Mariya is solving the quadratic equation by completing the square.

4x2-20x+3=0

4x2-20x=-3

A(x2-5x)=-3

What is the value of A?

The area of the region enclosed by one loop of the curve r = 3cos(7θ) can be found by calculating the definite integral of the function r with respect to θ over the appropriate interval. The result of this integral will give us the area of the region enclosed by one loop of the curve.

To find the area of the region enclosed by one loop of the curve, we need to evaluate the definite integral of the function r = 3cos(7θ) with respect to θ. Since the curve completes one loop for every interval of θ from 0 to π/7, we can set up the integral as follows:

A = ∫[0, π/7] (1/2) r^2 dθ

Using the given function r = 3cos(7θ), we substitute it into the integral and simplify:

A = ∫[0, π/7] (1/2) (3cos(7θ))^2 dθ

  = (9/2) ∫[0, π/7] cos^2(7θ) dθ

To evaluate this integral, we can use trigonometric identities or integration techniques such as u-substitution. After integrating, we obtain the value of A, which represents the area of the region enclosed by one loop of the curve.

It's important to note that the given curve r = 3cos(7θ) forms multiple loops, and to find the area of one loop specifically, we need to consider the appropriate interval for θ that corresponds to one complete loop.

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Answer:

y=-5cos(pi/8(x+2))-21

Step-by-step explanation:

y=acos(b(x-c))+d

Let's find the max and then the min first. We are aiming to find the amplitude, a, which is the half the distance from the max to the min.

Max is y=-16

Min is y=-26

a=(-16--26)/2=-8--13=5

a could also tell us if y=cos(x) has been reflected about the x-axis if it is -5, which it looks like it has considering cos(0)=1 not -1.

I'm looking at the point (-2,-26) which is the lowest point of our curve and comparing it to our high point of (0,1) on y=cos(x).

So far we have, y=-5cos(b(x-c))+d.

Let's look at d, vertical shift. The average of our max and min is (-26+-16)/2=-13-8=-21.

So our vertical shift is d=-21.

So now equation is y=-5cos(b(x-c))+-21.

Now since we are using (-2,-26) as our (0,1) point on y=cos(x) then c=-2.

So now equation is y=-5cos(b(x--2))+-21

Now we are going to plug in a point haven't mentioned to find b. So looking at graph I will use (6,-16).

-16=-5cos(b(6--2))+-21

Add 21 on both sides:

5=-5cos(b(6--2))

Divide both sides by -5:

-1=cos(b(6--2))

Simplify inside:

-1=cos(b8)

Multiplication is commutative:

-1=cos(8b)

8b=pi * cos(pi)=-1

b=pi/8

The equation is

y=-5cos(pi/8(x--2))+-21

Simplify:

y=-5cos(pi/8(x+2))-21

a) We cannot definitively say that exactly 38% of Americans approved the job of the president. b) The 95% confidence interval is (0.312, 0.448). c) The 95% confidence interval is (0.067, 0.123). d) The 95% confidence interval is (0.026, 0.050). e) At least 1459 people should be interviewed.

(a) Based on the poll result, we cannot definitively say that exactly 38% of Americans approved the job of the president. The poll only surveyed a sample of 150 randomly selected adults, and the result showed that 57 of them approved. This provides an estimate of the approval rate, but it may not represent the entire population of Americans. To make a definitive statement about the approval rate of Americans, a larger and more representative sample would be needed.

(b) To determine the 95% confidence interval for the approval rate of the president's job in the United States, we can use the formula for a confidence interval for a proportion. The formula is:

CI = p ± z * √((p(1 - p)) / n)

where p is the sample proportion (57/150), z is the critical value for a 95% confidence level (which corresponds to approximately 1.96), and n is the sample size (150).

Substituting the values into the formula:

CI = (57/150) ± 1.96 * √(((57/150)(1 - 57/150)) / 150)

Simplifying the expression:

CI ≈ 0.38 ± 0.068

Therefore, the 95% confidence interval for the approval rate of the president's job in the United States is approximately 0.312 to 0.448.

(c) Using the same formula as in part (b), but with a sample size of 400 and a sample proportion of 38/400, we can calculate the 95% confidence interval:

CI = (38/400) ± 1.96 * √(((38/400)(1 - 38/400)) / 400)

CI ≈ 0.095 ± 0.028

The 95% confidence interval for the approval rate is approximately 0.067 to 0.123.

