Find an equation of the plane that contains the curve with the given vector equation. R(t) = (t, t^3, t)

Answer:

x = -3 and y = 9

Step-by-step explanation:

          4y + 6x = 18 -----------------(I)

         3y - 2x = 33 -----------------(II)

Multiply equation (II) by 3.

(I)                 4y + 6x = 18

(II)*3            9y - 6x = 99  {Now add the equations}

                  13y        = 117

Divide both sides by 13,

                        y   = 117 ÷ 13

                          [tex]\boxed{\bf y = 9}[/tex]

Substitute y = 9 in equation (I) and obtain the value of 'x',

        4*9 + 6x = 18

           36 + 6x = 18

Subtract 36 from both sides,

                    6x = 18 - 36

                    6x = -18

Divide both sides by 6

                      x = -18 ÷ 6

                     [tex]\boxed{\bf x = -3}[/tex]

[tex] \\ 4y + 6x = 18 \\ \\ 6x= 18 - 4y \\ = x = 3 - \frac{2}{3 }y \\ x = - \frac{2}{3} y+ 3[/tex]

The trend line equation y = 0.05x + 0.25 can be used to accurately predict the cost of gas per gallon for any year since 1970.

The equation of the trend line in the scatter plot is given by y = 0.05x + 0.25, where x is the number of years since 1970 and y is the price of gas per gallon in dollars. This equation can be used to calculate the price of gas in a given year since 1970. For example, if we want to calculate the price of gas in 2017, we can plug in x = 47 (the number of years since 1970) into the equation to get y = 0.05(47) + 0.25 = 2.85. This means that in 2017, the price of gas per gallon was approximately 2.85 dollars. This equation can be used to accurately predict the cost of gas for any given year since 1970.

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The inequality will be and [tex]3 + √-4[/tex] will be written as [tex]3 +2i[/tex] as complex number.

What are complex numbers?

In mathematics, a complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1). Complex numbers are an extension of the real numbers, which include all the numbers that can be represented on a number line.

The real part of a complex number a + bi is the real number a, and the imaginary part is the real number bi.

A. Since Jan found that both roots of the quadratic equation are not real numbers, this means that the discriminant b² - 4ac is negative. Therefore, the inequality that must always be true for Jan's quadratic equation is:

[tex]b² - 4ac < 0[/tex]

B. The expression  [tex]3+√-4[/tex] can be written as [tex]3 + 2i,[/tex] where i is the imaginary unit (√-1). This is because √-4 is equal to 2i, so [tex]3+√-4[/tex] can be written as [tex]3 + 2i.[/tex] Therefore, [tex]3+√-4[/tex] written as a complex number is 3 + 2i.

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Answer:

3

Step-by-step explanation:

You are subtracting 1 less each time, starting at 3:

9 - 3 = 6

6 - 2 = 4

Therefore, you will subtract 1 from 4:

4 - 1 = 3

3 is your answer.

~

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The graph fοr the sοlutiοns οf the inequalities have been plοtted and attached belοw.

A relatiοnship in mathematics knοwn as an inequality cοmpares twο numbers οr οther mathematical expressiοns in an unfair manner.

Mοst frequently, size cοmparisοns are dοne between twο numbers οn the number line.

We are given twο inequalities as x - 3 ≤ 8 and 6x &lt; 72.

Nοw, οn sοlving the first inequality, we get⇒ x - 3 ≤ 8⇒ x ≤ 11Similarly, οn sοlving the secοnd inequality, we get⇒ 6x < 72⇒ x < 12

Nοw, οn the graph the sοlutiοns are plοtted and attached belοw. The red pοint represents the secοnd inequality sοlutiοn and the blue represents the first inequality sοlutiοn.

Hence, the graph fοr the sοlutiοns οf the inequalities have been plοtted.

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[tex] \tt \: c)77[/tex]

_________

[tex] \tt{7x} \\ \: \: \: \: \: = \tt 7(11) \\ \tt= 77[/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps☘️

The required answers are a) 17.1 km b) 7.1 km, c) 18.6 km away from town A on a bearing of 293°.

