NEED THE ANSWER PLEASE...

The change in variables s=x+3y, t=y to find the area of the ellipse x²+6xy+10y²≤1 yields; s² +t²≤ 1.

It follows from the equations given that;

Hence, it follows from substitution that;

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By applying a change of variables to the ellipse equation x2+6xy+10y2≤1, this simplifies to the standard ellipse equation from which the area, π*sqrt(10), can be determined using the standard formula for the area of an ellipse.

To find the area of the ellipse defined by the equation x2+6xy+10y2≤1, we can begin by using the provided change of variables, s = x + 3y and t = y. This transformation simplifies the equation of the ellipse to: (s^2) / 10 + (t^2) ≤ 1.

Now the expression looks like the standard equation of an ellipse, where 'a' (the radius on the x-axis) is sqrt(10) and 'b' (the radius on the y-axis) is 1. The area 'A' of an ellipse is given by 'A=πab'. Substituting 'a' and 'b' into the formula gives us 'A = π*sqrt(10)*1 = π*sqrt(10)'

This indicates that the area of the ellipse with the change of variables s=x+3y, t=y for the equation x2+6xy+10y2≤1 is π*sqrt(10).

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Answer:  0.80

Step-by-step explanation:

P (2 ≤ x ≤ 4) is: P(X = 2) + P(X = 3) + P(X = 4)

                   =     0.36    +    0.12    +     0.32

                   =             0.48             +     0.32

                   =                            0.80

It can also be calculated as: 1 - P(X = 1)

                                           =  1 -  0.20

                                           =     0.80

hence angle between vectors is option a ,173 degree

When an equation has both direction as well as magnitude it refers to vector equation.

Formula used [tex]\alpha =cos^{-1} (\frac{u.v}{|u||v|})[/tex]

u.v=1*(-1)+[tex]\sqrt{7} *(-4)[/tex]

|u|=[tex]\sqrt{ 1^{2} +\sqrt{7} ^{2}}[/tex]=[tex]2\sqrt{2}[/tex]

|v|=[tex]\sqrt{ (-1^{2}) +(-4^{2})[/tex]=[tex]\sqrt{17}[/tex]

[tex]\alpha =cos^{-1} (\frac{-1-4\sqrt{7} }{|2\sqrt{2} ||\sqrt{17} |})[/tex]

[tex]\alpha =cos^{-1} (\frac{-1-10.58}{2.82*4.12})[/tex]

[tex]\alpha =cos^{-1} (\frac{-11.58}{11.61})[/tex]

[tex]\alpha =cos^{-1} (-0.99)[/tex]

hence option a 173 degree is correct

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The cost of 11 laptops is 11c

Using the parameters given :

The expresssion to determine the cost is :

Now we have;

33 = 30000

11 = c

cross multiply

33c = 330000

c = 10000

The cost of 11 laptops : 11*c

Answer:

Option A is correct (17,11).

Step-by-step explanation:

6x - 9y = 3

3x - 4y =7

it can be represented in matrix form as[tex]\left[\begin{array}{cc}6&-9\\3&4\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}3\\7\end{array}\right][/tex]

A= [tex]\left[\begin{array}{cc}6&-9\\3&4\end{array}\right] [/tex]

X= [tex]\left[\begin{array}{c}x\\y\end{array}\right][/tex]

B= [tex] \left[\begin{array}{c}3\\7\end{array}\right][/tex]

i.e, AX=B

or X= A⁻¹ B

A⁻¹ = 1/|A| * Adj A

determinant of A = |A|= (6*-4) - (-9*3)

                                    = (-24)-(-27)

                                    = (-24) + 27 = 3

so, |A| = 3

Adj A=  [tex]\left[\begin{array}{cc}-4&9\\-3&6\end{array}\right] [/tex]

A⁻¹ =  [tex]\left[\begin{array}{cc}-4&9\\-3&6\end{array}\right] [/tex]/3

A⁻¹ =  [tex]\left[\begin{array}{cc}-4/3&3\\-1&2\end{array}\right] [/tex]

X= A⁻¹ B

X=  [tex]\left[\begin{array}{cc}-4/3&3\\-1&2\end{array}\right] *\left[\begin{array}{c}3\\7\end{array}\right][/tex]

X= [tex]\left[\begin{array}{c}(-4/3*3) + (3*7)\\(-1*3) + (2*7)\end{array}\right][/tex]

X= [tex]\left[\begin{array}{c}-4+21\\-3+14\end{array}\right][/tex]

X= [tex]\left[\begin{array}{c}17\\11\end{array}\right][/tex]

x= 17, y= 11

solution set= (17,11).

