = x

gof(x) = g(f(x)) = g(5x+2) = ((5x+2)-2)/5 = x/5

Since fog(x) = gof(x) = x, f and g are inverse functions.

8. To verify that f(x)=1/2x-7 and g(x)=2x+14 are inverse functions, we need to show that fog(x)=gof(x)=x for all x in the domain of f and g.

fog(x) = f(g(x)) = f(2x+14) = 1/2(2x+14) - 7 = x

gof(x) = g(f(x)) = g(1/2x-7) = 2(1/2x-7) + 14 = x

Since fog(x) = gof(x) = x, f and g are inverse functions.

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The  true triangle  statement regarding the diagram are:

1.  m∠5 + m∠6 = 180°  ________Linear Pair

2. ∠ 2+ ∠ 3 = ∠ 6________Exterior angle Property of Triangle

3. m∠2 + m∠3 + m∠5 = 180°________Triangle Sum Property

From the question,  Δ ABC with Exterior angles as ∠ 1 , ∠ 4 ,and ∠ 6

Note that the  Exterior angle Property of Triangle state that An exterior angle of a triangle is equal to the sum of the opposite interior angles.

Hence: For Exterior ∠ 1 :

∠ 1 = ∠ 5 + ∠ 3 ________Exterior angle Property of Triangle

Also,

For Exterior ∠ 4:

∠ 4 = ∠ 5 + ∠ 2 ________Exterior angle Property of Triangle

Also,

In regards to Exterior ∠ 6:

∠ 6 = ∠ 2 + ∠ 3 ________ Exterior angle Property of Triangle

Using Triangle Sum Property, it state that In a triangle sum of the measures of angles is equal to 180° Hence:  m∠2 + m∠3 + m∠5 = 180° ________Triangle Sum Property

The Linear Pair will be: The measure of a straight angle is 180 degrees, so a linear pair of angles must add up to 180 degrees.

Therefore,  m∠5 + m∠6 = 180°  ________Linear Pair

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See full question below

A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options.

m∠5 + m∠3 = m∠4

m∠3 + m∠4 + m∠5 = 180°

m∠5 + m∠6 =180°

m∠2 + m∠3 =

m∠6 m∠2 + m∠3 + m∠5 = 180°

The value of (7 4/9 -8)*3.6-1.6*(1/3-3/4)+ 1 2/5 ÷(0.35) is given as 241/54.

(7 4/9 -8) = -5/9.

3.6-1.6 = 2.0

1/3-3/4 = 1/3 - 3/4

= 4/12 - 9/12

= -5/12

we will have -5/9 *  2 = -10/9.

-10/9 *  -5/12

10/9 * -5/12 = (10 * 5) / (9 * 12) = 50/108

We can simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 2:

50/108 = 25/54

we will have

25/54 + 1 2/5 ÷(0.35)

1 2/5 ÷ 0.35 = (7/5) ÷ (35/100) = (7/5) * (100/35) = 4

Now, we can substitute this value into the expression:

25/54 + 4 = (25/54) + (216/54) = 241/54

Therefore, the value of the expression 25/54 + 1 2/5 ÷(0.35) is 241/54.

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(a) Indefinite integral of 5(5x^4 - 5sinx)dx is (5/3)x^5 + 5cosx + C. Definite integral over [0, π/2] is (125π/6) - 5.

We can evaluate the indefinite integral by applying the power rule and integration by substitution. The definite integral can be evaluated by substituting the limits of integration and simplifying.

(b) Indefinite integral of [(18x^2 - 1)(6x^3 - 9x^2 - 3)]^6dx is (18x^11 - 77x^9 + 126x^7 - 108x^5 + 49x^3 - 9x) / 11 + C.

To simplify the given expression, we can first expand the polynomial and then apply the power rule to integrate each term. The constant of integration can be added at the end.

(c) Definite integral of ∫tan^2(x)sec^2(x)dx over [0,π/4] is 1.

We can use the trigonometric identity sec^2(x) - 1 = tan^2(x) to simplify the integrand. Then we can apply the power rule and substitute the limits of integration to evaluate the definite integral.

(d) Indefinite integral of ∫(x+4)^2√(3x^2+4)dx is (1/15)(3x^2+4)^(3/2)(x+4) - (4/45)(3x^2+4)^(3/2) + C.

We can use substitution to simplify the integrand by setting u = 3x^2 + 4. After integrating, we can substitute back for u and simplify the constant of integration.

(e) Indefinite integral of ∫(120/(1+2x^2))dx is 60√2tan^(-1)(√2x) + C.