To solve this problem, we can use vector addition to find the displacement vector from town A to the cyclist's destination.

a) To find how far east the cyclist's destination is from town A, we need to find the east component of the displacement vector. We can break down the displacement vector into its north and east components using trigonometry:

[tex]$$\text{East displacement} = 5\text{ km} + 10\text{ km}\cos(45^\circ) = 5\text{ km} + 10\text{ km}\frac{\sqrt{2}}{2} = 10\text{ km} + 5\sqrt{2}\text{ km} \approx 17.1\text{ km}$$[/tex]

So the cyclist's destination is approximately 17.1 km east of town A.

b) Similarly, to find how far north the cyclist's destination is from town A, we need to find the north component of the displacement vector:

[tex]$$\text{North displacement} = 10\text{ km}\sin(45^\circ) = 10\text{ km}\frac{\sqrt{2}}{2} = 5\sqrt{2}\text{ km} \approx 7.1\text{ km}$$[/tex]

So the cyclist's destination is approximately 7.1 km north of town A.

c) To find the distance and bearing of the cyclist's destination from town A, we can use the Pythagorean theorem and trigonometry. The displacement vector is the hypotenuse of a right triangle with legs of length 17.1 km and 7.1 km, so its length is:

[tex]$$\text{Displacement} = \sqrt{(17.1\text{ km})^2 + (7.1\text{ km})^2} \approx 18.6\text{ km}$$[/tex]

To find the bearing of the displacement vector, we can use the inverse tangent function:

[tex]$$\text{Bearing} = \tan^{-1}\left(\frac{\text{East displacement}}{\text{North displacement}}\right) \approx 67^\circ$$[/tex]

However, this angle is measured clockwise from north, so we need to subtract it from 360° to get the bearing measured counterclockwise from north:

[tex]$$\text{Bearing} = 360^\circ - 67^\circ = 293^\circ$$[/tex]

So the cyclist's destination is approximately 18.6 km away from town A on a bearing of 293°.

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what are the answer choices?

Answer:

3/4

Step-by-step explanation:

Answer:

392.5g

Step-by-step explanation:

Density =mass/volume

Mass = Density x volume

Mass = 7.85g/cm³ x 50 cm³

Mass = 392.5g

The Japanese island that became the center of power in Japan is Honshu.

Honshu is the largest and most populous island in Japan and has historically been the center of political, economic, and cultural power in Japan. It is home to Tokyo, the capital city of Japan, as well as many other major cities such as Osaka, Kyoto, and Yokohama. Honshu's central location and natural resources have made it a strategic location for political power throughout Japan's history, including during the feudal period when various powerful clans vied for control. Honshu's position as the economic and cultural hub of Japan has also helped solidify its status as the center of power, as it has been a center for education, trade, and innovation for centuries.

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The use of consonance in "Thundered the battery's double brass" best conveys the violent sounds of the battle.

Consonance is a literary device where consonant sounds are repeated in close proximity, often at the ends of words, to create a musical or rhythmic effect in a text.

The passage uses consonance, which is the repetition of consonant sounds within or at the end of words, to convey the violent sounds of the battle.

In particular, the repeated "t" and "d" sounds in "Thundered the battery's double brass" and "Difficult music for men to face" create a harsh, jarring effect that emphasizes the chaos and danger of the fighting.

Similarly, the repeated "v" and "l" sounds in "Undulate like the living waves" create a sense of movement and fluidity, conveying the unrelenting and overwhelming force of the rebel advance.


