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The given equation, X = 36y², represents a parabola. In standard form, the equation can be rewritten as y² = (1/36)x. The vertex (V) is located at the origin (0, 0), the focus (F) is at (0, 1/4), and the directrix (d) is the horizontal line y = -1/4.

To rewrite the equation X = 36y² in standard form, we divide both sides by 36 to get y² = (1/36)x. This form represents a parabola with its vertex at the origin (0, 0).

In standard form, the equation of a parabola can be written as y² = 4px, where p is the distance from the vertex to the focus and also the distance from the vertex to the directrix. In this case, p = 1/4.

Therefore, the vertex (V) is located at (0, 0), the focus (F) is at (0, 1/4), and the directrix (d) is the horizontal line y = -1/4.

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The solution to the Cauchy-Euler initial value problem is -3/2

To solve the Cauchy-Euler initial value problem, we need to find the general solution of the differential equation and then use the initial conditions to determine the specific solution.

The given Cauchy-Euler differential equation is:

y" + xy' + y = 0

To solve this equation, we assume a solution of the form [tex]y(x) = x^r[/tex]

Differentiating twice with respect to x, we have:

[tex]y' = rx^{r-1}[/tex] and y" = [tex]r(r-1)x^{r-2}[/tex]

Substituting these expressions into the differential equation, we get:

[tex]r(r-1)x^{r-2} + x(rx^{r-1}) + x^r = 0[/tex]

[tex]r(r-1)x^{r-2} + r*x^r + x^r = 0[/tex]

[tex]x^{r-2}(r(r-1) + r + 1) = 0[/tex]

For a non-trivial solution, the expression in parentheses must equal zero:

r(r-1) + r + 1 = 0

Expanding and rearranging, we have:

[tex]r^2 - r + r + 1 = 0\\r^2 + 1 = 0[/tex]

The roots of this equation are complex numbers:

r = ±i

Therefore, the general solution of the Cauchy-Euler differential equation is:

[tex]y(x) = c_1x^i + c_2x^{-i}[/tex]

To simplify the solution, we can rewrite it using Euler's formula:

[tex]y(x) = c_1x^i + c_2x^{-i}\\ = c_1(cos(ln(x)) + i*sin(ln(x))) + c_2(cos(ln(x)) - i*sin(ln(x)))\\ = (c_1 + c_2)cos(ln(x)) + (c_1 - c_2)i*sin(ln(x))[/tex]

Now, let's apply the initial conditions to find the specific solution. We are given:

y(t) = 3 and y'(1) = 4

Substituting x = t into the solution, we have:

[tex](c_1 + c_2)cos(ln(t)) + (c_1 - c_2)i*sin(ln(t)) = 3[/tex]

To satisfy this equation, the real parts and imaginary parts on both sides must be equal.

From the real parts:

[tex](c_1 + c_2)cos(ln(t)) = 3[/tex]

From the imaginary parts:

[tex](c_1 - c_2)i*sin(ln(t)) = 0[/tex]

Since sin(ln(t)) ≠ 0 for any t, we must have ([tex]c_1 - c_2[/tex]) = 0.

This implies [tex]c_1 = c_2[/tex].

Substituting [tex]c_1 = c_2[/tex] into the real part equation, we get:

[tex]2c_1cos(ln(t)) = 3[/tex]

Solving for [tex]c_1[/tex], we find:

[tex]c_1 = 3/(2cos(ln(t)))[/tex]

Therefore, the specific solution of the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

Now, we can find y'(1) by differentiating the specific solution with respect to x and evaluating it at x = 1:

y'(x) = -(3/2)(ln(t)sin(ln(x)) + cos(ln(x)))

y'(1) = -(3/2)(ln(t)sin(ln(1)) + cos(ln(1)))

      = -(3/2)(ln(t)(0) + 1)

      = -3/2

Therefore, the solution to the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

y(t) = 3

y'(1) = -3/2

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a. The probability that a randomly selected bottle will have less than 354 ml of beer is approximately 0.3085.