We can use substitution to simplify the integrand by setting u = 1 + 2x^2. After integrating, we can substitute back for u and simplify the constant of integration.

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To find out how much lace Ms. Regan needs to purchase, we first need to calculate the circumference of the circular quilt. We know that the area of the quilt is 28.26 square feet, and we can use the formula A = πr^2 to find the radius of the quilt.

28.26 = 3.14 x r^2

r^2 = 9

r = 3

Now that we know the radius is 3 feet, we can use the formula C = 2πr to find the circumference of the quilt.

C = 2 x 3.14 x 3

C = 18.84 feet

Therefore, Ms. Regan needs to purchase 18.84 feet of lace to go around the outside of her circular quilt.

In summary, to find out how much lace Ms. Regan needs to purchase, we need to calculate the circumference of the circular quilt. We do this by first finding the radius using the formula A = πr^2. Once we know the radius, we can use the formula C = 2πr to find the circumference. In this case, the circumference is 18.84 feet, so Ms. Regan needs to purchase that amount of lace.

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So the expression that represents the average distance of a planet from the sun using radicals is: d = k/2√13 * √x

An exponent, also known as a power, is a mathematical notation that indicates the number of times a quantity is multiplied by itself. It is usually written as a small number (the exponent) placed to the right and above a larger number (the base). Exponents are used in many mathematical concepts, including logarithms, roots, and scientific notation.

Here,

The expression ((1)/∛(52)x)²) can be simplified using exponent rules:

((1)/∛(52)x)²) =((1)/∛(52)x)²) * ∛x²)

= 1/(∛52² * ∛x²)

The average distance of a planet from the sun measured in astronomical units can be represented using the formula:

d = k * √T

where d is the distance from the sun, T is the time it takes for the planet to orbit the sun, and k is a constant of proportionality.

We can rewrite this formula in terms of Earth weeks by noting that there are 52 weeks in a year, so T = (1/52)x years. Substituting this into the formula, we get:

d = k * √((1/52)x)

Simplifying this expression using exponent rules, we get:

d = k * √(1/52)* √x

So an equivalent expression using radicals to represent the average distance of a planet from the sun is:

d = k * √(1/(52)) * √x

which simplifies to:

d = k/√(52) * √x

or

d = k/2√13 * √x

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Answer:

x = 16

Step-by-step explanation:

(2x + 16) = 48

Subtract 16 with the positive 16 to cancel the numbers.

Subtract 16 with 48.

2x = 32

divide 32 by 2 to isolate the x.

32/2 = 16

x = 16

Note that the first coordinate of M is -1. (Option D)

Given :- coordinates of F = (-4,-2) = (x1,y1)

coordinates  of G = (2,-2) = (x2,y2)

coordinates of P = (2,-8) = (x3,y3)

and distance between point M and P is half of the distance between FG

To find :- first coordinate of point M

solution :- let the coordinate of M be (x4,y4)

as we know that distance between of the opposite point of parallel line segments are equal

so, second coordinate of M = -8

now by distance formula

FG = √(x2-x1)² + (y2-y1²)

= √[2-(-4)]² + [-2-(-2)]²

= √(2+4)² + (2-2)²

= √(6)² + (0)²

=√36

F G = 6

so, distance between point M and P = 1/2 × F G

= 1/2 × 6

= 3units

again, by distance formula

MP = √(x3-x⁴)² + (y3-y4)²

3 = √(2-x⁴)² + [-8-(-8)]²

squaring on both side

(3)² = (√(2-x⁴)² + [-8-(-8)]²)²

9 = (2-x⁴)² + [-8-(-8)]²

9 = (2-x⁴)²+(0)²

9 = (2-x⁴)²

√9 = 2-x⁴

3 = 2-x⁴

x⁴ = 2-3

x⁴ = -1

Hence the first coordinate of M is -1

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Full Question:

Although part of your question is missing, you might be referring to this full question:

Point M is located in the third quadrant.

The distance between point M and point P is half the distance between point F and point G.

• Segment MP is parallel to segment FG. What is the first coordinate of point M?

If Ofra tried to solve an equation 3x = 4.5, The statement "Ofra correctly solved the equation" is correct. So, correct option is C.

We can see this by substituting x = 1.5 into the original equation 3x = 4.5:

3(1.5) = 4.5

Simplifying the left-hand side, we get:

4.5 = 4.5

This is a true statement, which means that x = 1.5 is a valid solution to the equation 3x = 4.5.

