24. Show and explain clearly, how you would solve the following problem using John Mason's Model for problem solving: A lady rears goats and fowls in her backyard. She observed that there were 42 eyes and 66 legs in her backyard. How many fowls and how many goats are in the yard?

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Answer:

  x = π/12, 5π/12

Step-by-step explanation:

We can work this problem a couple of ways. We can convert it to single trig function, shifted horizontally. Or we can convert it to a double-angle equation.

__

We recall the sum of angles formula is ...

  sin(x+y) = sin(x)cos(y) +cos(x)sin(y)

Using this, we can rewrite the left side of the equation using some angle offset y, and some scale factor k.

  k·sin(x +y) = k·sin(y)·cos(x) +k·cos(y)·sin(x) = 2cos(x) +2sin(x)

finding the shift

Equating coefficients of cos(x) and sin(x), we can solve for k and y:

  k·sin(y) = 2

  k·cos(y) = 2

The ratio of these equations is ...

  (k·sin(y))/(k·cos(y)) = 2/2

  tan(y) = 1   ⇒   y = π/4

  k = 2/sin(π/4) = 2√2

shifted equation

So, our original equation becomes ...

  2√2·sin(x +π/4) = √6

Dividing by √2 and using the inverse sine function, we have ...

  x +π/4 = arcsin((√3)/2) = π/2 ±π/6

  x = π/4 ±π/6

  x = π/12, 5π/12

__

If we square both sides of the original equation, we get ...

  (2sin(x) +2cos(x))² = (√6)²

  4sin²(x) +8sin(x)cos(x) +4cos²(x) = 6

  2sin(x)cos(x) = (6 -4)/4 = 1/2 . .  . . use sin² +cos² = 1, subtract 4, divide by 4

Using the trig identity 2sin(x)cos(x) = sin(2x), we can find ...

  2x = arcsin(1/2) = π/2 ±π/3 +2kπ . . . . k = an integer;

  x = π/4 ±π/6 +kπ . . . . for integer k

  x = {π/12, 5π/12, 13π/12, 17π/12}

We know that squaring the equation can introduce extraneous solutions, so we need to try these out. For x in the third quadrant, the sine and cosine values are both negative, so the only useful solutions here are ...

  x = π/12, 5π/12

_____

Additional comment

Based on the above, we now know that any trig expression of the form ...

  a·sin(x) +b·cos(x)

can be rewritten to the form ...

  (√(a² +b²))·sin(x +arctan(b/a)) . . . . . a scaled and shifted sine function

The arctangent will have to take the signs of 'a' and 'b' into account in order to get the angle quadrant right.

Answer:

a)  81π  in³

b)  27  in³

c)  divide the volume of the slice of cake by the volume of the whole cake

d)  10.6%

e)  see explanation

Step-by-step explanation:

The cake can be modeled as a cylinder with:

[tex]\sf Radius=\dfrac{1}{2}diameter \implies r=4.5\:in[/tex]

[tex]\textsf{Volume of a cylinder}=\sf \pi r^2 h \quad\textsf{(where r is the radius and h is the height)}[/tex]

[tex]\begin{aligned}\sf \implies \textsf{Volume of the cake} & =\pi (4.5)^2(4)\\ & = \sf \pi (20.25)(4)\\ & = \sf81 \pi \:\: in^3\end{aligned}[/tex]

[tex]\begin{aligned}\textsf{Circumference of the cake} & = \sf \pi d\\& = \sf 9 \pi \:\:in\end{aligned}[/tex]

If each slice of cake has an arc length of 3 in, then the volume of each slice is 3/9π of the entire volume of the cake.

[tex]\begin{aligned}\implies \textsf{Volume of slice of cake} & = \sf \dfrac{3}{9 \pi} \times \textsf{volume of cake}\\\\& = \sf \dfrac{3}{9 \pi} \times 81 \pi\\\\& = \sf \dfrac{243 \pi}{9 \pi}\\\\& = \sf 27\:\:in^3\end{aligned}[/tex]

The volume of each slice of cake is 27 in³.