To calculate this probability, we convert the value of 354 ml to a z-score using the formula z = (x - μ) / σ, where x is the value we want to find the probability for (354 ml), μ is the mean (355 ml), and σ is the standard deviation (8 ml). By calculating the z-score, we can then look up the corresponding area under the normal distribution curve using a z-table. The z-score for 354 ml is approximately -0.125, and the corresponding area (probability) is 0.4508. Therefore, the probability of having less than 354 ml is 0.5 - 0.4508 = 0.0492 (or approximately 0.3085 when rounded to four decimal places).

b. The probability that a randomly selected 6-pack of beer will have a mean amount less than 354 ml is approximately 0.0194.

To calculate this probability, we need to consider the distribution of the sample mean. Since we are selecting a sample of size 6, the mean of the sample will have a standard deviation of σ / √n, where σ is the standard deviation of the population (8 ml) and n is the sample size (6). The standard deviation of the sample mean is therefore 8 ml / √6 ≈ 3.27 ml. We can then convert the value of 354 ml to a z-score using the same formula as in part a. The z-score for 354 ml is approximately -0.3061. By looking up this z-score in the z-table, we find the corresponding area (probability) of 0.3808. Therefore, the probability of the mean amount being less than 354 ml is 0.5 - 0.3808 = 0.1192 (or approximately 0.0194 when rounded to four decimal places).

c. The probability that a randomly selected 12-pack of beer will have a mean amount less than 354 ml is approximately 0.0022.

Similar to part b, we calculate the standard deviation of the sample mean for a sample size of 12, which is σ / √n = 8 ml / √12 ≈ 2.31 ml. By converting 354 ml to a z-score, we find a value of approximately -1.08. Looking up this z-score in the z-table, we find the corresponding area (probability) of 0.1401. Therefore, the probability of the mean amount being less than 354 ml is 0.5 - 0.1401 = 0.3599 (or approximately 0.0022 when rounded to four decimal places).

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Step-by-step explanation:

7y = 7

y = 1

2(1) + 4x = 14

4x = 12

x = 3

Answer:

dont know sorry

Step-by-step explanation:

Answer:

i dont kno bestie.. :/

Step-by-step explanation:

The centroid of the region bounded from above by the curve y = 4x - x² and below by the curve y = x is (2/3, 4/3).

The region is bounded from above by the curve y = 4x - x² and below by the curve y = x. We need to find the points of intersection between these two curves. Setting the equations equal to each other,

4x - x² = x

Rearranging,

x² - 3x = 0

Factoring,

x(x - 3) = 0

So, x = 0 or x = 3.

The region is bounded from x = 0 to x = 3. To find the y-values within this region, we evaluate the equations y = 4x - x² and y = x at these x-values.

For x = 0,

y = 4(0) - (0)² = 0

For x = 3,

y = 4(3) - (3)² = 12 - 9 = 3

Thus, the y-values within the region are y = 0 to y = 3. Now, we calculate the area of the region by integrating the difference of the upper and lower curves,

A = ∫[0,3] [(4x - x²) - x] dx

A = ∫[0,3] (3x - x²) dx

A = [3x²/2 - x³/3] evaluated from x = 0 to x = 3

A = [27/2 - 9/3] - [0 - 0]

A = [27/2 - 3] - 0

A = 21/2

Now, for the centroid,

x = (1/A) * ∫[0,3] x * [(4x - x²) - x] dx

Simplifying,

x = (1/A) * ∫[0,3] (3x² - x³) dx

x = (1/A) * [x³ - x⁴/4] evaluated from x = 0 to x = 3

x = (1/A) * [(3)³ - (3)⁴/4] - [0 - 0]

x = (1/A) * [(27) - (81)/4] - 0

x = (1/A) * [(108 - 81)/4]

x = (1/A) * (27/4)

x = 27/(4A)

x = 27/(4 * 21/2)

x = 2/3, and,

x = (1/A) * ∫[0,3] [(4x - x²) - x]² dx

Simplifying,

y = (1/A) * ∫[0,3] (16x² - 8x³ + x⁴) dx

y = (1/A) * [(16x³/3 - 8x⁴/4 + x⁵/5)] evaluated from x = 0 to x = 3

y = (1/A) * [(16(3)³/3 - 8(3)⁴/4 + (3)⁵/5)] - [0 - 0]

y = (1/A) * [(16 * 27/3 - 8 * 81/4 + 243/5)]

y = (1/A) * [(144/3 - 648/4 + 243/5)]

y = (1/A) * [(480 - 972 + 243)/60]

y = (1/A) * (480 - 972 + 243)/60

y = -83/(20A)