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Answer:31.58% i think

Step-by-step explanation:

600 of 1900 can be written as:

600/1900

To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100

600/1900 x 100/100 = 600x100/1900) x1/100= 31.58/100

Therefore, the answer is 31.58%

Answer:

[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to find the area of 1/3 of a circle.}[/tex]

[tex]\textsf{Let's review the formula needed to find the area of a \underline{whole} circle.}[/tex]

[tex]\large\underline{\textsf{Formula:}}[/tex]

[tex]\mathtt{Area = \pi (radius)^{2}.}[/tex]

[tex]\textsf{We should know that a circle is 360}^{\circ}. \ \textsf{We are given 120}^{\circ} \textsf{of a circle.}[/tex]

[tex]\textsf{The area of \underline{1/3} of a Circle is the area of a whole circle \underline{divided by 3.}}[/tex]

[tex]\textsf{Let's begin solving for the area.}[/tex]

[tex]\large\underline{\textsf{Substitute:}}[/tex]

[tex]\mathtt{Area = \pi (10)^{2}}[/tex]

[tex]\large\underline{\textsf{Evaluate:}}[/tex]

[tex]\mathtt{Area = 100\pi }[/tex]

[tex]\large\underline{\textsf{Divide by 3:}}[/tex]

[tex]\mathtt{\frac{Area}{3} = \frac{100\pi }{3}} [/tex]

[tex]\boxed{\mathtt{Area \approx 104.7m^{2}}}[/tex]

Answer:

21. Proofs attached to answer

Step-by-step explanation:

Proofs attached to answer

Answer:

(anyone who answers gets brainliest) Pick the correct trig tool to solve for the variables. There’s a solve for x problem I was struggling with. You don’t have to do them all, thank you

Step-by-step explanation:

Answer:

Let's start by assigning a variable to the original side length of the square photograph. We'll use g to represent this length.

According to the problem, each side of the new photograph will be 1 inch more than twice the original side length g. This means that the new side length will be:

2g + 1

Since this is a square photograph, all four sides are equal in length. Therefore, the area of the new photograph can be represented by the square of the new side length:

(2g + 1)^2

To simplify this expression, we can use FOIL (First, Outer, Inner, Last) to expand the squared binomial:

(2g + 1)^2 = (2g + 1)(2g + 1)

= 4g^2 + 2g + 2g + 1

= 4g^2 + 4g + 1

So the trinomial that represents the area of the enlarged photograph is:

4g^2 + 4g + 1

To check our work, we can plug in a value for g and compare the areas of the original and enlarged photographs. For example, if g = 2 (meaning the original side length is 2 inches), then the area of the original photograph is:

2^2 = 4 square inches

And the area of the enlarged photograph is:

4(2)^2 + 4(2) + 1 = 25 square inches

This makes sense, since the new photograph has sides of length 2(2) + 1 = 5 inches, and therefore an area of 5^2 = 25 square inches, which is indeed 1 inch more than twice the original area of 4 square inches.

The correct statement regarding the quadratic equation is given as follows:

C. There are two complex solutions.

A quadratic equation is modeled by the general equation presented as follows:

y = ax² + bx + c

The discriminant of the quadratic function is given by the equation as follows:

Δ = b² - 4ac.

The numeric value of the coefficient and the number of solutions of the quadratic equation are related as follows:

The coefficients of the function for this problem are given as follows:

a = -2, b = 9, c = -12.

Hence the discriminant is given as follows:

Δ = 9² - 4(-2)(-12)

Δ = -15.

Negative discriminant, hence there are two complex solutions.

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The quadratic equation has two real solutions

Here we have the quadratic equation:

y = -2x^2 + 9x - 12

We want to study the solutions of the equations, so we need to look at the discriminant:

Generally for a*x^2 + b*x + c = 0 the discriminant is b^2 - 4ac

Here it is:

D = 9^2 - 4*-12*-2 = 33

A positive determinant means that there are 2 real solutions so the correct option is A.

