I REALLY NEED HELP PLEASE, I WILL FOLLOW AND FAV THE BRAINIEST ONE HERE.

t2 as a linear combination of the In P2 can be written as:                                 t^2 = (-1/7)(1 - 5t) - (4/7)(-2 + t + t^2). The change-of-coordinates matrix from B to S is: [ -1/35 2/7 -1/7 ]

                        [ -1/35 1/7 -4/7 ]

                          [ -1/35 0 0 ]

Let P1(t) = 1 - 5t and P2(t) = -2 + t + t^2 be the basis polynomials for B.

To write t^2 as a linear combination of P1(t) and P2(t), we need to find constants a and b such that:

t^2 = a P1(t) + b P2(t)

Substituting in the expressions for P1(t) and P2(t), we get:

t^2 = a(1 - 5t) + b(-2 + t + t^2)

Rearranging terms, we get:

t^2 = (b - 5a) t^2 + (t + 5a - 2b)

Equating coefficients of t^2 and t on both sides, we get:

b - 5a = 1

5a - 2b = -2

Solving for a and b, we get:

a = -1/7

b = -4/7

Therefore, we can write t^2 as:

t^2 = (-1/7)(1 - 5t) - (4/7)(-2 + t + t^2)

To find the change-of-coordinates matrix from the basis B to the standard basis S = {1, t, t^2}, we need to express each basis vector of S as a linear combination of the basis polynomials in B.

We have:

1 = -1/35 (9 P1(t) - 20 P2(t))

t = 2/7 P1(t) + 1/7 P2(t)

t^2 = -1/7 P1(t) - 4/7 P2(t)

Therefore, the change-of-coordinates matrix from B to S is:

[ -1/35 2/7 -1/7 ]

[ -1/35 1/7 -4/7 ]

[ -1/35 0 0 ]

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The derivative of the expression x²y⁵ with respect to x is 2xy⁵.

To differentiate the expression x²y⁵ with respect to x, we will use the Power Rule for differentiation. The Power Rule states that the derivative of xⁿ, where n is a constant, is nxⁿ⁻¹. In our case, the expression is x²y⁵, which can be written as (x²)(y⁵). Since y⁵ is a constant with respect to x, we will treat it as such during differentiation.

Now, applying the Power Rule to x², we get 2x^(2-1), which is 2x. Therefore, the derivative of the expression x²y⁵ with respect to x is (2x)(y⁵) or 2xy⁵.

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The linear model, F(t) = 9.44t + 84, does not fit the data well.

To determine if the linear model is a good fit for the data, we can compare the model's predictions with the actual data points shown in the scatter plot. The scatter plot shows the retail sales in billions of dollars for different years since 1995. The linear model F(t) = 9.44t + 84 is a linear equation with a slope of 9.44 and a y-intercept of 84.

Upon comparing the linear model's predictions with the actual data points, we can see that the linear model does not accurately capture the trend in the data. The data points do not form a straight line, but instead exhibit a curved pattern. The linear model may not capture the non-linear relationship between the years since 1995 and the retail sales accurately.

Therefore, the linear model, F(t) = 9.44t + 84, is not a good fit for the data, as it does not accurately represent the trend exhibited by the scatter plot of retail sales in drug stores in the U.S. since 1995

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Answer:

4

Step-by-step explanation:

The expanded form of equations, 3.1 is A11b1 + A12b2 + A13b3 + f1 = 0, A21b1 + A22b2 + A23b3 + f2 = 0, A31b1 + A32b2 + A33b3 + f3 = 0, 3.2 is A11, A12, A13, A21, A22, A23, A31, A32, and A33 and 3.3 is A11δ11 + A22δ22 + A33δ33 = B11m + B22m + B33m, where δ is the Kronecker delta function.

In mathematics and science, equations are frequently expressed in a compact form to represent complicated systems or connections. However, to comprehend their separate components or solve them numerically, these equations must frequently be expanded. To extend the equations and describe them more thoroughly, we substituted the variables i, j, and k with their corresponding values 1, 2, and 3.

We have enlarged the matrix equation Aijbj + fi = 0 in equation 3.1 to reflect three different equations, each corresponding to a row in the matrix. This allows us to separately solve the variables in each row and derive a solution for the full matrix.