The volume of the whole cake is 81π in³.

To calculate the probability that the first slice of cake will have the marble, divide the volume of a slice by the volume of the whole cake:

[tex]\begin{aligned}\implies \sf Probability & = \sf \dfrac{27}{81 \pi}\\\\& = \sf 0.1061032954...\\\\ & = \sf 10.6\% \:\:(1\:d.p.)\end{aligned}[/tex]

Probability is approximately 10.6%  (see above for calculation)

If the four slices of cake are cut and passed out before anyone eats or looks for the marble, the probability of getting the marble is the same for everyone. If one slice of cake is cut and checked for the marble before the next slice is cut, the probability will increase as the volume of the entire cake decreases, until the marble is found.  So it depends upon how the cake is cut and distributed as to whether Hattie's strategy makes sense.

Answer:

[tex]b = 2[/tex]

Step-by-step explanation:

[tex]( - 12 - 1 - 5) + (3b - b) \\ - 18 + 2b \\ 2b - 18 \\ b = 2[/tex]

Answer:

5/17

Step-by-step explanation:

If we do the multiplication first, we get:

[tex]\frac{4-(12\div-12)}{13+(-8\div-2)}[/tex]

If we do the division next, we get:

[tex]\frac{4+1}{13+4} = 5/17[/tex]

Answer:

170

Step-by-step explanation:

[tex]x+40=210\\x=170[/tex]

If a number, x, is increased by 40 (+40) and is now equal to 210 (=210), then the number, x, is equal to 170.

The constant of the proportional relationship graphed in this problem is of 10.

A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:

y = kx

In which k is the constant of proportionality.

In this problem, the points on the graph are: (0,0), (2,20), (4,40), ..., hence the constant is:

k = 40/4 = 20/2 = 10.

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Your question translates to computing [tex]3^{2020} \pmod {13}[/tex].

Recall Euler's theorem: if [tex]\gcd(a,n)=1[/tex] (that is, [tex]a[/tex] and [tex]n[/tex] are relatively prime), then [tex]a^{\varphi(n)}\equiv1\pmod n[/tex], where [tex]\varphi(n)[/tex] denotes Euler's totient function, which counts the number of positive integers relatively prime to [tex]n[/tex].

Since 13 is prime, we have [tex]\phi(13)=12[/tex]. Then by Euler's theorem,

[tex]3^{12} \equiv 1 \pmod{13}[/tex]

Now, observe that 2020 = 168×12 + 4, so that

[tex]3^{2020} \equiv 3^{168\times12+4} \equiv \left(3^{12}\right)^{168} \times 3^4 \equiv 1^{168} \times 3^4 \equiv 3^4 \pmod{13}[/tex]

and since 3⁴ = 81 = 6×13 + 3, we end up with

[tex]3^{2020} \equiv 81 \equiv 3 \pmod{13}[/tex]

so the remainder upon dividing 3²⁰²⁰ by 13 is 3.

Answer:

The mid-point is (9,-9/2)

Step-by-step explanation:

You would use the mid-point formula for this

[tex](\frac{x_{2} +x_{1} }{2} , \frac{y_{2}+y_{1} }{2} )[/tex]

if you plug that in it is ([tex]\frac{13-2}{2}, \frac{-7+5}{2}[/tex])

resulting in (11/2,-2/2) = (11/2,-1)

Answer:

2nd one is the correct

Answer:

a) 5 pounds

b) 365 dollars

Answer:

4/11 of an inch (about .4 of an inch)

Step-by-step explanation:

[tex]l \times 33 \times 5.75 = 69[/tex]

[tex]189.75l = 69[/tex]

[tex]l = \frac{69}{189.75} = \frac{4}{11} = .3636[/tex]

So the length of the popcorn box is 4/11 of an inch, or about .4 of an inch.

