Find f. (Use C for the constant of the first antiderivative and D for the constant of the second antiderivative.) f ''(x) = 24x3 − 15x2 + 8x

Answer:

[tex] {x}^{2} - 36 = 5x[/tex]

[tex] {x}^{2} - 5x - 36 = 0[/tex]

[tex](x - 9)(x + 4) = 0[/tex]

[tex]x = 9[/tex]

The constant c in the joint PDF of two random variables X, Y given by fX,Y(x,y) = c, where 0≤x≤y≤2, is 1/2.

Consider the joint PDF of two random variables X, Y given by fX,Y(x,y) = c, where 0≤x≤y≤2. To find the constant c, we need to make sure that the joint PDF integrates to 1 over the given region.

Here's a step-by-step explanation:

1. Set up the double integral for fX,Y(x,y):
  ∬fX,Y(x,y) dx dy = ∬c dx dy

2. Determine the integration limits:
  Since 0≤x≤y≤2, we have:
  - x goes from 0 to y (inner integral)
  - y goes from 0 to 2 (outer integral)

3. Set up the double integral with the correct limits:
  ∬c dx dy = ∫(from 0 to 2) ∫(from 0 to y) c dx dy

4. Perform the integration:
  First, integrate with respect to x:
  ∫(from 0 to 2) [cx] (from 0 to y) dy = ∫(from 0 to 2) cy dy

  Next, integrate with respect to y:
  [c/2 * y^2] (from 0 to 2) = c * (4/2) = 2c

5. Equate the double integral to 1 and solve for c:
  2c = 1 => c = 1/2

Your answer: The constant c in the joint PDF of two random variables X, Y given by fX,Y(x,y) = c, where 0≤x≤y≤2, is 1/2.

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The size of the files on Bobby’s drive (in gigabytes) as a function of time (in seconds) is given by y=0.4x+6

Given that Bobby was transferring files from his computer onto a flash drive.

The flash drive already had 6 gigabytes of data before the transfer

an additional 0.4 gigabyte were transferred onto the drive each second.

Let x be the number of seconds

y be the size of the files on Bobby’s drive (in gigabytes) as a function of time

y=0.4x+6

Hence, the graph of y=0.4x+6 is given in the attachment

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To find the PDF of Z = X + Y when X and Y have the given joint PDF, f(x,y) = 1/900 for 0≤x,y≤3, and 0 otherwise.

Step 1: Identify the range of Z. Since X and Y range from 0 to 3, the minimum value for Z is 0 (when X = 0 and Y = 0) and the maximum value for Z is 6 (when X = 3 and Y = 3).

Step 2: Find the marginal PDFs of X and Y. Since X and Y are uniformly distributed, we have f_X(x) = 1/3 for 0≤x≤3 and f_Y(y) = 1/3 for 0≤y≤3.

Step 3: Compute the convolution of the marginal PDFs.

To find the PDF of Z = X + Y, we need to compute the convolution of f_X(x) and f_Y(y): f_Z(z) = ∫ f_X(x) * f_Y(z-x) dx

Now, let's compute the convolution for different ranges of Z:

a) 0≤z≤3: f_Z(z) = ∫(1/3)(1/3) dx from x=0 to x=z f_Z(z) = (1/9)[x] from 0 to z f_Z(z) = z/9

b) 3

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If h(x) = 6 5f(x) , where f(5) = 6 and f '(5) = 4, then h'(5). h'(5) =20

To find h'(5) given h(x) = 6 + 5f(x), f(5) = 6, and f'(5) = 4, follow these steps:

1. Differentiate h(x) with respect to x: h'(x) = 0 + 5f'(x) (since the derivative of a constant is 0, and we use the chain rule for the second term).

2. Now, h'(x) = 5f'(x).

3. Plug in the given values: h'(5) = 5f'(5).

4. Since f'(5) = 4, substitute this value: h'(5) = 5 * 4.

5. Compute the result: h'(5) = 20.

So, h'(5) = 20.

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a random variable x, the number of successes, follows a Poisson process, then the probability of success for any two intervals of the same size.