We enlarged the equation Aikk = Bij + Ckkδij = Bimm in equation 3.3 to represent three independent equations, each corresponding to a diagonal element in the matrix. Here, δij is the Kronecker delta, which allows us to distinguish between diagonal and off-diagonal components. This is frequently beneficial in solving matrices-based problems since diagonal elements have specific features and can be solved more readily than off-diagonal elements.

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the length of the repel line needed is approximately 44 feet (rounded to the nearest whole number).

what is length ?

Length is a physical dimension that describes the extent of an object or distance between two points. In geometry, length refers to the distance between two points, and it is usually measured in units of length such as meters, centimeters, feet, inches,

In the given question,

(a) The correct trigonometric ratio to use in this problem is the sine ratio, which relates the opposite side to the hypotenuse in a right triangle. In this case, we are given the angle A and we want to find the length of the opposite side, which is the distance from the top of the ravine to Lucy. Therefore, we can use the sine ratio as follows:

sin(A) = opposite/hypotenuse

(b) We can set up the equation using the given information as follows:

sin(17°) = opposite/150

where opposite is the length of the repel line that we want to find.

(c) To solve for the length of the repel line, we can rearrange the equation as follows:

opposite = sin(17°) x 150

opposite = 0.2924 x 150

opposite ≈ 44

Therefore, the length of the repel line needed is approximately 44 feet (rounded to the nearest whole number).

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If a cylindrical can of cocoa has the dimensions radius of 4 in , height of 3 in then the area of label is 75.36 square inches

We have a cylindrical can of cocoa.

The radius of the can R = 4 in

The height of the can H = 3 in

We know the formula for finding the lateral surface area of the cylinder is given by:

A = 2πRH

A = 2π×4×3

A = 24π

A=24×3.14

A=75.36 square inches

Hence, the approximate area available for the label is 75.36 square inches

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So, the answers are: dx/dt = [tex]18t^2 + 3[/tex], dy/dt = 5 - 8t, dy/dx = [tex](5 - 8t) / (18t^2 + 3)[/tex]

To find dx/dt, we need to take the derivative of x with respect to t:

dx/dt = [tex]18t^2 + 3[/tex]

To find dy/dt, we need to take the derivative of y with respect to t:

dy/dt = 5 - 8t

To find dy/dx, we can use the chain rule:

dy/dx = dy/dt / dx/dt
      = (5 - 8t) / (18t^2 + 3)[tex](18t^2 + 3)[/tex]

Hi! I'd be happy to help you with your question. Let's find dx/dt, dy/dt, and dy/dx using the given functions [tex]x = 6t^3 + 3t  \\and \\y = 5t - 4t^2.[/tex]

1. Find dx/dt: This is the derivative of x with respect to t.
  Differentiate x = 6t^3 + 3t with respect to t:
  dx/dt = [tex]d(6t^3 + 3t)/dt = 18t^2 + 3[/tex]

2. Find dy/dt: This is the derivative of y with respect to t.
  Differentiate y = 5t - 4t^2 with respect to t:
  dy/dt = [tex]d(5t - 4t^2)/dt = 5 - 8t[/tex]

3. Find dy/dx: This is the derivative of y with respect to x.
  To find this, we can use the chain rule: dy/dx = (dy/dt) / (dx/dt)

  Substitute the values we found for dy/dt and dx/dt:
  dy/dx = [tex](5 - 8t) / (18t^2 + 3)[/tex]

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The most likely true statement is that the median is greater than the mean because the data is skewed to the left.

Based on the information provided, Indira makes a box-and-whisker plot of her data and finds that the distance from the minimum value to the first quartile is greater than the distance between the third quartile and the maximum value. Which is most likely true? The answer is: the median is greater than the mean because the data is skewed to the left.

Here's a step-by-step explanation,

1. The distance from the minimum value to the first quartile being greater than the distance between the third quartile and the maximum value indicates that there is more data spread out on the left side of the plot.

2. This spread causes the data to be skewed to the left.

3. When data is skewed to the left, the median (Q2) is typically greater than the mean (average), as the mean gets pulled towards the longer tail on the left side.

So, the most likely true statement is that the median is greater than the mean because the data is skewed to the left.

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If your opponent goes first, you are guaranteed to win with 8 gems if

your opponent takes 1, 2, or 3 gems on their first turn.