Since A = 21/2, we can substitute it in,

y = -83/(20 * 21/2)

y = -83/(210/2)

y = -83/(105)

y = -4/5

Therefore, the centroid of the region is (2/3, 4/3).

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Answer:

12 oz

Step-by-step explanation:

2.56 ÷ 8 = 0.32 per oz

3.66 ÷ 12= 0.305 per oz

The upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm

To find the upper boundary for a 95% confidence interval for the mean length of corn ears, we can use the formula:

Upper Boundary = Mean + (Critical Value * Standard Error)

The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be obtained from the standard normal distribution, which is approximately 1.96.

The standard error is calculated by dividing the standard deviation by the square root of the sample size:

Standard Error = s / [tex]\sqrt{(n)}[/tex]

Given that the mean length was 31.5 cm (Mean) and the standard deviation was s = 5.8 cm, and the sample size was n = 26, we can calculate the upper boundary as follows:

Standard Error = 5.8 / [tex]\sqrt{26}[/tex] ≈ 1.138

Upper Boundary = 31.5 + (1.96 * 1.138) ≈ 33.8

Therefore, the upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm.

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Answer:

Step-by-step explanation:

So in ratios you can mostly all of the time scale your answer. So by determining how much increase there is in the baby's thigh bone each week you can pretty much answer these questions.

keep in mind: Proportional means having the same ratio. A scale factor is the ratio of the model measurement to the actual measurement in simplest form.

Example from https://www.mathsisfun.com/numbers/ratio.html

A ratio says how much of one thing there is compared to another thing.

ratio 3:1

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

Use the ":" to separate the values:   3 : 1

Or we can use the word "to":   3 to 1

Or write it like a fraction:    31  

A ratio can be scaled up:

ratio 3:1 is also 6:2

Here the ratio is also 3 blue squares to 1 yellow square,

even though there are more squares.

Answer:

114.51

Step-by-step explanation:

I'm not to sure what you meant by 'each' so I solved it like there was only one cylinder. hope this helped

y=14 I hope this helps!!

Answer: Evaluate the findings to compare to his hypothesis

Step-by-step explanation: Since the biologist already has the findings and has a hypothesis, he now has to compare both of them together.

The graph is nearly symmetrical.

Instead of using rigorous mathematics to solve this issue, let's simply look at it.

Most of the values are on the left side of a graph when it is skewed to the right.

The majority of values are on the right side of a graph when it is skewed left.

Perfect symmetry occurs when both sides are identical with regard to the median. Here, the means and medians are equal.

Nearly symmetrical would be very nearly perfect symmetry, with very minor variations on either side. Median and mean would be almost equal.

Now that we have counted the dots and have carefully examined them, we can rule out skewed right and skewed left. Is the graph now completely symmetrical? No!

Therefore, "nearly symmetrical" is the right response.

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Complete question =

The dot plot shows the number of words students spelled correctly on a pre-test. Which statement best describes the shape of the graph?

A.) The graph is skewed right.

B.) The graph is nearly symmetrical.

C.) The graph is skewed left.

D.) The graph is perfectly symmetrical.

Answer:

0.416667 ft

Step-by-step explanation:

Answer: 250 mL Juice container

Step-by-step explanation:

Given

Half liter juice costs $2.86 i.e.

[tex]\dfrac{1}{2}\ L\rightarrow\$2.86\\\\1\ L\rightarrow\dfrac{2.86}{\frac{1}{2}}=\$5.72\\\\5\ L\rightarrow\$28.6[/tex]

A 250 mL juice costs $1.05 i.e.