A) is the same

The correct answer is A) is the same.

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence. Using it, we can only make predictions about the likelihood of an event happening, or how likely it is. The probability can be between 0 and 1.

In a Poisson process, the probability of success within a certain time interval is determined only by the length of the interval and the rate of success. Therefore, any two intervals of the same size will have the same probability of success, regardless of when the intervals occur.

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The radius of the circle is [tex]\frac{4\sqrt3}{3}[/tex] cm.

What is triangle?

A triangle is a form of polygon with three sides; the intersection of the two longest sides is known as the triangle's vertex. There is an angle created between two sides. One of the crucial elements of geometry is this.

Here consider the equilateral triangle XYZ.

Then XY=YZ=XZ = 8 cm.

Now using formula then,

Radius of the circle = [tex]\frac{a}{2\sqrt3}[/tex]

Where a = side length of triangle.

Then,

Radius = [tex]\frac{8}{2\sqrt3}=\frac{4}{\sqrt3}=\frac{4\sqrt3}{3}[/tex]

Hence the radius of the circle is [tex]\frac{4\sqrt3}{3}[/tex] cm.

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Answer:

3400

Step-by-step explanation:

3.4 x 1000 =3400

to go from km a big unit to meters a smaller unit, you multiply by 1000

The value of the population standard deviation necessary to produce a standard error of 3 points is 18 points. The value of the population standard deviation necessary to produce a standard error of 12 points is 72 points. The value of the population standard deviation necessary to produce a standard error of 2 points is 12 points.

To calculate the value of the population standard deviation (σ) necessary to produce each of the following standard error values for a sample of n = 36 scores, we can use the formula:

σM = σ / √n
where σM is the standard error of the mean, σ is the population standard deviation, and n is the sample size.

1. If σM = 12 points, then:
12 = σ / √36
12 = σ / 6
σ = 12 x 6
σ = 72 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 12 points is 72 points.

2. If σM = 3 points, then:
3 = σ / √36
3 = σ / 6
σ = 3 x 6
σ = 18 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 3 points is 18 points.

3. If σM = 2 points, then:
2 = σ / √36
2 = σ / 6
σ = 2 x 6
σ = 12 points

Therefore, the value of the population standard deviation necessary to produce a standard error of 2 points is 12 points.

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The parabola with equation y=ax^2+bx whose tangent line at (1, 1) has equation y=4x-3; y = 3x^2 - 2x

The tangent line at (1, 1) has equation y = 4x - 3, which means that the slope of the tangent line at that point is 4. We know that the derivative of y = ax^2 + bx is y' = 2ax + b, which gives us the slope of the tangent line at any point on the parabola. So, we can set 2ax + b equal to 4 (the slope of the tangent line) and substitute x = 1 and y = 1 (the point on the tangent line and parabola, respectively).
2a(1) + b = 4
a(1)^2 + b(1) = 1
Simplifying the second equation, we get b = 1 - a. Substituting this into the first equation and simplifying, we get:
2a + 1 - a = 4
a = 3
Therefore, b = 1 - a = -2. The equation of the parabola is y = 3x^2 - 2.
To find the parabola with equation y = ax^2 + bx whose tangent line at (1, 1) has the equation y = 4x - 3, we will first determine the values of a and b.
Since the tangent line touches the parabola at (1, 1), we can substitute these values into both the parabola and tangent line equations:
1 = a(1)^2 + b(1) (Parabola equation)
1 = 4(1) - 3 (Tangent line equation)
From the tangent line equation, we see that it is already satisfied. Now we need to find the derivative of the parabola equation with respect to x to find the slope of the tangent line:
dy/dx = 2ax + b
At the point (1, 1), the slope of the tangent line is equal to the slope of the parabola:
4 = 2a(1) + b
We already know from the parabola equation that:
1 = a + b
Now, we have a system of linear equations:
4 = 2a + b
1 = a + b
Solving the system, we find that a = 3 and b = -2. Therefore, the equation of the parabola is:
y = 3x^2 - 2x

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Using a trigonometric relation we can see that the value of x is 3,549.3 ft

We can see that we have a right triangle, where x is the hypotenuse.