Therefore, Ofra did not make any mistakes in solving the equation. She correctly set up the equation 3x = 4.5 by multiplying both sides by 3 to isolate x, and then calculated the value of x to be 1.5, which is the correct solution.

Option (c) is the correct answer.

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Complete question is:

Ofra tried to solve an equation.

3x = 4.5, Setting up x = 1.5 Calculating

Where did Ofra make her first mistake?

Choose 1 answer:

a) Setting up

b)  Calculating

c) Ofra correctly solved the equation.

Let's use "a" to represent the length of the equal sides of the isosceles triangle, and let's use "b" to represent the length of the third side. We're told that one of the equal sides is 2x + 1ft long, so we can set up an equation:

2a + b = 55

We're also told that the other two sides are both 3x - 14ft long, so we can set up another equation:

a = 3x - 14

Now, we can substitute the second equation into the first equation and solve for "b":

2a + b = 55

2(3x-14) + b = 55

6x - 28 + b = 55

b = 83 - 6x

Now, we can substitute both equations into the equation a = 3x - 14 and solve for "x":

3x - 14 = 2x + 1 + 3x - 14

6x - 27 = 0

x = 4.5

Finally, we can substitute "x" into our equations to find the lengths of the sides:

a = 3x - 14 = 3(4.5) - 14 = 0.5

b = 83 - 6x = 83 - 6(4.5) = 55

So the length of the equal sides is 0.5ft, and the length of the third side is 55ft. Therefore, the lengths of the sides of the isosceles triangle are 0.5ft, 0.5ft, and 55ft.

The statement on finding the areas of a trapezoid and a kite are True.

To determine the area of a trapezoid, it can be broken down into separate geometrical shapes. One possible breakdown would include a rectangle with two adjacent right triangles or an isosceles triangle with one right triangle configuration. By calculating each smaller compartment's size and summing them together, one can obtain the total area for the trapezoid.

Similarly, in order to find the surface area of a kite shape, drawing a diagonal creates two adjoining triangles that are easily computed individually then summed.

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Options for this question :
True

False

To solve the equation, we need to first simplify both sides:

(4x - 6)/5 + 1 = (x + 1)/5 - 2/5

Multiplying both sides by 5 to eliminate the denominator:

4x - 6 + 5 = x + 1 - 2

Simplifying further:

4x - 1 = x - 1

Subtracting x from both sides:

3x - 1 = -1

Adding 1 to both sides:

3x = 0

Dividing both sides by 3:

x = 0

Therefore, the solution to the equation is x = 0.

Answer:  x=28

Step-by-step explanation:

Given:      <A=68

Find:     x

Reasoning:  

<B = 2x+x

<B= 3x

<C=x     they say the sides across from <C is same as other side so the

             angles are the same

Solution:

All angles of a triangle =180

<A + <B + <C =180    >substitute

68 + 3x + x =180      > combine like terms

68 + 4x = 180           > subtract 68 from both sides

4x=112                       >divide both sides by 4

x=28

Answer: The amount of money you would have in 5 years if you could save your entire salary with a 1.2% increase every year would be $87,357.41.

Explanation:

The initial term, a₁, is $74,000, and the common ratio, r, is 1 + 1.2% = 1.012. To find the sum of the first 5 terms, we use the formula for a finite geometric series:

S₅ = a₁(1 - r⁶)/(1 - r)

Plugging in the values, we get:

S₅ = $74,000(1 - 1.012⁵)/(1 - 1.012) = $87,357.41 (rounded to the nearest cent)

Therefore, if you save your entire salary, you would have approximately $87,357.41 in 5 years with a 1.2% increase every year.

Answer:

60%

Step-by-step explanation:

Take 96 and divide it by 160.

(easier if done on a calculator.)

For example: Find A/B as a percentage: take "A" and divide it by "B"

Answer:

C. The product of two irrational numbers is irrational.

Example: √3•√3=3

Answer:

18 cent

Step-by-step explanation:

Answer:

61 degrees

Step-by-step explanation:

Triangle interior measures add up to 180 degrees.

61 + 58 + x = 180

119 + x = 180

x = 61

hope this helps :) !!!

(Explanation below)

x=2

x = 2 is the solution of the equation

Two or more expressions with an Equal sign is called as Equation.

The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]

We have to find the value of x

Subtracting 3 from both sides:

[tex](3X-5)^(^1^/^4^) = 1[/tex]

Raising both sides to the fourth power:

3X - 5 = 1^4

3X - 5 = 1

Adding 5 to both sides:

3X = 6

Dividing by 3:

X = 2

Therefore, x = 2 is the solution of the equation

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Answer:

c/π=d

explanation:

d × π = c

divide c to isolate d

Answer: I would multiply pie by a diameter until it equals 16.