Answer:

a. (17,11)

Step-by-step explanation:

The given system is ;

[tex]6x-9y=3[/tex]

[tex]3x-4y=7[/tex]

The augmented matrices is

[tex]\left[\begin{array}{ccc}6&-9&|3\\3&-4&|7\end{array}\right][/tex]

Divide Row 1 by 6

[tex]\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\3&-4&|7\end{array}\right][/tex]

Subtract 3 times Row 1 from Row 2

[tex]\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\0&\frac{1}{2}&|\frac{11}{2}\end{array}\right][/tex]

Divide Row 2 by [tex]\frac{1}{2}[/tex]

[tex]\left[\begin{array}{ccc}1&-\frac{3}{2}&|\frac{1}{2}\\0&1}&|11\end{array}\right][/tex]

Add [tex]\frac{3}{2}[/tex] times Row 2 to Row 1

[tex]\left[\begin{array}{ccc}1&0&|17\\0&1}&|11\end{array}\right][/tex]

Hence the solution is (17,11)

Answer:

Ben has mixed up the digit for millions and hundred thousands: he wrote 5 in the place for millions and 6 in the place for hundred thousands, but it should be the other way. I think that he should exchange those two digits in the faulty report.

B, C, and D is the answer

This question most likely pertains to a mathematical figure or diagram. The student is asked to identify true statements applying to the figure, which might illustrate a problem-solving process or the form of a disjunctive syllogism. Without the actual figure, a more detailed analysis isn't possible.

Without having the actual figure to analyze it's difficult to state definitively, but from the context provided, it sounds like this could be a diagram or figure related to problem-solving in mathematics. Figures are often used in mathematics to illustrate complex concepts and provide visual representation to support understanding. It seems like the figure might be used to demonstrate the process of solving mathematical problems or possibly, the process of argumentation using a disjunctive syllogism. Based on this information, the student may have to identify true statements about the figure in question. Potential statements could relate to the type of problem the figure represents, the process it illustrates, or specific features or attributes depicted in the figure itself.

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Answer:

1. C

2. A

3. C

Step-by-step explanation:

1. The equation is most easily solved through the quadratic formula since it’s a term is greater than 1.

2. The equation is most easily factored since it’s a term is 1.

3. The equation is most easily solved by the quadratic formula since it’s a term is greater than 1.

Answer:

B

Step-by-step explanation:

The domain is the set of x values for which the function is defined.

The range is the set of y values for which the function is defined.

Attached is the graph of the exponential function.

It is the basic graph of exponential function of y = 5^x which is shifted 6 units above (because of +6 at the end).

Looking at the graph, the domain is the set of all x values.

The range is anything above 6.

Correct answer is B.

Answer: b

Step-by-step explanation:

A P E X

Answer:

a. The amount of caffeine left is 52.77 mg

b. It will take about 5.42 hours

Step-by-step explanation:

* Lets solve it as an exponential decay

- Exponential decay:  If a quantity decrease by a fixed percent at

 regular intervals, the pattern can be depicted by this functions

  y = a(1 - r)^x

# a = initial value (the amount before measuring growth or decay)

# r = growth or decay rate (most often represented as a percentage

  and expressed as a decimal)

# x = number of time intervals that have passed

* Now lets solve the problem

∵ The initial amount of caffeine is 100 mg

∴ a = 100 mg

∵ The caffeine decreases by about 12% each hour

∴ r = 12/100 = 0.12

* Lets solve a.

