To solve the given initial value problem using the method of Laplace transforms, we need to perform the following steps:
Step 1: Take the Laplace transform of both sides of the given differential equation.
Step 2: Solve for the Laplace transform of y.
Step 3: Take the inverse Laplace transform to obtain y.
Step 4: Use the initial conditions to find the constants in the solution obtained in Step 3.1.
Taking the Laplace transform of both sides of the given differential equation: L{y" - 12y' + 72y} = L{40e⁴}L{y" - 12y' + 72y} = 40L{e⁴}.
Taking Laplace transform of y" term L{y"} - 12L{y'} + 72L{y} = 40L{e⁴}.
Using the Laplace transform property of derivatives,
we get:s²Y(s) - sy(0) - y'(0) - 12[sY(s) - y(0)] + 72Y(s) = 40/(s - 4)
Simplifying the above equation, we get: s²Y(s) - s - 10 - 12sY(s) + 12 + 72Y(s) = 40/(s - 4)⇒ s²Y(s) - 12sY(s) + 72Y(s) = 40/(s - 4) + s + 2
Using partial fraction decomposition, we can write the right-hand side of the above equation as:40/(s - 4) + s + 2 = [10/(s - 4)] - [10/(s - 4)²] + s + 2
Now, the given equation becomes:
s²Y(s) - 12sY(s) + 72Y(s) = [10/(s - 4)] - [10/(s - 4)²] + s + 2
Taking the Laplace transform of y(0) = 1 and y'(0) = 10, we get: Y(s) = (10s + 2 + 1)/[s² - 12s + 72]
Applying partial fraction decomposition to find Y(s),
we get: Y(s) = [3/(s - 6)] - [1/(s - 6)²] + [7/(s - 6)²] + [1/(s - 6)]
Taking the inverse Laplace transform of Y(s), we get: y(t) = [3e⁶t - 3te⁶t + 7te⁶t + e⁶t]
Using the initial conditions y(0) = 1 and y'(0) = 10, we get: y(0) = 1 = 1 + 0 + 0 + 1, y'(0) = 10 = 18 - 3 + 7 + 1
Therefore, the solution to the given initial value problem is: y(t) = [3e⁶t - 3te⁶t + 7te⁶t + e⁶t]
Answer: y(t) = [3e⁶t - 3te⁶t + 7te⁶t + e⁶t]
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Solve the system of equations.
5y - 4x = -7
2y + 4x = 14
X=
y =
Step-by-step explanation:
7y = 7
y = 1
2(1) + 4x = 14
4x = 12
x = 3
The initial size of a bacteria culture is 1000. After one hour the bacteria count is 8000. After how many hours will the bacteria population reach 15000? Assume the population grows exponentially.
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.
A researcher wishes to estimate, with 90 % confidence, the population proportion of adults who eat fast food four to six times per week. Her estimate must be accurate within 2% of the population proportion. Find the minimum sample size needed.
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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PLSS HELP IMMEDIATELY!!! i’ll give brainiest if u don’t leave a link!
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.
Cierra is buying juice. She needs 5 liters. A half liter juice cost $2.86. A 250-milliliter container of juice costs $1.05. What should Cierra buy so she gets 5 liters at the lowest price?
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
Use the normal distribution of SAT critical reading scores for which the mean is 505 and the standard deviation is 118. Assume the variable x is normally distributed. (a) What percent of the SAT verbal scores are less than 600? (b) If 1000 SAT verbal scores are randomly selected, about how many would you expect to be greater than 575? Click to view page 1 of the standard normal table. Click to view page 2 of the standard normal table. (a) Approximately 79 % of the SAT verbal scores are less than 600. (Round to two decimal places as needed.) (b) You would expect that approximately 722 SAT verbal scores would be greater than 575.
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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find a parametric representation for the surface.
part of the surface of the sphere x² + y² + z² = 4 that lies above the cone z = √x²+y².
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 restaurant sells an 8-oz drink for $2.56 and a 12 oz drink for $3.66. Which drink is the better buy? i need help fast :(
Answer:
12 oz
Step-by-step explanation:
2.56 ÷ 8 = 0.32 per oz
3.66 ÷ 12= 0.305 per oz
Hey Guys,.I just wanted to check. Is this correct? :V
Answer:
It's correct.
Step-by-step explanation:
- - - - - - - - - - - - - - - - - - - -
Use the Divergence Theorem to compute the net outward flux of the vector field F = (x², - y², z²) across the boundary of the region D, where D is the region in the first octant between the planes z = 9 - x - y and z = 6 - x - y.
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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Please Help. What expression is equivalent to 6( t - 5 ) + 3
A. 6t - 2
B. 6t - 12
C. 3 ( 2t - 11 )
D. 3 ( 2t + 9 )
plz help me and answer correctly for branliest
Answer:
It is complementary since their sum is equal to 90°
what is the approximate radius of a sphere with a volume of 900 cm squared
A 12 cm
B 36 cm
C 18cm
D 6cm
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]
please help with this?!?
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
Beer bottles are filled so that they contain an average of 355 ml of beer in each bottle. Suppose that the amount of beer in a bottle is normally distributed with a standard deviation of 8 ml. [You may find it useful to reference the z table.]
a. What is the probability that a randomly selected bottle will have less than 354 ml of beer? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
b. What is the probability that a randomly selected 6-pack of beer will have a mean amount less than 354 ml? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
c. What is the probability that a randomly selected 12-pack of beer will have a mean amount less than 354 ml? (Round intermediate calculations to at least 4 decimal places, "z" value to 2 decimal places, and final answer to 4 decimal places.)
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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Number 5 please helpppppppppp 10 points
5 in = ___________ ft *Write your answers like this: whole number, one space, numerator, /, denominator. Example: 1 1/2 * PLEASE AWNSER FAST <3
Answer:
0.416667 ft
Step-by-step explanation:
Verify the equation: (cos x + 1)/(sin^3 x) = (csc x)/(1 - cos x)
Answer:
dont know sorry
Step-by-step explanation:
Rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.
X = 36y²
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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Suppose that $575.75 is invested in a savings account with an APR of 12% compounded monthly. What is the future value of the account in 5 years?
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
someone help this question is worth 50 points! What ratios are equivalent to the ratio 24:4
A.) 6:1
B.) 12:2
C.) 4:24
D.) 48:8
E.) 18:3
F:) 1:6
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
which statement best discribes the shape of the graph? the graph is skewed left. the graph is skewed right. the graph is nearly symmetrical. the graph is perfectly symmetrical.
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.
find the volume of the solid that results when the region bounded by y=x−−√, y=0 and x=36 is revolved about the line x=36.
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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Pls help and if you can show me how you do it :)
Find the number less than 40, that is
divisible by 5, and when divided by 6
has a remainder of 2.
What is the answer to this question?
If y varies directly as x, and y = 6 when x = 4, find y when x = 12.
y =
y=14 I hope this helps!!
Help please please pray that you pray
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
what is 1/3 plus 1/2 in fraction form
Answer:
5/6
Step-by-step explanation:
Hope this helped!!!
The table shows the amounts A (in billions of dollars) budgeted for national defense for the years 1998 to 2004.
Solve for the value of a
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