We know one angle of the triangle and the opposite cathetus of said angle.

Then we need to use the trigonometric relation:

sin(a) = (opposite cathetus)/hypotenuse.

Replacing the things that we know we will get.

sin(25°) = 1500ft/x

Solving that for x we will get:

x = 1500ft/sin(25°)

x = 3,549.3 ft

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The value of the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3 is [tex]27e^{12} - 36.[/tex]

What is integration?

Integration is a fundamental concept in calculus that involves finding the integral of a function.

To evaluate the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3, we integrate with respect to z, y, and then x.

∭bzex ydv = [tex]\int\limits^3_0 \int\limits^3_0\int\limits^4_0 bzex y\ dx\ dy\ dz[/tex]

Integrating with respect to x, we get:

[tex]\int\limits^4_0 bzex\ y\ dx\ = bzex\ y\ |^4_0 = bze 4^y - bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv = [tex]\int\limits^3_0 \int\limits^3_0(bze 4^y - bz) dy dz[/tex]

Integrating with respect to y, we get:

[tex]\int\limits^3_0 (bze4^y - bz) dy = (1/4)bze4^y - bzy|_0^3 = (1/4)bz(e^{12} - 1) - 3bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv = [tex](1/4)bz(e^{12} - 1) - 3bz) \int\limits^3_0 dz[/tex]

Integrating with respect to z, we get:

[tex]\int\limits^3_0 (1/4)bz(e^{12} - 1) - 3bz) dz = (9/4)bz(e ^{12} - 1) - 9bz[/tex]

Substituting this result into the triple integral, we get:

∭bzex ydv =[tex](9/4)bz(e^{12} - 1) - 9bz)[/tex]

Now, substituting the limits of integration, we get:

∭bzex ydv = [tex](9/4)(4)(e_{-1} ^{12} - 1) - 9(4) = 27e^{12} - 36[/tex]

Therefore, the value of the triple integral ∭bzex ydv over the box b determined by 0≤x≤4, 0≤y≤3, and 0≤z≤3 is [tex]27e^{12} - 36.[/tex]

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The probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days is 0.0228.

Since the sample size n = 100 is sufficient, we can apply the central limit theorem to roughly approximate the distribution of the sample mean.

Let X represent the total paid time a single blue-collar worker missed during a three-month period. Given that the population mean is 1.3 days and the population standard deviation is 1.0 days, X N(1.3, 1.02) follows.

Let Y be the sample mean of X for a sample of 100 blue-collar workers selected at random. So, according to the central limit theorem, Y = N(1.3, 1.02/100).

We are looking for P(Y > 1.5). By standardized Y, we obtain:

Z is defined as (Y - ) / (n /√(n)) = (1.5 - 1.3) / (1.0 / √(100)). = 2

The probability of the event that average amount of the paid time loss is 0.0288.

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Complete question - In an article in the Journal of Management, Joseph Martocchio studied and estimated the costs of employee absences. Based on a sample of 176 blue-collar workers, Martocchio estimated that the mean amount of paid time lost during a three-month period was 1.3 days per employee with a standard deviation of 1.0 days. Martocchio also estimated that the mean amount of unpaid time lost during a three-monthperiod was 1.4 day per employee with a standard deviation of 1.2 days.

Suppose we randomly select a sample of 100 blue-collar workers. Based on Martocchio's estimates:

(a)What is the probability that the average amount of paid time lost during a three-month period for the 100 blue-collar workers will exceed 1.5 days?

It is true that A biased estimate of an odds ratio can exist even if this estimate is very precise.

A biased estimate of an odds ratio can exist even if this estimate is very precise. Bias refers to a systematic error in the estimation process, which can affect the accuracy of the estimate even if it is based on a large sample size or other factors that might increase precision. Therefore, it is important to identify and address sources of bias in the estimation of odds ratios to ensure that the estimates are as accurate as possible.