Differentiate the function/equation with respect to x and solve for the derivative, dy/dx. The value of dy/dx at the given point is the slope of the tangent line to the curve at that point. Then use the point-slope formula to get the equation of the tangent.

1.

[tex]y = \dfrac{x^2}{x+2} \implies \dfrac{dy}{dx} = \dfrac{2x\times(x+2) - x\times1}{(x+2)^2} = \dfrac{x(x+4)}{(x+2)^2}[/tex]

When x = 2, the derivative is

[tex]\dfrac{dy}{dx}\bigg|_{x=2} = \dfrac{2(2+4)}{(2+2)^2} = \dfrac34[/tex]

Then the equation of the tangent line at (2, 1) is

[tex]y - 1 = \dfrac34 (x - 2) \implies \boxed{y = \dfrac{3x}4 - \dfrac12}[/tex]

2.

[tex]x^3 + 2y^2 = 10y \implies 3x^2 + 4y \dfrac{dy}{dx} = 10 \dfrac{dy}{dx} \implies \dfrac{dy}{dx} = \dfrac{3x^2}{10-4y}[/tex]

When x = 2 and y = 1, the derivative is

[tex]\dfrac{dy}{dx}\bigg|_{(x,y)=(2,1)} = \dfrac{3\times2^2}{10-4\times1} = 2[/tex]

Then the tangent at (2, 1) has equation

[tex]y - 1 = 2 (x - 2) \implies \boxed{y = 2x - 3}[/tex]

[tex]\quad \huge \quad \quad \boxed{ \tt \:Answer }[/tex]

[tex]\qquad \tt \rightarrow \: x = 32°[/tex]

____________________________________

[tex] \large \tt Solution \: : [/tex]

Diagonals of a Rhombus bisect each other at 90°

[tex]\qquad \tt \rightarrow \: x + 58 + 90 = 180[/tex]

[ Sum of interior angles of a triangle ]

[tex]\qquad \tt \rightarrow \: x + 148 = 180[/tex]

[tex]\qquad \tt \rightarrow \: x = 180 - 148[/tex]

[tex]\qquad \tt \rightarrow \: x = 32 \degree[/tex]

Answered by : ❝ AǫᴜᴀWɪᴢ ❞

The diagonals of the rhombus intersect at right angle. Then the value of x will be 32°.

It is a polygon with four sides. The total interior angle is 360 degrees. In a rhombus, opposite sides are parallel and equal.

Figure ABCD is a rhombus.

Its diagonals intersect at right angle.

Let the another angle be x. Then we have

x + 58° + 90° = 180°

        x + 148° = 180°

                   x = 32°

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Answer:

h(t)= 5r2-1

That the answer for the question

The slope of the line that passes through (1, -3) and (0, 2) is: B. m = -5.

Slope (m) = rise / run = change in y / change in x.

Given the points, (1, -3) and (0, 2):

Slope (m) = (-3 - 2)/(1 - 0)

Slope (m) = -5/1

Slope (m) = -5

Therefore, the slope of the line is: B. m = -5.

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Ivanna's score was at the 36th percentile, will be 99.64 and Aldo's score was at the 19th percentile, will be 99.81.

The analysis of mathematical representations is algebra, and the handling of those symbols is logic.

Last year, over 10,000 students took an entrance exam at a certain state university.

Let the maximum score be 100.

Ivanna's score was at the 36th percentile, will be

⇒ [(10,000 – 36) / 10,000] x 100

⇒ 99.64

Aldo's score was at the 19th percentile, will be

⇒ [(10,000 – 19) / 10,000] x 100

⇒ 99.81

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Answer:

The answers are=

The x-coordinate of the vertex is (38, 0) and the y-coordinate of the vertex is (19, 1768.9).

It is defined as the graph of a quadratic function that has something bowl-shaped.

(x - h)² = 4a(y - k)

(h, k) is the vertex of the parabola:

a = √[(c-h)² + (d-k²]

(c, d) is the focus of the parabola:

It is given that:

The graph of the parabolic path is shown in the picture.