a. ∵ x = 5 ⇒ the time interval

∵ The amount of caffeine left = a(1 - r)^x

∴ The amount of caffeine left = 100(1 - 0.12)^5

∴ The amount of caffeine left = 100(0.88)^5= 52.77 mg

* To find the time x use the linear logarithmic function

b. ∵ The amount of caffeine is 50 mg

∴ 50 = 100(1 - 0.12)^x ⇒ divide both sides by 100

∴ 50/100 = (0.88)^x

∴ 0.5 = (0.88)^x ⇒ take ln for each side

∴ ln(0.5) = ln(0.88)^x

∵ ln(a)^n = n ln(a)

∴ ln(0.5) = x ln(0.88) ⇒ divide both sides by ln(0.88)

∴ x = ln(0.5)/ln(0.88) = 5.4 years

* It will take about 5.42 hours

After 5 hours, approximately 54.47 mg of caffeine would remain in your system. It would take approximately 3.72 hours for the caffeine level to reach 50 mg.

To find the remaining amount of caffeine after 5 hours, we need to calculate the decreasing amount of caffeine each hour. The decrease in caffeine can be calculated by multiplying the previous amount of caffeine by 0.88 (1 - 0.12). So, after 5 hours, the remaining caffeine can be found using the formula:

To find the time it takes to have 50 mg of caffeine remaining, we can set up an equation and solve for time:

Therefore, after 5 hours, approximately 54.47 mg of caffeine would remain in your system. It would take approximately 3.72 hours for the caffeine level to reach 50 mg.

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Answer:

The cost of a candy bar is [tex]\$1.5[/tex]

Step-by-step explanation:

Let

x----> the cost of a soda

y----> the cost of a candy bar

we know that

[tex]3x+4y=10.17[/tex]-----> equation A

[tex]2x+5y=10.28[/tex]-----> equation B

Solve the system of equations by graphing

Remember that the solution of the system of equations is the intersection point both graphs

The solution is the point [tex](1.39,1.5)[/tex]

see the attached figure

therefore

The cost of a soda is [tex]\$1.39[/tex]

The cost of a candy bar is [tex]\$1.5[/tex]

Answer:

Step-by-step explanation:

105 years old. Since if he 35 the mother is 105>>------->(35 x 3= 105)

85 years old. 25•3=75 the difference between 75 and 25 is 50, which means she is 50 years older than him. 35+50=85.

Answer:

i really have no clue but if i put this i get points so good luck on your test

Answer:

[tex]\boxed{\text{Min = 400; Max = 960}}[/tex]

Step-by-step explanation:

F = 8x + 5y

At (0, 100):  F = 8×0    + 5×100 =     0 + 500 = 500

At (0, 80):   F = 8×0    + 5×80   =     0 + 400 = 400 — MINIMUM

At (80, 60): F = 8×80  + 5×60  = 640 + 300 = 940

At (80, 0):   F = 8×80  + 5×0    = 640  +     0 = 640

At (120, 0):  F = 8×120 + 5×0   = 960  +     0 = 960 — MAXIMUM

The minimum value is [tex]\boxed{ 400}[/tex]and the maximum value is [tex]\boxed{ \text{ 960}}[/tex].

Answer:
Minimum; 400
Maximum; 960

☆

EDGE2022; Good Luck :D

Answer:

4 quarts

Step-by-step explanation:

Answer:

  82

Step-by-step explanation:

The sum resolves to ...

  13 + 18 + 23 + 28 = 82

_____

for n=2, the term is 5·2 +3 = 10+3 = 13

The coefficient of n is 5, so you know the next term will be 5 more than this:

for n=3, the term is 5·3 +3 = 15+3 = 18

Additional terms are each 5 more than the one before. Since there are so few terms, you can add them up directly as quickly as you can use any other formula. For a longer series, you might find the average term (here, 41/2), then multiply by the number of terms (here, 4). The result is the sum of the series:

  4·(41/2) = 82

Answer:

Step-by-step explanation:

139 +10 = 149

149 / 8 = 18.625

round  to nearest hundredth = 18

Answer:

the solution is (40, 21)

Step-by-step explanation:

Let the two numbers be x and y.

Then x + y = the sum = 61, and

         x - y = the difference = 19.

Solve the system of linear equations

x + y = 61

x - y  = 19

Combining these two equations:

x + y = 61

x - y  = 19

---------------

2x    =  80, so x must be 40.

Substituting 40 for x in the first equation, we get 40 + y = 61.  

Combining the constants, we get y = 21.