In general, if there are n gems, you can guarantee a win if your opponent

takes n/2 or fewer gems on their first turn when you go second.

Let's consider the starting configuration of gems with 8 gems total.

If you go first, the maximum number of gems you can take on your first

turn is 4 (either 4 diamonds or 4 rubies or 2 of each).

If you take 4 diamonds, your opponent can take all 4 rubies, leaving you

with no choice but to take the remaining 4 diamonds on your next turn,

which means your opponent will take the last 4 rubies and win. Similarly,

if you take 4 rubies on your first turn, your opponent can take all 4

diamonds and win.

If you take 2 diamonds and 2 rubies on your first turn, your opponent can

mirror your move and take 2 diamonds and 2 rubies, leaving you with 2

diamonds and 2 rubies left. At this point, no matter what you do, your

opponent can take the remaining gems and win.

So, if you go first, there is no way to guarantee a win with 8 gems.

Now let's consider the case where your opponent goes first. If your

opponent takes 1, 2, or 3 gems on their first turn, you can mirror their

move and take the same number of gems, leaving 4, 5, or 6 gems left

respectively.

At this point, no matter what your opponent does, you can take enough

gems to ensure that you take the last gem and win. For example, if there

are 4 gems left, you can take 2 diamonds (or 2 rubies) to leave 2 gems,

and then take the remaining 2 gems on your next turn. Similarly, if there

are 5 gems left, you can take 1 diamond and 1 ruby to leave 3 gems, and

then take the remaining 3 gems on your next turn. And if there are 6

gems left, you can take 2 diamonds and 2 rubies to leave 2 gems, and

then take the remaining 2 gems on your next turn.

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Demonstrating the polynomial identity, As proved algebraically in section 1, it is possible to demonstrate that this identity holds true for any values of x and y.

The operations of addition, subtraction, multiplication, and non-negative integer exponents can be used to solve the expression made up of variables and coefficients in a polynomial.

The largest power of a variable that appears in an expression is the polynomial's degree.

Proving (x - y)² = (x² - 2xy + y²) algebraically:

Taking a look at the equation's left-hand side (LHS) first:

(x - y)² = (x - y)(x - y) // Using the formula for squaring a binomial

= x(x - y) - y(x - y) // Expanding the product of (x - y)(x - y)

= x² - xy - yx + y² // Simplifying by distributing the negative sign

= x² - 2xy + y² // Combining like terms

the polynomial identity being demonstrated (x - y)² = (x² - 2xy + y²) numerically:

Let's take x = 5 and y = 3 as an example:

LHS = (5 - 3)² = 2² = 4

RHS = 5² - 2(5)(3) + 3² = 25 - 30 + 9 = 4

We have thus demonstrated the numerical validity of the polynomial identity for the selected values of x and y. As proved algebraically in section 1, it is possible to demonstrate that this identity holds true for any values of x and y.

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The correct question is given below:

Prove that (x -y)² = (x^2 - 2xy + y^2) is true through an algebraic proof, identifying each step.

Demonstrate that your polynomial identity works on numerical relationships :- (x -y)² = (x²- 2xy + y²)

Answer:

Step-by-step explanation: There are four possible digits that the code number can begin with: 1, 3, 5, or 7. Since each of these digits is equally likely to be selected, the probability of the code number beginning with a 7 is 1/4 or 0.25. Therefore, the probability that the code number for a new customer will begin with a 7 is 0.25 or 25%.

Answer: m∠V = 28°

Step-by-step explanation:

     We know that a triangle's angles add up to 180. We will create an equation to solve for x. Then, we will substitute it back into the expression for angle V and simplify.

Given:

(9x - 8) + (2x + 2) + (3x + 4) = 180°

Simplify:

9x - 8 + 2x + 2 + 3x + 4 = 180°

Reorder:

9x + 2x + 3x - 8 + 2 + 4 = 180°

Combine like terms:

14x - 2 = 180°

Add 2 to both sides of the equation:

14x = 182°

Divide both sides of the equation by 13:

x = 13

---

m∠V = 2x + 2

m∠V = 2(13) + 2

m∠V = 28°

30 students needs to be added for the food to be enough for 75 days

A hostel contains 150 students

They have food that will last them for 90 days

If the food is supposed to last for 75 days, the number of students that will be added can be calculated as follows

150= 90

1= x

cross multiply

x= 150×90

x= 13,500

= 13500/75

= 180

180 students will be fed for 75 days

We initially had 150 students, subtract 150 from 180

180-150

= 30

Hence 30 students needs to be added so the food lasts for 75 days

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a. 150 families in the sample earned less than $45,000.

b. We can nοt determine hοw many families earned mοre than $ 55,000 exactly.