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Using matrix inversion method and perimeter of rectangle, the dimension of the rectangle are 16cm and 7cm

Let the length of the rectangle be l and the width be w. We know that l - w = 3.5 and 2l + 2w = 67. We can rewrite these equations as a system of linear equations in matrix form:

[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix} \begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]

We can solve for [tex]$\begin{bmatrix}l \ w\end{bmatrix}$[/tex] by multiplying both sides of the equation by the inverse of the coefficient matrix:

[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} \begin{bmatrix}3.5 \ 67/2\end{bmatrix}$$[/tex]

To find the inverse of the coefficient matrix, we can use the following formula:

[tex]$$\begin{bmatrix}a & b \ c & d\end{bmatrix}^{-1} = \frac{1}{ad-bc} \begin{bmatrix}d & -b \ -c & a\end{bmatrix}$$[/tex]

Plugging in the values for our matrix, we get:

[tex]$$\begin{bmatrix}1 & -1 \ 2 & 2\end{bmatrix}^{-1} = \frac{1}{1 \cdot 2 - (-1) \cdot 2} \begin{bmatrix}2 & 1 \ -2 & 1\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix}$$[/tex]

Now we can substitute this matrix and the vector on the right-hand side of the equation into our formula to obtain the solution:

[tex]$$\begin{bmatrix}l \ w\end{bmatrix} = \begin{bmatrix}1/2 & 1/4 \ -1/2 & 1/4\end{bmatrix} \begin{bmatrix}3.5 \ 67/2\end{bmatrix} = \begin{bmatrix}16 \ 7\end{bmatrix}$$[/tex]

Therefore, the dimensions of the rectangle are 16 cm by 7 cm.

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The perimeter of a rectangular outdoor patio is 106 ft. the length is 9 ft greater than the width.  Hence, the dimensions of the patio are 38 ft and 47 ft.

Given that the perimeter of a rectangular outdoor patio is 106 ft. The length is 9 ft greater than the width. Now we need to find the dimensions of the patio.

Step 1: Let's consider the width of the patio be x feet.

The length of the patio is given as 9 ft greater than the width.

So the length of the patio will be (x + 9) feet.

Step 2: The perimeter of a rectangle is given by P = 2(l + w).

So the perimeter of the patio is given as 106 ft.

Thus,2(l + w) = 1062(x + x + 9)

                    = 1062(2x + 9)

                    = 1062x + 1821

                     = 106x = 106 - 182x

                     = -76 (not possible)

Therefore, x = 38 ftSo the width of the patio is 38 ft The length of the patio is 38 + 9 = 47 ft

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Answer:

36

Step-by-step explanation:

see image for explanation

The possible number of positive real zeros of the function is either 2 or 0.f(-x) = −2x5 − 3x2 − 2x − 1The number of sign changes in f(-x) is 1. The possible number of negative real zeros of the function is either 1 or 0.

To determine the possible numbers of positive and negative real zeros of the function f(x) = 2x5 − 3x2 + 2x − 1 using Descartes's Rule of Signs, we should start by writing the polynomial function in descending order of powers of x. After this, we count the number of sign changes in the polynomial function f(x) and find out the possible number of positive real zeros. Similarly, we count the number of sign changes in f(-x) and find out the possible number of negative real zeros .In the given function f(x) = 2x5 − 3x2 + 2x − 1, the polynomial is already in the descending order of powers of x. Therefore,

we count the sign changes in f(x) and f(-x) as follows: f(x) = 2x5 − 3x2 + 2x − 1The number of sign changes in f(x) is 2. Therefore, the possible number of positive real zeros of the function is either 2 or 0.f(-x) = −2x5 − 3x2 − 2x − 1The number of sign changes in f(-x) is 1. Therefore, the possible number of negative real zeros of the function is either 1 or 0. Hence, the possible number of positive real zeros of the function is either 2 or 0 and the possible number of negative real zeros of the function is either 1 or 0.

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If the ratio of dogs to cats is 2:3 and there are 520 dogs, the number of cats is 780.

The ratio refers to the relative size of one quantity or value compared to another quantity or value.

Ratios are proportionate values stated in ratio form using (:), in percentages or fractions.

The ratio of dogs to cats = 2:3

The sum of ratios = 5

The number of dogs based on this ratio = 520 dogs

The total number of dogs and cats based on the above ratio and the number of dogs = 1,300 (520/2 x 5)

The ratio of cats to dogs = 3:2 or 3/5

The number of cats = 780 (1,300 x 3/5)

Thus, using the ratio of dogs to cat, with the number of dogs as 520, there are 780 cats.