Then the solution is (40, 21).

The numbers whose sum is 61 and difference 19 are 40 and 21. This is obtained by using algebraic expression for the given condition.

Given that sum of numbers is 61 and difference is 19.

Let the larger number be x and smaller number be y.

Then we can write that, x+y=61 (sum) and x-y=19 (difference)

By solving the algebraic equations we can find the numbers.

(19+y) + y = 61

19+2y = 61

2y = 61 - 19 = 42

y=42/2=21 ⇒ y=21 is the smaller number

x = 19+y = 19+21 =40 ⇒ x=40 is the larger number

Hence the numbers whose sum is 61 and difference 19 are 40 and 21.

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Answer:

Step-by-step explanation:

1. Equation of a circle:

(x - h)² + (y - k)² = r²

where (h, k) is the center and r is the radius.

If (h, k) is (2, 8) and r = 10:

(x - 2)² + (y - 8)² = 100

2. The vertex and focus have the same x-coordinate, so this is a vertical parabola.  Equation of a vertical parabola is:

y = 1/(4p) (x - h)² + k

where (h, k) is the vertex and p is the distance from the vertex to the focus.

If (h, k) is (2, 2) and p = 5-2 = 3:

y = 1/12 (x - 2)² + 2

3. The directrix is a vertical line, so this is a horizontal parabola.  Equation of a horizontal parabola is:

x = 1/(4p) (y - k)² + h

The distance between the directrix and the vertex is the same as p.

If (h, k) is (5, 2) and p = 5-3 = 2:

x = 1/8 (y - 2)² + 5

Answer:

[tex]\texttt{Probability that the first ball will be red and the second will be blue = }\frac{9}{55}[/tex]

Step-by-step explanation:

Total number of balls = 3 + 2 + 6 = 11

Probability is the ratio of number of favorable outcome to total number of outcomes.

[tex]\texttt{Probability that the first ball will be red = }\frac{\texttt{Total number of red balls}}{\texttt{Total number of balls}}\\\\\texttt{Probability that the first ball will be red = }\frac{3}{11}[/tex]

Now we have 10 balls in which 6 are blue.

[tex]\texttt{Probability that the second ball will be blue = }\frac{\texttt{Total number of blue balls}}{\texttt{Total number of balls}}\\\\\texttt{Probability that the first ball will be red = }\frac{6}{10}=\frac{3}{5}[/tex]

[tex]\texttt{Probability that the first ball will be red and the second will be blue = }\frac{3}{11}\times \frac{3}{5}\\\\\texttt{Probability that the first ball will be red and the second will be blue = }\frac{9}{55}[/tex]

The leading coefficient is the number in front of the variable with the highest degree(exponent)

The highest degree (exponent) is 5, so the leading coefficient = -0.2

ANSWER

The leading coefficient of the given polynomial is -0.2

EXPLANATION

The given polynomial function is

[tex]f(x)=3x^2-0.2x^5+7x^3[/tex]

We rewrite in standard form to obtain,

[tex]f(x)=-0.2x^5+7x^3+3x^2[/tex]

Note that writing in standard form means writing in descending powers of x.

The coefficient of the leading term is -0.2.

Answer:

tan(Sin^-1 x/2)=  [tex]\frac{x/2}{\sqrt{1-x^{2}/4 } }[/tex]

Step-by-step explanation:

Let sin^-1 x/2= θ

then sinθ= x/2

on the basis of unit circle, we have a triangle with hypotenuse of length 1,   one side of length x/2 and opposite angle of θ.

          tan(Sin^-1 x/2) = tanθ

           tanθ= sinθ/cosθ

as per trigonometric identities cosθ= √(1-sin^2θ)

           tanθ= sinθ/ √(1-sin^2θ)

substituting the value sinθ=x/2 in the above equation

             tanθ= [tex]\frac{x/2}{\sqrt{1-x^{2}/4 } }[/tex]

now substituting the value sin^-1 x/2= θ in above equation

             tan(sin^-1 x/2) =  [tex]\frac{x/2}{\sqrt{1-x^{2}/4 } }[/tex]

!