The term “incοme” generally refers tο the amοunt οf mοney, prοperty, and οther transfers οf value received οver a set periοd οf time in exchange fοr services οr prοducts.

Here, we have

Given:

Suppοsed yοu study family incοme in a randοm sample οf 300 families. Yοu find that the mean family incοme is $55,000; the median is $45,000; and the highest and lοwest incοmes are $250,000 and $2400, respectively.

a. 150 families in the sample earned less than $45,000 because the median is the middle value in the οrdered data.

Median = 45,000/300

=  150

b. We can nοt determine hοw many families earned mοre than $ 55,000 exactly. It will be less than half. Because $55,000 is the mean value, it is nοt based οn the οrder.

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The general solution for the Cauchy-Euler equation is a linear combination of the two solutions:

[tex]y(t) = C_1 * t^{-9} + C_2 * t^2[/tex]

To find the general solution to the given Cauchy-Euler equation for t > 0, first, we'll rewrite the equation using the given terms:

[tex]t^2 \frac{d^2y}{dt^2}+ 8t(dy/dt) - 18y = 0[/tex]

Now, we'll use the substitution y(t) = t^m, where m is a constant, to transform the equation into a simpler form:

By using this substitution, we get:

[tex]dy/dt = m * t^{m-1}\\d^{2}y/dt^2= m * (m-1) * t^{m-2}[/tex]

Substitute these expressions back into the original Cauchy-Euler equation:

[tex]t^2 * m * {m-1} * t^{m-2}+ 8t * m * t^{m-1} - 18 * t^m = 0[/tex]

Simplify by dividing both sides by t^(m-2):

[tex]m * (m-1) + 8m - 18t^2 = 0[/tex]

Now, we have a characteristic equation in terms of m:

[tex]m^2 + 7m - 18 = 0[/tex]

Factoring this equation gives:

(m+9)(m-2) = 0

This yields two possible values for m: m1 = -9, m2 = 2

Therefore, the general solution for the Cauchy-Euler equation is a linear combination of the two solutions:

[tex]y(t) = C_1 * t^{-9} + C_2 * t^2[/tex]

Where C1 and C2 are constants determined by any initial conditions.

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Therefore, the standard deviation of the data set is approximately 4.14.

The standard deviation is calculated by taking the square root of the variance, which is the sum of the squared deviations divided by the sample size minus 1.

So, first we need to calculate the variance:

variance = sum of squared deviations / (sample size - 1)

variance = 103 / (7 - 1)

variance = 17.17

Now we can find the standard deviation:

standard deviation = √(variance)

standard deviation = √(17.17)

standard deviation = 4.14 (rounded to two decimal places)

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The area of the shaded region is 36.3  to 3  significant figures

What is the area?

A two-dimensional figure's area is the amount of space it takes up. In other terms, it is the amount that counts the number of unit squares that span a closed figure's surface.

Step one: find the two diagonals of the kite.

The Horizontal diagonal can be obtained using the cosine rule:

AC² = OA²- OC² - 2 *OA*OC* cosθ

       = 10²+ 10² - 2* 10 *10 * cos(120)

AC² = 200

=> AC=√200 = 14.1

The vertical diagonal of the kite can be obtained by Pythagoras' Theorem:

Please note the law in circle geometry which states that a radius and a tangent always meet at right angles.

This implies that triangle OBC is a right-angled triangle, with angle OCB being 90 degrees, and COB being 60 degrees. This is because the diagonal divides the 120-degree angle into half.

cos(60)= 10/OB

=> OB= 10/ cos(60) = 20 m

Step two: Use the dimensions of the two diagonals of the Kite to find the area:

The area of a Kite is obtained using this formula:

area = pq/2, , where p and q are the two diagonals.

area =( 14.1*20)/2 = 141 m²

Step three: Calculate the area of the sector of the circle.