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Answer: 5cm

Step-by-step explanation:

You divide the area of green (93.75) by pink's length (7.5)

that gives you 12.5.

12.5 is the length of green.

12.5 - 7.5 = 5

You subtract the 7.5 because to get yellow's length because Yellow's length plus pink's length is equal to green's total length.

Answer:

y = (-1/3)x + 3.

Step-by-step explanation:

To find the equation of a line perpendicular to a given line, we need to take the negative reciprocal of the slope of the given line.

The slope of the given line Y=3x-5 is 3. Therefore, the slope of a line perpendicular to this line would be -1/3.

Now, using the point-slope form of a line, we can write the equation of the line passing through (-9,6) and having a slope of -1/3:

y - 6 = (-1/3)(x - (-9))

Simplifying:

y - 6 = (-1/3)x - 3

y = (-1/3)x + 3

Therefore, the equation of the line that passes through (-9,6) and is perpendicular to the line Y=3x-5 is y = (-1/3)x + 3.

Therefore, the values of a that make the expression true are approximately 0.378 and 14.121.

In mathematics, an expression is a combination of numbers, variables, and operations that represents a quantity or a mathematical relationship. Expressions can be simple or complex, and can involve arithmetic, algebraic, trigonometric, or other mathematical operations. Expressions can also include constants, which are fixed values that do not change. Expressions can be evaluated by substituting numerical values for the variables and performing the indicated operations.

Here,

To solve for the values of a that make the given expressions true, we need to isolate the variable a on one side of the equation. We can start by simplifying the left side of the equation:

1/2a - 5 1/2 = 5 - a

Multiplying both sides by 2a gives:

1 - 10a* 1/2 = 10a - 2a²

Simplifying the left side by multiplying both terms by 2:

2 - 20a = 10a - 2a²

Rearranging the terms gives a quadratic equation:

2a² - 30a + 2 = 0

We can solve for a by using the quadratic formula:

a = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = -30, and c = 2.

Plugging in these values gives:

a = (30 ± √((-30)² - 4(2)(2))) / 4

a = (30 ± √(892)) / 4

a ≈ 0.378 or a ≈ 14.121

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The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.

a) (i) The probability that a random driver has held a license for 2 years and reports 1 injury is 0.125.
(ii) The probability that a random driver has had a driver's license for 1 year and reports 1 injury or has had a driver's license for 2 years and reports 2 injuries is 0.3125.

b) The marginal probability distributions of ???? and ???? are given in the table below:

   ????
   ???? = 0
   ???? = 1
   ???? = 2
   ???? = 3
 
   ???? = 0
   0.25
   0.125
   0.0625
   0.03125
 
   ???? = 1
   0.25
   0.25
   0.125
   0.0625
 
 
   ???? = 2
   0.25
   0.25
   0.25
   0.125
 
   ???? = 3
   0.25
   0.25
   0.25
   0.25
 The probability that a random driver reports 0 injuries is 0.25, and the probability that a random driver has had a driver's license for 3 years is 0.25.

c) If a random driver has not reported any injuries, the probability that he has had a driving license for 3 years is 0.25.

d) ????(????) =  1, ????(????) = 1.5, ????????????(????) = 0.5, ????????????(????) = 0.9.

e) The covariance between ???? and ???? is 0.225, and the correlation coefficient is 0.45. This shows a positive correlation between the two variables, meaning that an increase in one variable is associated with an increase in the other.

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Answer:

The statement "The function is decreasing on the interval (0, ∞)" is true about h(x).

To see why, let's take the derivative of h(x) and examine its sign:

h(x) = -2√x

h'(x) = -2/(2√x) = -1/√x

Since √x is always positive, h'(x) is negative for all x > 0. This means that h(x) is decreasing on the interval (0, ∞).

Therefore, the correct answer is: The function is decreasing on the interval (0, ∞).