[tex]250\ mL=0.25\ L\rightarrow \$1.05\\\\1\ L\rightarrow \dfrac{1.05}{0.25}=\$4.2\\\\\Rightarrow 5\ L\rightarrow \$21[/tex]

The cost of 250 mL Juice packet is low for 5 L quantity, therefore, Cierra must buy 250 mL Juice container

The parametric representation for the surface is x = ρsin(φ)cos(θ), y = ρsin(φ)sin(θ), z = ρcos(φ) with the restrictions 0 ≤ ρ ≤ 2, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π/4.

To find a parametric representation for the surface that lies above the cone z = √(x² + y²) and is part of the sphere x² + y² + z² = 4, we can express the surface in terms of spherical coordinates.

In spherical coordinates, the sphere x² + y² + z² = 4 can be represented as:

ρ² = 4

ρ = 2

Since we want to consider only the part of the sphere above the cone, we restrict the values of ρ to be between 0 and 2.

The cone z = √(x² + y²) in spherical coordinates is expressed as:

z = ρcos(φ)

Combining these equations, we can find the parametric representation for the desired surface:

x = ρsin(φ)cos(θ)

y = ρsin(φ)sin(θ)

z = ρcos(φ)

However, we need to restrict the values of ρ and φ to only the part of the surface above the cone. This means that ρ should range from 0 to 2, and φ should range from 0 to the angle that corresponds to the cone z = √(x² + y²).

Let's find the range of φ by substituting the equation for the cone into the equation for z:

z = ρcos(φ)

√(x² + y²) = ρcos(φ)

Since x² + y² = ρ²sin²(φ) (using the spherical coordinate expressions for x and y), we can rewrite the equation as:

√(ρ²sin²(φ)) = ρcos(φ)

ρsin(φ) = ρcos(φ)

tan(φ) = 1

Solving for φ, we find φ = π/4.

Therefore, the parametric representation for the surface is:

x = ρsin(φ)cos(θ)

y = ρsin(φ)sin(θ)

z = ρcos(φ)

with the restrictions:

0 ≤ ρ ≤ 2

0 ≤ θ ≤ 2π

0 ≤ φ ≤ π/4

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Answer:

It's correct.

Step-by-step explanation:

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

To apply the Divergence Theorem, we need to first find the divergence of the vector field F:

div(F) = ∂/∂x(x²) + ∂/∂y(-y²) + ∂/∂z(z²)

= 2x - 2y + 2z

Next, we find the bounds for the region D by setting the two plane equations equal to each other and solving for z:

9 - x - y = 6 - x - y

z = 3

So the region D is bounded below by the xy-plane, above by the plane z = 3, and by the coordinate planes x = 0, y = 0, and z = 0. Therefore, we can set up the integral using the Divergence Theorem as follows:

∫∫F · dS = ∭div(F) dV

= ∭(2x - 2y + 2z) dV

= ∫₀³ ∫₀^(3-z) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx

We can simplify this integral using the limits of integration to get:

∫∫F · dS = ∫₀³ ∫₀^(3-x) ∫₀^(3-x-y) (2x - 2y + 2z) dz dy dx

= ∫₀³ ∫₀^(3-x) [(2x - 2y)(3-x-y) + (2/3)(3-x-y)³] dy dx

= ∫₀³ [∫₀^(3-x) (2x - 2y)(3-x-y) dy + ∫₀^(3-x) (2/3)(3-x-y)³ dy] dx

Evaluating the two inner integrals, we get:

∫₀^(3-x) (2x - 2y)(3-x-y) dy = -x²(3-x) + (3/2)x(3-x)²

∫₀^(3-x) (2/3)(3-x-y)³ dy = (2/27)(3-x)⁴

Substituting these back into the integral and evaluating, we get:

∫∫F · dS = ∫₀³ [-x²(3-x) + (3/2)x(3-x)² + (2/27)(3-x)⁴] dx

= 9/5

Therefore, the net outward flux of the vector field F across the boundary of the region D is 9/5.