Area of the sector is obtained using this formula

Area =θ/360 * πr²  = 120/360 * 3.14 * 10²  = 104.66 m²

Step Four: Subtract the area of the sector from the area of the kite:

Area of the shaded region will be  141 m²  -  104.66 m²  = 36.34 m²

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By Limit comparison test the series ∞∑n=11n√n²+5 converges.

To use the limit comparison test, we need to find a series whose convergence or divergence is known and that is similar to the given series.

Let's consider the series ∞∑n=11n√n²+5 and choose a series that we know converges, such as ∞∑n=1 1/n².

We can now take the limit as n approaches infinity of the ratio of the nth term of the given series to the nth term of the chosen series:

limₙ→∞ (n√(n²+5))/(1/n²)

Simplifying the expression inside the limit, we get:

limₙ→∞ (n√(n²+5))/(1/n²) = lim(n→∞) n³√(1+5/n²)/1 = lim(n→∞) n³/√(n⁶+5n⁴)

Using L'Hopital's rule, we can take the derivative of the numerator and denominator separately with respect to n to get:

limₙ→∞n³/√(n⁶+5n⁴) = lim(n→∞) 3n²/3n⁵/²= lim(n→∞) 3n¹/²)/3n⁵/²) = 0

Since the limit is finite and nonzero, the given series and the chosen series have the same convergence behavior. Therefore, since we know that ∞∑n=1 1/n² converges (by the p-series test with p=2),

we can conclude that the given series ∞∑n=11n√n²+5 also converges.

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The eigenvalue with the larger absolute value for the Jacobian matrix at the point (30568, 386008) is approximately 5269.407, which is positive. No need to enter it as a negative number.

The system of equations is

f(x,y) = 9x^2 + 3x + y - 30 = 0

g(x,y) = 3x^2 + xy - 10^6 = 0

The Jacobian matrix J is

J(x,y) = [ df/dx df/dy ]

[ dg/dx dg/dy ]

where

df/dx = 18x + 3

df/dy = 1

dg/dx = 6x + y

dg/dy = x

Evaluated at the point (30568, 386008), we have

df/dx = 18(30568) + 3 = 550149

df/dy = 1

dg/dx = 6(30568) + 386008 = 582216

dg/dy = 30568

So, J(30568, 386008) =

[550149    1]

[582216  30568]

The eigenvalues of J(30568, 386008) are the solutions to the characteristic equation

det(J - λI) = 0

where I is the identity matrix and det denotes the determinant.

The characteristic equation is

(550149 - λ)(30568 - λ) - 582216 = 0

Expanding and simplifying this expression, we get

λ^2 - 855717λ + 166573528 = 0

Using the quadratic formula, we get

λ = (855717 ± √(855717^2 - 4(166573528))) / 2

λ ≈ 5269.4073 or λ ≈ 315.5927

The eigenvalue with the larger absolute value is 5269.4073. Since it is positive, we don't need to enter it as a negative number. Rounding to 4 decimal places, we get

5269.4073 ≈ 5269.407

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--The given question is incomplete, the complete question is given

" When verifying the stability of the potential coexistence points, you calculated the eigenvalues for each requested point. For x = 8.47*10-8 and the point (30568, 386008), choose the eigenvalue with the larger absolute value. Here, f(x,y) = 9x^2 + 3x + y - 30 = 0 and g(x,y) = 3x^2 + xy - 10^6 = 0What is the value of this eigenvalue, entering it as a negative number if it is negative? Round your answer to 4 decimal places. Your Answer:"--

The different combinations of fudge that Silas can purchase are:

8 pounds of peanut butter fudge and 0 pounds of chocolate fudge

6 pounds of peanut butter fudge and 4 pounds of chocolate fudge

4 pounds of peanut butter fudge and 8 pounds of chocolate fudge

2 pounds of peanut butter fudge and 12 pounds of chocolate fudge

0 pounds of peanut butter fudge and 16 pounds of chocolate fudge

Chocolate (x)  Peanut Butter (y)         Cost

0                              8                                  $24

4                              6                                  $24

8                              4                                  $24

12                              2                                  $24

16                               0                                  $24

We can see that there are five different combinations of fudge that Silas can purchase if he only buys whole pounds of fudge:

8 pounds of peanut butter fudge and 0 pounds of chocolate fudge

6 pounds of peanut butter fudge and 4 pounds of chocolate fudge

4 pounds of peanut butter fudge and 8 pounds of chocolate fudge

2 pounds of peanut butter fudge and 12 pounds of chocolate fudge

0 pounds of peanut butter fudge and 16 pounds of chocolate fudge

We can also verify that the cost of each combination is $24.