(d) Similarly, with a sample size of 1000 and a sample proportion of 38/1000, the 95% confidence interval is:

CI = (38/1000) ± 1.96 * √(((38/1000)(1 - 38/1000)) / 1000)

CI ≈ 0.038 ± 0.012

The 95% confidence interval for the approval rate is approximately 0.026 to 0.050.

(e) To determine the sample size required to reduce the size of the 95% confidence interval to 4% or less (±2%), we can rearrange the formula for the confidence interval as follows:

n = (\(z^2\) * p(1 - p)) / \(E^2\)

where n is the required sample size, z is the critical value for a 95% confidence level (1.96), p is the estimated proportion (0.38), and E is the desired margin of error (0.02).

Plugging in the values:

n = (\(1.96^2\) * 0.38(1 - 0.38)) / \(0.02^2\)

Simplifying the expression:

n ≈ 1458.88

Therefore, to achieve a 95% confidence interval with a margin of error of 4% or less, at least 1459 people should be interviewed.

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Answer:

MP= (-6,-1)

Step-by-step explanation:

midpoint formula: \((\frac{x^{1}+x^{2} }{2})(\frac{y^{1} + y^{2} }{2} )=MP\)

(-7,-7) (\(x^{1}\),\(y^{1}\))

(-5,5)  (\(x^{2}\),\(y^{2}\))

\((\frac{-7+-5}{2}) \\ (\frac{-7+5}{2})\)= -7+-5/2= -12/2= -6

-7+5/2= -2/2= -1

Answer:

$18 less than the cost of 5 hot dog combos

Step-by-step explanation:

First, we see from the minus sign that this will be subtraction.  PEMDAS (order of operations) says we do the multiplication first, then the subtraction.

Answer:

The answer is 20

Step-by-step explanation:

This is because 4(5)= 20

Hope this helps :)

Answer:

20

Step-by-step explanation:

If a number is next to x, that means multiply

4x

x=5

4*5=20

Answer:

w^2+9w+20

Step-by-step explanation:

Apply the FOIL METHOD: (a + b) (c + d) = ac + ad + bc + bd

a = 5        

b = w        

c = w            

d = 4

= 5w + 5 x 4 + ww + w x 4

= 5w + 5 x 4 + ww + 4w

Simplify 5w + 5 x 4 + ww + 4w:

w^2 + 9w + 20

To find out the intervals for which the given function is positive, we need to analyze the sign of the function in different intervals.

Here is the solution: Let f(x) be the given function.

Then,\(f(x) = (x + 4)(x − 6)(x − 8)\)

We can see that the function f(x) changes its sign when x = −4, 6, and 8.

So, these are the critical points that divide the real line into different intervals.

We can check the sign of f(x) in each interval using the following table:x-∞-4-40668f(x)+-+-+-+

\(:x-∞-4-40668f(x)+-+-+-+\)

Therefore, the intervals for which the function is positive are (−4, 0) and (6, 8).

Hence, the correct options are (−4, 0) and (6, 8).

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\(\textit{area of a sector of a circle}\\\\ A=\cfrac{\theta \pi r^2}{360}~~ \begin{cases} r=radius\\ \theta =\stackrel{degrees}{angle}\\[-0.5em] \hrulefill\\ A=12\\ \theta =75 \end{cases}\implies 12=\cfrac{(75)\pi r^2}{360}\implies 12=\cfrac{5\pi r^2}{24} \\\\\\ 288=5\pi r^2\implies \cfrac{288}{5\pi }=r^2\implies \sqrt{\cfrac{288}{5\pi }}=r\implies 4.3\approx r\)

Answer:

  P(t) = 23400/(1 +e^(-35.1t))

Step-by-step explanation:

You want the solution of the logistic differential equation dP/dt = P(35.1-0.0015P) with initial condition P(0) = 936.

The generic solution to p' = p(a -bp) with p(0) = c is ...

  p = (a/b)/(1 +(a/(bc) -1)e^(-at))

For a = 35.1, b = 0.0015, c = 936, we have the coefficients ...

  a/b = 35.1/0.0015 = 23400

  a/(bc) -1 = 23400/936 -1 = 24

That means the solution equation is ...

  P(t) = 23400/(1 +24e^(-35.1t))

P(t) = e^(35.1t - 0.00075P² + C). The logistic differential equation you provided is dP/dt = P(35.1 - 0.0015P) with an initial condition P(0) = 936. To solve this equation, we will use the method of separation of variables.