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Doing some changes of units, we can see that the total volume is V = 1.5 gal

We know that the recipe that Janelle follows is the following one:

So we need to do some changes of units, we know that:

1 pint = 0.125 gal

Then:

5 pints = 5*(0.125 gal) = 0.625 gal

Then for the orange juice we have:

1 cup = 0.0625 gal

Then for the 14 cups of apple and grape juice we have:

14*(0.0625 gal) = 0.875 gal

Adding that we have the total volume:

0.625 gal + 0.875 gal = 1.5 gal

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The image of triangle EFG after rotation 90 degrees countercloeckwise is shown below.

We know that when we rotate a point P(x, y) 90 degrees counterclockwise about the origin then the coordinates of point after rotation becomes (-y, x)

Here the coordinates of the triangle EFG are:

E(4, -8)

F(4, -1)

G(3, -9)

We need to rotate triangle EFG 90 degrees counterclockwise.

With the help of above statement the coordinates of rotated triangle would be,

E'(8, 4)

F'(1, 4)

G'(9,3)

The transformed triangle is shown below.

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The linear equation y = -3x - 3, have a slope of -3 while the y intercept is -3.

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The standard form of a linear equation is:

y = mx + b

Where m is the rate of change (slope); and b is the y intercept

Given the equation:

y = -3x - 3

The slope of the graph is -3 while the y intercept is -3.

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The correct choice that completes the sentence is: Besides vertical asymptotes, the zeros of the denominator of a rational function give rise to holes oblique so hat , the correct choice is b

Explanation:-

A rational function is a fraction with polynomials in the numerator and denominator. The denominator of a rational function cannot be zero because division by zero is undefined. Therefore, the zeros of the denominator are important in analyzing the behaviour of a rational function.

When a rational function has a zero in the denominator, it creates a vertical asymptote. The function becomes unbounded as it approaches the vertical asymptote from both sides. However, if the numerator has a zero at the same point where the denominator has a zero, the function may have a hole in the graph instead of a vertical asymptote.

In addition to vertical asymptotes and holes, the zeros of the denominator of a rational function can also give rise to oblique asymptotes. An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this case, the function approaches a slanted line as x goes to infinity or negative infinity.

Therefore, analyzing the zeros of the denominator of a rational function is crucial in understanding its behaviour and graph. It can help identify vertical asymptotes, holes, and oblique asymptotes, providing valuable insights into the function's behavior.

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Answer:

vertex = (- 4, - 2 )

Step-by-step explanation:

given a parabola in standard form

f(x) = ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the vertex is

[tex]x_{vertex}[/tex] = - [tex]\frac{b}{2a}[/tex]

f(x) = 2x² + 16x + 30 ← is in standard form

with a = 2 , b = 16 , then

[tex]x_{vertex}[/tex] = - [tex]\frac{16}{2(2)}[/tex] = - [tex]\frac{16}{4}[/tex] = - 4

substitute x = - 4 into f(x) for corresponding y- coordinate

f(- 4) = 2(- 4)² + 16(- 4) + 30

      = 2(16) - 64 + 30

      = 32 - 34

     = - 2

vertex = (- 4, - 2 ) or x = - 4 , y = - 2

Answer:

Step-by-step explanation:

The y-intercept of the line of fit is 41.16, which means that when the machine is first filled, it starts with approximately 41.16 fluid ounces of diet soda.

In the context of the data, the y-intercept represents the initial amount of diet soda in the machine before any soda is purchased. This information can be useful for determining how much soda is being purchased by customers over time, as it provides a baseline for comparison.

Therefore, the correct answer is: The y-intercept is 41.16. The machine starts with 41.16 ounces of diet soda.

Answer:

y-intercept is 41.16. The machine starts with 41.16 ounces of diet soda.

Step-by-step explanation:

The method of moments estimate for [tex]\lambda[/tex] is 1 / sample mean.

To find the method of moments estimate for [tex]\lambda[/tex], we equate the sample mean to the theoretical mean.