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$400 will yield an interest of $78 in 1/2 years at the rate of 39%  per annum. We can use the formula for simple interest to calculate the rate.

Simple interest is calculated based on the initial amount (principal) and time period, without considering any additional interest on the accumulated interest.

Using the formula for simple interest:

Given:

Principal amount (P) = $400

Simple interest (I) = $78

Time (T) = 1/2 years

To calculate the rate (R):

Simple Interest (I) = (Principal amount (P) × Rate (R) × Time (T)) / 100

Plugging in the given values:

$78 = ($400 × R × 1/2) / 100

Multiplying both sides by 100 to get rid of the fraction:

$78 * 100 = $400 * R * 1/2

7800 = $200 * R

Dividing both sides by $200 to isolate R:

R = 7800 / 200

R = 39

Thus, the rate of interest per annum is 39%.

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Answer:

1) To find the $y-$intercept, we set $x=0$ in the equation:

\begin{align*}

y &= x^{2}-6x-16 \\

y &= 0^{2}-6(0)-16 \\

y &= -16

\end{align*}

Therefore, the $y$-intercept is $(0,-16)$.

2) To find the $x$-intercepts, we set $y=0$ in the equation and solve for $x$:

\begin{align*}

y &= (3x+2)(x-5) \\

0 &= (3x+2)(x-5) \\

\end{align*}

Using the zero product property, we have:

\begin{align*}

3x+2 &= 0 \quad \text{or} \quad x-5=0 \\

x &= -\frac{2}{3} \quad \text{or} \quad x=5\\

\end{align*}

Therefore, the $x$-intercepts are $(-\frac{2}{3},0)$ and $(5,0)$.

3) If a quadratic function written in standard form $y=a x^{2}+bx+c$ has a negative $a$ parameter, then the parabola opens downwards.

4) To find the $x$-intercepts, we set $y=0$ in the equation and solve for $x$:

\begin{align*}

y &= x^{2}+4x-21 \\

0 &= x^{2}+4x-21 \\

\end{align*}

Using factoring or the quadratic formula, we get:

\begin{align*}

(x+7)(x-3) &= 0 \\

x &= -7 \quad \text{or} \quad x=3 \\

\end{align*}

Therefore, the $x$-intercepts are $(-7,0)$ and $(3,0)$.

To find the $y$-intercept, we set $x=0$ in the equation:

\begin{align*}

y &= 0^{2}+4(0)-21 \\

y &= -21

\end{align*}

Therefore, the $y$-intercept is $(0,-21)$.

Step-by-step explanation:

The answer is in the picture.

The correct answer is: A s is a function of r as the thickness of the paper currency depends on its denomination.

Based on the given information, s is a function of r.

The thickness (s) of any piece of U.S. paper currency is determined by its denomination (r). This means that for a given denomination (r), there is a specific thickness (s) associated with it. However, the reverse may not be true as different denominations of U.S. paper currency can have the same thickness. For example, a $1 bill and a $100 bill may have the same thickness, but they have different denominations.

Therefore, s is a function of r as the thickness of the paper currency depends on its denomination.

Therefore, the correct answer is: A. s is a function of r.

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Answer:

the height is 14m

Step-by-step explanation:

[tex]h=2*\frac{A}{a+b} =2 *\frac{126}{8+10} =14[/tex]

Answer: 21.4%

Step-by-step explanation: Find the decrease in price per share from 2008 to 2016

=$56- $44

=$12 decrease

Divide by the price per share in 2008

=$12/$56

=0.2142

=21.4% decrease

Answer:  sin(2tan^-1(u/6)) = (2u) / [(u² + 36) * √(u² + 36)]

Step-by-step explanation:  We can use the trigonometric identity:

tan(2θ) = (2 tan θ) / (1 - tan² θ)

to rewrite sin(2tan^-1(u/6)) as an algebraic expression.

