Solve the initial value problem below using the method of Laplace transforms. y" - 12y' + 72y = 40 e 4 y(0) = 1, y'(0) = 10

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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Answer: Let’s assume that the bacteria population grows exponentially according to the formula P(t) = P0 * e^(kt), where P0 is the initial population, k is the growth rate, t is time in hours, and e is the mathematical constant approximately equal to 2.71828. We know that at time t = 0, the population is P(0) = 1000. After one hour, the population is P(1) = 8000. We can use this information to solve for the growth rate k. Substituting the values into the formula, we get: 8000 = 1000 * e^(k * 1) Dividing both sides by 1000, we get: 8 = e^k Taking the natural logarithm of both sides, we get: ln(8) = k Now that we have solved for k, we can use the formula to find out when the population will reach 15000. 15000 = 1000 * e^(ln(8) * t) Dividing both sides by 1000, we get: 15 = e^(ln(8) * t) Taking the natural logarithm of both sides, we get: ln(15) = ln(8) * t Dividing both sides by ln(8), we get: t = ln(15)/ln(8) ≈ 1.71 hours So it will take approximately 1.71 hours for the bacteria population to reach 15000. Received message.

Answer:

a=7

Step-by-step explanation:

These two angles are complementary and the sum of their measures is 90°.

Therefore, we can create the equation: (5a+3)+52=90.

1. combine like terms. the equation becomes 5a+55=90.

2. -55 to both sides of the equation. the equation becomes 5a=35.

3. /5 to both sides of the equation. we can reach the conclusion that a=7

Answer:

5/6

Step-by-step explanation:

Hope this helped!!!

Answer:

0.416667 ft

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]

Answer:

It's correct.

Step-by-step explanation:

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

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:

It is complementary since their sum is equal to 90°

Answer:

well religion does not work here sorry

Step-by-step explanation:not everyone prays im sorry

Answer:

im confused

Step-by-step explanation:

dont delete this

Answer:

12 oz

Step-by-step explanation:

2.56 ÷ 8 = 0.32 per oz

3.66 ÷ 12= 0.305 per oz

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.

Answer:

FV= $1,045.96

Step-by-step explanation:

Giving the following information:

Initial investment (PV)= $575.75

Number of periods (n)= 15*5= 60 months

Interest rate (i)= 0.12 / 12= 0.01

To calculate the future value (FV), we need to use the following formula:

FV= PV*(1+i)^n

FV= 575.75*(1.01^60)

FV= $1,045.96

Step-by-step explanation:

just put the ratios into a fraction if x:y then x/y

A.)6/1=6

B.)12/2=6

C.)4/24=1/6

D.)48/8=6

E.)18/3=6

F.)1/6

24/4=6 so A, B, D, and E are equivilant to the ratio 24/4

Hope that helps :)

Answer:

6:1 ,12:2, 48:8

Step-by-step explanation:

24:4

24 ÷ 4 =6 and 4÷4 =1

6:1

24:4

24÷2 =12 , 4÷2=2,

12:2

48:8

24×2=48, 4×2=8

48:8

Step-by-step explanation:

7y = 7

y = 1

2(1) + 4x = 14

4x = 12

x = 3

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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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 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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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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Answer:

dont know sorry

Step-by-step explanation:

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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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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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