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Answer:

about 5.99 or D. 6 cm

Step-by-step explanation:

you can use this formula

[tex]V=4/3 * \pi *r^{3}[/tex]

The minimum sample size needed is 423.

To find the minimum sample size needed to estimate the population proportion with a given level of confidence and a desired margin of error, we can use the formula:

n = (Z^2 * p * q) / E^2

where:

n is the minimum sample size

Z is the Z-score corresponding to the desired confidence level

p is the estimated proportion of the population

q is 1 - p (complement of the estimated proportion)

E is the desired margin of error

In this case, the researcher wants to estimate the population proportion of adults who eat fast food four to six times per week with a 90% confidence level and an accuracy within 2% (margin of error of 0.02).

Since the estimated proportion is not given, we can use a conservative estimate of p = 0.5, which maximizes the sample size. This is because when the estimated proportion is unknown, assuming p = 0.5 results in the largest sample size required.

The Z-score corresponding to a 90% confidence level is approximately 1.645.

Plugging the values into the formula:

n = (1.645^2 * 0.5 * 0.5) / 0.02^2

n ≈ 422.94

Rounding up to the nearest whole number, the minimum sample size needed is 423.

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Answer:

It is complementary since their sum is equal to 90°

Answer:

196.1

Step-by-step explanation:

Area of a circle is [tex]\pi r^{2}[/tex] so in order to find the radius you divide the diameter by 2 to get 7.9

Then you do [tex]7.9^{2}[/tex] x [tex]\pi[/tex] to get around 196.1

Answer:

5/6

Step-by-step explanation:

Hope this helped!!!

The volume of the solid obtained by revolving the region bounded by y = x - √x, y = 0, and x = 36 around the line x = 36 can be found using the method of cylindrical shells. The resulting volume is approximately 3,012 cubic units.

To calculate the volume, we integrate the formula for the volume of a cylindrical shell, which is given by V = 2π∫[a,b] x * h(x) dx, where [a,b] represents the range of x values.
In this case, the lower bound of integration is 0 and the upper bound is 36, since the region is bounded by y = 0 and x = 36. The height of the cylindrical shell, h(x), is given by the difference between the x-coordinate of the curve y = x - √x and the line x = 36.
To obtain the x-coordinate of the curve, we set x - √x = 0 and solve for x. This gives us x = 0 or x = 1.
Next, we calculate the difference between x and 36, which gives us  the height of the cylindrical shell. Then, we substitute the expressions for x and h(x) into the volume formula and integrate with respect to x.
After performing the integration, we find that the volume of the solid is approximately 3,012 cubic units.

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Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.

For a normal distribution of SAT critical reading scores with a mean of 505 and a standard deviation of 118, approximately 79% of the SAT verbal scores are less than 600. If 1000 SAT verbal scores are randomly selected, it is expected that approximately 722 of them would be greater than 575.

To determine the percentage of SAT verbal scores that are less than 600, we need to find the area under the normal distribution curve to the left of 600. We can use the standard normal distribution table or a statistical software to find the corresponding z-score.

First, we calculate the z-score using the formula:

z = (x - μ) / σ

Substituting the values:

z = (600 - 505) / 118

z ≈ 0.8051

Using the standard normal distribution table, we can find the area to the left of z = 0.8051, which is approximately 0.7910.

To determine the percentage, we multiply the result by 100, giving us approximately 79% of SAT verbal scores that are less than 600.

For part (b), we can apply the same approach. We calculate the z-score for x = 575:

z = (575 - 505) / 118

z ≈ 0.5932

Using the standard normal distribution table, we find the area to the left of z = 0.5932, which is approximately 0.7242. This means that approximately 72.42% of SAT verbal scores are less than 575.

To estimate the number of SAT verbal scores greater than 575 in a sample of 1000, we multiply the percentage by the sample size:

Number of scores greater than 575 = 0.7242 * 1000 ≈ 722.

Therefore, we would expect that approximately 722 SAT verbal scores out of 1000 would be greater than 575.

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