First, separate the variables P and t by dividing both sides by P(35.1 - 0.0015P):

(1/P) dP = (35.1 - 0.0015P) dt.

Now, integrate both sides with respect to their respective variables:

∫(1/P) dP = ∫(35.1 - 0.0015P) dt.

The left side is a simple integration, resulting in ln|P| + C₁. The right side requires integration by parts, which yields 35.1t - 0.00075P² + C₂. Combining the constants of integration, we get:

ln|P| = 35.1t - 0.00075P² + C.

Now, exponentiate both sides to remove the natural logarithm:

P = e^(35.1t - 0.00075P² + C).

To find the constant C, use the initial condition P(0) = 936:

936 = e^(35.1(0) - 0.00075(936)² + C).

Solve for C, and then plug it back into the equation:

P(t) = e^(35.1t - 0.00075P² + C).

This equation represents the solution to the given logistic differential equation with the specified initial condition. Note that the actual values for C and the final equation form may require further calculations based on the integration steps.

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Answer:

24,000 letters.

Explanation:

In 60 minutes, 4 postal clerks sort 1,200 letters.

Since 1 clerk sorts 300 letters in 1 hour (i.e. 60 minutes):

Thus, in 8 hours, 10 clerks will sort 24,000 letters.

Alternate Approach

60 minutes = 1 hour

• In 1 hour, 4 postal clerks sort 1200 letters.

• In 1 hour, 10 postal clerks sorts x letters.

Expressing this as a ratio:

Thus, in 1 hour, 10 postal clerks will sort 3,000 letters.

Therefore, the number of letters 10 postal clerks will sort in 10 hours will be:

Thus, in 8 hours, 10 clerks will sort 24,000 letters.

Anwser: 41°

Okay so every triangle is 180° degrees in total.

We need to find the angle of x in the triangle BDC.

We know that the angle of x is 180 - ( x + 15)

We also have another triangle named BAD

Which angles are 90 + 56 + x

the angle of D in the triangle BDC is 180 -58 = 124

So 124 + 15 = 139

180 - 139 = 41

Answer:

y-intercept: 2

x-intercept: 8

Step-by-step explanation:

I just graphed it into desmos. Wherever it crossed the axis, it's the answer.

Answer:

C.    16.9

Step-by-step explanation:

Answer:

C.

Step-by-step explanation:

Answer: Her brother is 5 years old.

Step-by-step explanation:

s = 2b

4b + 3b = 35

7b = 35

First, I used s to represent Sarah's age. We are told that she is twice her brother, I used b to represent her brother's age. The sum of twice her age is 4 times her brother's age (2b + 2b) and 3 times her brother's age is 3b.

Answer:

1 feet

Step-by-step explanation:

15-14

We can determine coordinates if point t is between points u and v by checking if u < t < v or v < t < u.

To determine if point t is between points u and v, we need to compare their coordinates. If u < v, then point t is between points u and v if and only if u < t < v. On the other hand, if v < u, then point t is between points u and v if and only if v < t < u.

Whether or not point t is between points u and v depends on the relationship between the coordinates of u and v. If u < v, t must fall between them, and if v < u, t must also fall between them.

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The solution to the system of equations is (1.6,-1.6).

What is the system of equations?

Two or more equations in algebra must be solved jointly (i.e., the solution must satisfy all the equations in the system). The number of equations must match the number of unknowns for a system to have a singular solution.

Here, we have

Given equations:

y = 1.5x - 4....(1)

y = -x....(2)

Substitute y = -x in the first equation

-x = 1.5x - 4

Collect like terms

-1.5x - x = -4

This gives

-2.5x = -4

Divide both sides by -2.5

x = 1.6

Substitute x = 1.6 in y = -x

y = -1.6

Hence, the solution to the system of equations is (1.6,-1.6)

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So the change of rate of x is -1/3 .

The Derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a fundamental tool of calculus.

Given that object is decreasing at constant rate of 2 units per second so we have \(y = 2x^2 + 1\)

\(\frac{dy}{dt} = 4x \frac{dx}{dt}\)

given dy/dt = -2

when x = 3/2

-2 = 4(3/2) dx/dt

-2 = 6 dx/dt

dx/dt = -2/6

         = -1/3

Hence the change of rate of x is -1/3 .