To find the method of moments estimate for lambda, we first calculate the first moment (or mean) of the distribution. For the exponential distribution, the mean is equal to 1/[tex]\lambda[/tex].

The theoretical mean of an exponential distribution is given by:

E(Y) = 1/[tex]\lambda[/tex]

The sample mean is the sum of the observations divided by the sample size:

y-bar = (1/n) * (y1 + y2 + ... + yn)

Next, we equate the sample mean to the theoretical mean (calculated from the first moment) and solve for [tex]\lambda[/tex].

Setting these two equal, we get:

1/[tex]\lambda[/tex] = y-bar

Solving for lambda, we get:

[tex]\lambda[/tex] = 1/y-bar

Therefore, the method of moments estimate for [tex]\lambda[/tex] is 1 divided by the sample mean.

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To find the equation of the tangent plane to the given surface at the specified point.

Given surface: z = 6(x - 1)^2 + 6(y + 3)^2

Specified point: (2, -2, 14)

Step 1: Find the partial derivatives with respect to x and y. ∂z/∂x = 12(x - 1) ∂z/∂y = 12(y + 3)

Step 2: Evaluate the partial derivatives at the specified point (2, -2, 14). ∂z/∂x|_(2,-2,14) = 12(2 - 1) = 12 ∂z/∂y|_(2,-2,14) = 12(-2 + 3) = 12

Step 3: Use the tangent plane equation: z - z₀ = ∂z/∂x(x - x₀) + ∂z/∂y(y - y₀), where (x₀, y₀, z₀) is the specified point. z - 14 = 12(x - 2) + 12(y + 2)

Step 4: Simplify the equation. z - 14 = 12x - 24 + 12y + 24 z = 12x + 12y + 14

So, the equation of the tangent plane to the given surface at the specified point (2, -2, 14) is z = 12x + 12y + 14.

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Answer:

answer is (-2,5)

Step-by-step explanation:

gonna make 2 by 5 into Percent

x=0.4y-4

3x-9y=-51

step 2 Add 3 into first column and make it negative

-3x=-1.2y+12

then move y to other side

-3x+1.2y=12

3x-9y=-51

-7.8y=-39

y=5

U got y now

add it into y in first column

x= 2/5(5)-4

then it be 2-4= -2

x=-2

+) Replace x = (2/5)y - 4 into 3x - 9y = -51

[tex] 3 \times ( \frac{2}{5} y - 4) - 9y = - 51 \\ [/tex]

[tex] \frac{6}{5} y - 12 - 9y = - 51[/tex]

[tex]\frac{6}{5} y - 9y = - 51 + 12 = - 39[/tex]

[tex] \frac{ - 39}{5} y = - 39[/tex]

[tex]y = (- 39) \div \frac{( - 39)}{5} = ( - 39) \times \frac{5}{( - 39)} [/tex]

[tex]y = 5[/tex]

[tex]x = \frac{2}{5} y - 4 = \frac{2}{5} \times 5 - 4 = 2 - 4 [/tex]

[tex]x = - 2[/tex]

Ans: (x;y) = (-2;5)

Ok done. Thank to me >:33

The closed-form expression for the given summation is (3n^2 + 7n) / (2n^2). Using this formula, the sums for n = 10, 100, 1000, and 10,000 are 37/20, 307/200, 3007/2000, and 30007/20000, respectively.

The given expression can be rewritten using the summation formulas as

∑nj=1 3j+2/n2 = (3(1)+2)/n2 + (3(2)+2)/n2 + ... + (3(n)+2)/n2

Let's simplify this expression by factoring out the common term of 1/n2

= (3/n2)(1 + 2 + ... + n) + (2/n2)(1 + 1 + ... + 1)

= (3/n2)(n(n+1)/2) + (2/n2)(n)

= (3n(n+1) + 4n) / (2n2)

= (3n^2 + 7n) / (2n^2)

Therefore, we have the closed-form expression for S(n) as

S(n) = (3n^2 + 7n) / (2n^2)

Using this formula, we can find the sums for n = 10, 100, 1000, and 10,000

S(10) = (3(10^2) + 7(10)) / (2(10^2)) = 37/20

S(100) = (3(100^2) + 7(100)) / (2(100^2)) = 307/200

S(1000) = (3(1000^2) + 7(1000)) / (2(1000^2)) = 3007/2000

S(10000) = (3(10000^2) + 7(10000)) / (2(10000^2)) = 30007/20000

Therefore, the sums for n = 10, 100, 1000, and 10,000 are 37/20, 307/200, 3007/2000, and 30007/20000, respectively.