(I know this probably isn’t the professional way but it should work.

2(2.5) + 5(23) = 150

3(2.5) + 4(23) = 144.5

Both equations are satisfied, so our solution is correct.

To solve the problem, let's first assign some variables. Let x be the cost of one pine tree and y be the cost of one hydrangea bush. We can then use these variables to set up a system of equations:

2x + 5y = 150 (equation 1)

3x + 4y = 144.5 (equation 2)

We can solve this system of equations using various methods. Here, we will use the substitution method.

From equation 1, we can solve for x in terms of y:

2x = 150 - 5y

x = (150 - 5y)/2

We can then substitute this expression for x into equation 2:

3((150 - 5y)/2) + 4y = 144.5

Multiplying both sides by 2 to eliminate the fraction:

3(150 - 5y) + 8y = 289

Expanding and simplifying:

450 - 15y + 8y = 289

-7y = -161

y = 23

We can now substitute this value for y into either equation 1 or 2 to solve for x:

2x + 5(23) = 150

2x = 5

x = 2.5

Therefore, one pine tree costs $2.50 and one hydrangea bush costs $23.

To check our work, we can substitute these values into both equations:

2(2.5) + 5(23) = 150

3(2.5) + 4(23) = 144.5

Both equations are satisfied, so our solution is correct.

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a) P(x) = R(x) - C(x) = 6x - (0.001x^2 + 18x + 40) = -0.001x^2 - 12x - 40

b) R(200) = 6(200) = 1200
  C(200) = 0.001(200)^2 + 18(200) + 40 = 4000
  P(200) = R(200) - C(200) = 1200 - 4000 = -2800

c) R'(x) = 6
  C'(x) = 0.002x + 18
  P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18) = -0.002x - 12

d) R'(200) = 6
  C'(200) = 0.002(200) + 18 = 18.4
  P'(200) = -0.002(200) - 12 = -12.4

Here are the answers to each part:

a) P(x) is the profit function, which is calculated as the difference between the revenue function and the cost function: P(x) = R(x) - C(x). In this case, P(x) = 6x - (0.001x^2 + 18x + 40).

b) To find R(200), C(200), and P(200), plug x = 200 into each function:
R(200) = 6(200) = 1200
C(200) = 0.001(200^2) + 18(200) + 40 = 7600
P(200) = 1200 - 7600 = -6400

c) To find R'(x), C'(x), and P'(x), we need to find the derivative of each function with respect to x:
R'(x) = d(6x)/dx = 6
C'(x) = d(0.001x^2 + 18x + 40)/dx = 0.002x + 18
P'(x) = R'(x) - C'(x) = 6 - (0.002x + 18)

d) To find R'(200), C'(200), and P'(200), plug x = 200 into each derivative function:
R'(200) = 6
C'(200) = 0.002(200) + 18 = 18.4
P'(200) = 6 - 18.4 = -12.4

I hope this helps! Let me know if you have any further questions.

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The value of x in the given circle is 18.1 units.

Given is a circle, where two radii are given one chord is given,

We need to find the value of the x which is also the radius,

We know all the radii in a circle are equal,

So, here the radius = 7.9+10.2 = 18.1 units.

Hence the value of x in the given circle is 18.1 units.

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The situation represents exponential decay.

The rate of growth or decay, r, is equal to 0.02.

So the depth of the lake each year is 0.98 times the depth in the previous year.

It will take between 11 and 12 years for the depth of the lake to reach 26. 7 meters.

The situation represents exponential decay, as the depth of the lake decreases by a constant percentage each year. The rate of decay is 2% per year, so the rate of growth or decay, r, is equal to 0.98 (1 - 0.02). This means that the depth of the lake each year is 0.98 times the depth in the previous year.

To find the number of years it will take for the depth of the lake to reach 26.7 meters, we can use the formula for exponential decay:\

D = D₀ *[tex]e^{(-rt)[/tex]

where D is the current depth, D₀ is the initial depth, r is the rate of decay, and t is the number of years.

Substituting the given values, we get:

26.7 = 30 * [tex]e^{(-0.02t)[/tex]

Solving for t, we get:

t = ln(26.7/30) / (-0.02) ≈ 11.33

Therefore, it will take between 11 and 12 years for the depth of the lake to reach 26.7 meters.

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