From the graph:

The x-coordinate of the vertex is (38, 0)

The x-coordinate represents time in seconds

The y-coordinate of the vertex is (19, 1768.9)

The y-coordinate represents the height in meters

Thus, the x-coordinate of the vertex is (38, 0) and the y-coordinate of the vertex is (19, 1768.9).

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Answer:

7.1%

Step-by-step explanation:

Those that did not complete the race either gave up or were disqualified. This means that 18 + 3 = 22 people did not complete the race. The percentage is 22/310 × 100% = 7.1%

Step-by-step explanation:

hope you can understand

Step-by-step explanation:

x^3 is a perfect cube, 8 is a perfect cube, so we use difference of cubes.

[tex] {a}^{3} - {b}^{3} = (a - b)( {a}^{2} + ab + {b}^{2} )[/tex]

Cube root of x^3 is x.

Cube root of 8 is 2

So

a=x

b= 2.

[tex](x - 2)( {x}^{2} - 2x + 4)[/tex]

Set these equations equal to zero

[tex]x - 2 = 0[/tex]

[tex]x = 2[/tex]

[tex] {x}^{2} - 2x + 4 = 0[/tex]

If we do the discriminant, we get a negative answer so we would have two imaginary solutions,

Thus the only real root is 2.

If you want imaginary solutions, apply the quadratic formula.

[tex]1 + i \sqrt{ 3 } [/tex]

and

[tex]1 - i \sqrt{3} [/tex]

The following expressions (1+cosβ)(1−cosβ)sinβ is equivalent to sin³β

In a Right angled triangle , trigonometric ratios can be used to determine the value of angles and sides of the triangle.

The trigonometric expression given in the question is

(1+cosβ)(1−cosβ)sinβ

(a+b)(a-b) = a² - b²

( 1 - cos²β)sinβ

By the trigonometric Identity

1-cos²β = sin² β

 sin² β x sin β

sin³β

Therefore Option B is the correct answer.

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Answer:

A.) x = 21.4

Step-by-step explanation:

When solving the inequality, you can treat the sign like an equal sign. To isolate the "x" variable, you can add "4" to both sides. This eliminates the -4 from the right side.

You are left with the inequality:

x > 19.5

If "x" must be greater than 19.5, the only answer that satisfies this is A.) x = 21.4. All of the other answers are less than 19.5.

[tex]\bf{x -4 > 15.5}[/tex]

[tex]\bf{Add \ 4 \ to \ both \ sides.}[/tex]

[tex]\bf{x-4+4 > 15.5+4 }[/tex]

[tex]\bf{x > 19.5 === > Answer }[/tex]

[tex]\red{\boxed{\green{\boxed{\boldsymbol{\purple{Pisces04}}}}}}[/tex]

They will earn If they work every week If the year is $22100.

We have given that,

you currently make 425

We have to determine what is the gross they will earn If they work a week If the year.

We know that

A worker currently makes $425 per week

1year=52 weeks

So by proportion find the amount that the worker will earn in one year

[tex]\frac{425}{x}\times \frac{\$}{weeks}[/tex]

[tex]=\frac{x}{52}\times \frac{\$}{weeks}[/tex]

[tex]x=52\times 425[/tex]

x=22,100

Therefore, The answer is $22100.

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Answer:

Step-by-step explanation:

regular decagon = all side equal and all angles congruent, so your answer is 144°

Answer:

survey (A)

A questionnaire (or someone asking questions) is considered to be a survey. It is "surveying" someone's opinion on data.

hope this helps!!

The size of the colony after 10 days is  852.

The growth rate of the colony can be represented with an exponential equation with the form:

FV = P(1 + r)^n

Where:

55(1.73)^5 = 852

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Answer:

usually at the login screen it gives you an option to reset your password if you forgot, it will appear in blue as "forgot password?" or "trouble logging in?"