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[tex]6 + 36x + 72x^2 + 96x^3[/tex] / 3! + ... this is the Maclaurin series for f(x). Note that since e^x has a well-known Maclaurin series, we were able to simplify the original expression before finding the series.

The problem asks us to find the Maclaurin series for the function:

[tex]f(x) = 6e^x e^4x[/tex] sigma^infinity_n=0 (1^n / n!)

To do this, we first need to recognize that the expression inside the sigma notation is actually the Maclaurin series for e^x:

sigma^infinity_n=0 (1^n / n!) = e^x

We can substitute this expression into the original function to get:

[tex]f(x) = 6e^x e^4x e^x[/tex]

Now we can simplify this expression using the laws of exponents:

[tex]f(x) = 6e^x * e^(4x) * e^x[/tex]

f(x) = 6e^(6x)

Now we need to express this function as a Maclaurin series. We can start by writing out the first few terms of the series:

[tex]f(x) = 6e^(6x)[/tex]

[tex]= 6(1 + 6x + (6x)^2 / 2! + (6x)^3 / 3! + ...)[/tex]

[tex]= 6 + 36x + 72x^2 + 96x^3 / 3! + ...[/tex]

This is the Maclaurin series for f(x). Note that since e^x has a well-known Maclaurin series, we were able to simplify the original expression before finding the series.

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The largest possible value of f(8) is 14.

Since f is continuous on [4,8] and differentiable on (4,8), we can apply the Mean Value Theorem (MVT) on the interval [4,8]. The MVT states that there exists a c in (4,8) such that

f(8) - f(4) = f'(c)(8-4)

or equivalently,

f(8) = f(4) + f'(c)(8-4).

Since f(4) = -6 and f'(x) ≤ 10 for all x in (4,8), we have

f(8) = -6 + f'(c)(8-4) ≤ -6 + 10(8-4) = 14.

Therefore, the largest possible value of f(8) is 14. This maximum value can be achieved by a function that is increasing at the maximum rate of 10 on the interval (4,8) and passes through the point (4,-6).

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The polar form of the equation x = 4 is r = 4 / cos(θ).

To convert the equation x = 4 to polar form:

To convert the equation x = 4 to polar form using variables r and θ (theta),

Follow these steps:

Step 1: Recall the polar to rectangular coordinate conversion formulas:
x = r * cos(θ)
y = r * sin(θ)

Step 2: Replace x in the given equation with the corresponding polar conversion formula:
r * cos(θ) = 4

Step 3: Solve for r:
r = 4 / cos(θ)

So, the polar form of the equation x = 4 is r = 4 / cos(θ).

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Answer:

2

Step-by-step explanation:

(½ × 8 + 6) ÷ 5

Order of operations: BODMAS - (Brackets, Orders, Division, Multiplication, Addition, Subtraction)

So Brackets first, and within the bracket we do mulplication first, then addition.

(4 + 6) ÷ 5

10 ÷ 5

Division is left so obviously

Ans : 2

The area of the composite shape is 40.57 square m and the perimeter is 34.57 meters

The surface area of composite shapes can be found by breaking the composite shape down into simpler shapes and then finding the surface area of each individual shape.

Here, we have

Area = Area of rectangle + circle

So, we have

Area = 4 * 7 + 22/7 * (4/2)^2

Area = 40.57 square m

So, the area is 40.57 square m

For the perimeter, we have

Perimeter = 2 * (4 + 7) + 2 * 22/7 * (4/2)

Perimeter = 34.57

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