Step 1: Let θ = tan^-1(u/6). Then we have:

tan θ = u/6

Step 2: Substitute θ into the formula for tan(2θ):

tan(2θ) = (2 tan θ) / (1 - tan² θ)

tan(2 tan^-1(u/6)) = (2 tan(tan^-1(u/6))) / [1 - tan²(tan^-1(u/6))]

tan(2 tan^-1(u/6)) = (2u/6) / [1 - (u/6)²]

tan(2 tan^-1(u/6)) = (u/3) / [(36 - u²) / 36]

Step 3: Simplify the expression by using the Pythagorean identity:

1 + tan² θ = sec² θ

tan² θ = sec² θ - 1

1 - tan² θ = 1 / sec² θ

tan(2 tan^-1(u/6)) = (u/3) / [(36 - u²) / 36]

tan(2 tan^-1(u/6)) = (u/3) * (6 / √(36 - u²))²

tan(2 tan^-1(u/6)) = (u/3) * (36 / (36 - u²))

Step 4: Rewrite the expression in terms of sine.

Recall that:

tan θ = sin θ / cos θ

sin θ = tan θ * cos θ

cos θ = 1 / √(1 + tan² θ)

Using this identity, we can rewrite the expression for tan(2tan^-1(u/6)) as:

sin(2tan^-1(u/6)) = tan(2tan^-1(u/6)) * cos(2tan^-1(u/6))

sin(2tan^-1(u/6)) = [(u/3) * (36 / (36 - u²))] * [1 / √(1 + [(u/6)²])]

simplify to get:

sin(2tan^-1(u/6)) = (2u) / [(u² + 36) * √(u² + 36)]

This is the Cartesian equation of the curve. To identify it, we can simplify it further: x³ + y² = 9y. This is the equation of a limacon with a loop, also known as a cardioid. Therefore, the answer is: Limacon.

Given the polar equation r = 9tan(θ)sec(θ), we can find the Cartesian equation for the curve by using the relationships x = rcos(θ) and y = rsin(θ).

r = 9tan(θ)sec(θ)
x = rcos(θ) = 9tan(θ)sec(θ)cos(θ)
y = rsin(θ) = 9tan(θ)sec(θ)sin(θ)

Since tan(θ) = sin(θ)/cos(θ) and sec(θ) = 1/cos(θ), we can rewrite the equations as:

x = 9(sin(θ)/cos(θ))(1/cos(θ))cos(θ) = 9sin(θ)
y = 9(sin(θ)/cos(θ))(1/cos(θ))sin(θ) = 9sin²(θ)/cos(θ)

Now we can eliminate θ using the identity sin²(θ) + cos²(θ) = 1:

cos²(θ) = (x/9)²
sin²(θ) = 1 - cos²(θ) = 1 - (x/9)²

Substitute sin²(θ) into the equation for y:

y = 9(1 - (x/9)²)/cos(θ) = 9 - x²

The Cartesian equation for the curve is y = 9 - x², which is a parabola.

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The first and second representation shows y as a function of x.

Explain function

An equation in mathematics is a statement that shows the equality between two expressions. It comprises one or more variables and can be solved to determine the value(s) of the variable(s) that satisfy the equation. Equations are widely used to represent relationships between quantities and solve problems in many fields, such as physics, engineering, and finance.

According to the given information

The first and second representation shows y as a function of x.

In the table and graph, for each value of x, there is only one corresponding value of y. This means that there is a well-defined rule that maps each x-value to a unique y-value, which is the definition of a function. Therefore, y is a function of x in this representation.

The other two representations are not functions of x.

In the equation x² + y² = 144, for each value of x, there are two possible values of y that satisfy the equation (one positive and one negative). Therefore, y is not a function of x in this representation.

In the set {(0, 4), (1, 6), (2, 8), (0, 9)}, there are two points with x-coordinate 0, which means that there are multiple y-values associated with the same x-value. Therefore, this set does not represent y as a function of x

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The equivalent expression to show the number of red and white balloons in each bunch is 8(2 + 4)

Number of red balloons = 16

Number of white balloons = 32

Number of bunches of balloons = 8

Red balloons in each bunch = 16/8

= 2

White balloons in each bunch = 32/8

= 4

Where,

a = the number of bunches of balloons.

b = number of red balloons in each bunch

c = number of white balloons in each bunch

Equivalent expression in the form a(b + c)

So therefore, the equivalent expression can be written as;

8(2 + 4)

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