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Step-by-step explanation:

\( \frac{ {4}^{7} }{ {4}^{6} } \)

= 4

.......

Answer:

A) -2 - i√3 , -2 + i√3

Step-by-step explanation:

Solve using quadratic formula

x² + 4x + 7 = 0

The Almighty Formula

= -b ± √b² - 4ac/2a

Where ax + bx² + c = 0

From the above question

a = 1, b = 4, c= 7

Hence,

-4 ± √4² - 4 × 1 × 7/2 × 1

-4 ± √16 - 28/2

=( -4 ± √-12)/2

Since

b² - 4ac < 0

We have two complex roots

Simplifying

( -4 ± √-12)/2

= -4/2 ± √-12/2

= -2 ± 2√3i/2

= -2 ± √3i

Therefore,

-2 - √3i , -2 + √3i

or

-2 - i√3 , -2 + i√3

Option A , is the correct answer

Step-by-step explanation:

ans;

6x=34-4

6x=30

x=30/6

:x=5#

Answer:

DABC

Step-by-step explanation:

Answer:

Andrew is driving at a faster speed and they both started at the same place.

The mean of the distribution of sample means is 0.45.

The mean of the distribution of sample means for a population with parameter p = 0.45 and sample size n = 671 is calculated using the formula: mean = p.

In this case, the mean of the distribution is simply equal to the population parameter p:

mean = 0.45

Rounded to 2 decimal places, the mean of the distribution of sample means is 0.45.

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Answer:

20 sweatshirts and 5 hats

Step-by-step explanation:

So, let's assign our variables.

x = sweatshirts

y = hats

Now, we can set up our system of equations.

x + y = 25

8x + 2.5y = 172.5

In this instance, you can use a variety of methods. I will use substitution.

x + y = 25

x = 25 - y

8(25 - y) + 2.5y = 172.5

200 - 8y + 2.5y = 172.5

200 - 5.5y = 172.5

-5.5y = -27.5

y = 5.5

Now, we can plug in y to solve for x.

8x + 2.5(5.5) = 172.5

8x + 13.75 = 172.5

8x = 158.75

x = 19.84

Now, this was very sloppy work, and now we must make sense of these answers as pertaining to the context of the equation. First, let's try 20 and 5.

20 + 5 = 25

8(20) + 2.5(5) = 172.5 It works!

Answer:

Step-by-step explanation:

Here, we want to graph the given equations

1. y = 2x - 3

The general form of the equation of a straight line is;

y = mx + b

m is the slope and b is the y-intercept

To plot the line, we need the x intercept and the y-intercept

At the x-intercept, the value of y is zero

at the y-intercept . the value of x is zero

From the equation, we have the y-intercept as -3

To get the x-intercept, set y = 0

0 = 2x - 3

2x = 3

x = 3/2

x = 1.5

so to graph the line, we have to join the points ;

(0,-3) and (1.5,0)

b) y = -3x + 2

The y-intercept is 2

To get the x intercept, set y = 0

0 = -3x + 2

3x = 2

x = 2/3

so, we have to join the points (2/3,0) and (0,2)

We have the lines plotted as an attachment

Answer:

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

After 3 half-lives the remaining amount becomes:

Convert 1/8 to percent:

We can use the trigonometric identity sin^2(x) = (1 - cos(2x))/2 and simplify sin^4(x) as (sin^2(x))^2 = [(1 - cos(2x))/2]^2.

So, the integral becomes:

∫9sin^4(x)cos(x) dx = ∫9[(1-cos(2x))/2]^2cos(x) dx

Expanding the square and distributing the 9, we get:

= (9/4) ∫[1 - 2cos(2x) + cos^2(2x)]cos(x) dx

Now, we can simplify cos^2(2x) as (1 + cos(4x))/2:

= (9/4) ∫[1 - 2cos(2x) + (1 + cos(4x))/2]cos(x) dx

= (9/4) ∫(cos(x) - 2cos(x)cos(2x) + (1/2)cos(x) + (1/2)cos(x)cos(4x)) dx

Integrating term by term, we get:

= (9/4) [sin(x) - sin(2x) + (1/2)sin(x) + (1/8)sin(4x)] + C

where C is the constant of integration.

Therefore,

∫9sin^4(x)cos(x) dx = (9/4) [sin(x) - sin(2x) + (1/2)sin(x) + (1/8)sin(4x)] + C.

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