Answer:

...so cos t = cot t can only happen when cos t is 0 or sin t is 1. (But sin t = 1 only happens when cos t = 0, so cos t = 0 is enough.)  

arccos(0) is pi/2, plus or minus any multiple of pi.

Step-by-step explanation:

cotθ = cosθ  

cosθ/sinθ = cosθ  

sinθ = 1  

θ = π/2

pi also = to 3.14 etc.

Answer:

Step-by-step explanation:

[tex]cot \theta = cos \theta\\ \frac {cos \theta}{sin \theta} = cos \theta[/tex]

An easy solution is if [tex]cos \theta = 0[/tex], or [tex]\theta = \frac \pi 2[/tex], plus or minus any number of half circles.

Answer:

Chris is correct. (Choice letter C)

Step-by-step explanation:

We can eliminate choices A and B from the get-go because the problem stated that both hikers are traveling at 1mi/hr. This means that the slope of the equation would equate to 1, so just x.

That leaves us with C and D. The two equations both have a slope of 1, but C has a y-intercept of -2 and D has a y-intercept of 2, which represents the starting location of the second hiker. The first hiker is the one who is ahead by 2 miles, so it can't be D. This leaves us with choice C. Chris is correct.

Answer:

C. Chris said, “The equation y = x – 2 represents the distance traveled by the second hiker.”

Step-by-step explanation:

Answer:

The probability is [tex]\frac{5}{16}[/tex]

Step-by-step explanation:

Given is that a fair coin is tossed 5 times in a row.

This gives total number of outcomes as = [tex]2\times2\times2\times2\times2=32[/tex]

As its needed that heads comes exactly 2 times, so favorable outcomes are = 5C2

[tex]\frac{5!}{2!(5-2)!}[/tex]

Solving this we get 10

Therefore, the probability of getting heads exactly 2 times is =

[tex]\frac{10}{32}=\frac{5}{16}[/tex]

Answer:

5/16

Step-by-step explanation:

the answer is

10 × (4+3) = 70

8+6+12+10 = 36

6+12+10 = 28

28/36

7/9

Hope this helps ❤️

For this case we have that by definition, the surface area of a cone is given by:

[tex]SA = \pi * r * S + \pi * r ^ 2[/tex]

Where:

SA: It is the surface area

A: It's the radio

S: It's slant height

Then, replacing the values of the figure in the formula we have:

[tex]SA = \pi * 6 * 15 + \pi * 6 ^ 2\\SA = 90 \pi + 36 \pi\\SA = 126 \pi \ units ^ 2[/tex]

ANswer:

Option A

Answer:

The equation of the circle in standard form is: (x - 2)² + (y - 4)² = 9

Step-by-step explanation:

* Lets revise the standard form of the equation of the circle

- If the center of the circle is point (h , v) and the radius of the

 circle is r, then the standard form of the equation of the circle

 is (x - h)² + (y - v)² = r²

- (x , y) a general point on the circle

* Lets look to the picture

- The center of the circle is point (2 , 4)

- The highest point on the circle is (2 , 7) and the lowest point

 on the circle is (2 , 1)

∴ The diameter of the circle = 7 - 1 = 6

∵ The radius = 1/2 the diameter

∴ The radius of the circle = 1/2 × 6 = 3

* Now we can write the equation of the circle

∵ h = 2 and v = 4

∵ r = 3

∴ (x - 2)² + (y - 4)² = 3²

∴ The equation of the circle in standard form is:

   (x - 2)² + (y - 4)² = 9

the equation would be y = 30 + x and if your brother will read for 15 minutes then you'll read for 45 minutes i believe.

The equation for the number of minutes you read, y, when your brother reads x number of minutes is y = x + 30. Therefore, if your brother reads for 15 minutes, you will read for 45 minutes.

To write an equation for the number of minutes you read, y, when your brother reads x number of minutes and you decide to read 30 minutes longer than him, you can represent this relationship as y = x + 30. This equation signifies that your reading time (y) is equal to your brother's reading time (x) plus an additional 30 minutes.

If your brother reads for 15 minutes (x=15), you can determine how many minutes you will read by substituting 15 into the equation for x, obtaining y = 15 + 30 = 45 minutes. Therefore, if your brother reads for 15 minutes, you will read for 45 minutes to stay ahead.