Find the volume of the region between the planes x + y + 3z = 4 and 3x + 3y + z = 12 in the first octant The volume is (Type an integer or a simplified fraction.)

It took Torrin 15.75 minutes to ride to school

Torrin house is 3.54 km away from school

An exponential equation is an expression that shows how numbers and variables using mathematical operators.

Let d represent the distance from Torrin home to school.

Let x represent the time it takes Torrin riding at 13.5 km/h and y represent the time it takes Torrin riding at 10.5 km/h

Torrin took a total of 36 min (0.6 hour) to ride to school and back, Hence:

x + y = 36    (1)

Also:

13.5 = d/x

d = 13.5x

10.5 = d/y

d = 10.5y

13.5x = 10.5y      (2)

Solving equation 1 and 2 simultaneously:

x = 0.2625 hours = 15.75 minutes

y = 0.3375 hour = 20.25 minutes

d = 10.5y = 10.5(0.3375) = 3.54 km

It took Torrin 15.75 minutes to ride to school

Torrin house is 3.54 km away from school

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The measure of the angle EDC is 17 degrees

From the question, we have the following parameters that can be used in our computation:

(9x - 1) = (2x + 13)

Evaluating the like terms

So, we have

7x = 14

Divide by 7

x = 2

So, we have

EDC = 9x - 1

Substitute the known values in the above equation, so, we have the following representation

EDC = 9(2) - 1

Evaluate

EDC = 17

Hence, the measure is 17 degrees

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Answer:

80 hours

Step-by-step explanation:

sorry if I do not understand but I think that is what is being asked

Answer:

Step-by-step explanation:

count all the the numbers and add the j,q,k,a together then figure out how many even numbers there are and put the total of all the numbers so it would be like 5/13 if your looking for a fraction, thats just for one suit if you need all the suits together then it would be 20/52

The time until the minimum price is reached is D)9.6 months

To find the time until the minimum price is reached, we need to find the value of t that minimizes the function P(t) = 130t^2 - 2500t + 6900.

One way to do this is to take the derivative of P(t) with respect to t, and set it equal to zero to find the critical point(s):

P'(t) = 260t - 2500 = 0

t = 2500/260 = 9.6 months

So the critical point is at t = 9.6 months. To check that this is a minimum, we can take the second derivative of P(t):

P''(t) = 260

Since P''(t) is positive for all t, we know that the critical point at t = 9.6 months is a minimum.

Therefore, the answer is D) 9.6 months.

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The value of the integral is 9. To evaluate the integral: ∫[1,8] [tex]x^{(-2/3)}[/tex] dx

We can use the power rule of integration. Specifically, we have:

∫ [tex]x^{(-2/3)}[/tex] dx = 3[tex]x^{(1/3)}[/tex] / (1/3) + C = 9[tex]x^{(1/3)}[/tex] + C

where C is the constant of integration.

Applying this formula to the given integral, we have:

∫[1,8] [tex]x^{(-2/3)}[/tex] dx = [9x^(1/3)] [1,8] = 9([tex]8^{(1/3)}[/tex] - [tex]1^{(1/3)}[/tex]= 9(2 - 1) = 9

Therefore, the value of the integral is 9.

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Answer:

B

Step-by-step explanation:

It has to reflect from one side to the other therefor left to right.

Answer:

B

Step-by-step explanation:

Both of them are congruent, meaning that they are the same but are reflected.

The value of f(x) is [tex]f(x) =\frac{2}{5}x^5+\frac{5}{2}x^{2} +x-\frac{39}{10}[/tex]

The equation in which the derivative of the given function is included is known as the differential equation. We have to find out a particular solution to the given ODE. We will use the power rule of integration to solve this question.

We have the function :

f"(x) = [tex]8x^3+5[/tex]

Integrate on both sides with respect to x.

[tex]f'(x) = 8\int\limits x^3dx + \int\limits 5dx\\\\f'(x) = 2x^4+5x+C_1[/tex]

Integrate on both sides with respect to x.

[tex]f(x) = 2\int\limitsx^4dx+5\int\limits xdx+\int\limits C_1dx\\\\f(x) = \frac{2}{5}x^5+\frac{5}{2}x^2+C_1x+C_2\\ \\[/tex]

f'(1) = 8

 8 = 2 + 5 + [tex]C_1[/tex]

[tex]C_1=0[/tex]

f(1) =0

[tex]0 = \frac{2}{5} +\frac{5}{2}+1+C_2\\ \\C_2=-\frac{39}{10\\}\\[/tex]

[tex]f(x) =\frac{2}{5}x^5+\frac{5}{2}x^{2} +x-\frac{39}{10}[/tex]

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The  natural annual percent of  increase of its population is 1.6%, under the given condition that in the given country possess a crude birth rate of 24 per 1,000 and crude death rate of 8 per 1,000.

Then the correct option is Option B.

For the purpose of evaluating the natural annual percent of increase in a population we have to  subtract crude death rate from  crude birth rate and then dividing by 10.

So for the given case,

the natural annual percent of increase in the population would be

((24-8)/10)

= 1.6%

The  natural annual percent of  increase of its population is 1.6%, under the given condition that in the given country possess a crude birth rate of 24 per 1,000 and crude death rate of 8 per 1,000.



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The complete question

If a country has a crude birth rate of 24 per 1,000 and a crude death rate of 8 per 1,000, the natural annual percent increase of its population is

a) 0.6%

b)1.6%

c) 3%

d)16%

e) 32%

Answer:

[tex]f'(x)= \frac{13}{(2x+3)^2}\\[/tex]

Step-by-step explanation:

[tex]f(x)= \frac{3x-2}{2x+3} \\[/tex]

[tex]f'(x)=\frac{dy}{dx} = \frac{d}{dx}(\frac{3x-2}{2x+3})\\ f'(x)= \frac{(2x+3)\frac{d}{dx}(3x-2)-(3x-2)\frac{d}{dx}(2x+3) }{(2x+3)^{2} } \\f'(x)= \frac{(2x+3)(3)-(3x-2)(2)}{(2x+3)^{2} } \\[/tex]

[tex]f'(x)= \frac{6x+9-6x+4}{(2x+3)^{2} }\\ f'(x)= \frac{13}{(2x+3)^2}\\[/tex]

The absolute maximum value of f(t) on the interval [-1,5] is 3

To find the absolute maximum and absolute minimum values of the function f(t) = t - √(t-1) on the interval [-1,5], we need to first find the critical points and endpoints of the interval.

Taking the derivative of f(t), we get:

f'(t) = 1 - 1/2(t-1)[tex]^{(-1/2)[/tex]

Setting this equal to zero and solving for t, we get:

1 ([tex]\frac{-1}{2}[/tex]) (t-1)([tex]\frac{-1}{2}[/tex])  = 0

[tex]\frac{1}{2}[/tex](t-1)([tex]\frac{-1}{2}[/tex])  = 1

(t-1)([tex]\frac{-1}{2}[/tex]) = 2

t-1 = 1/4

t = 1.25

The critical point is at t = 1.25.

Now, we need to check the function at the endpoints and the critical point to determine the absolute maximum and absolute minimum values.

f(-1) = -1 - √(-1-1) = -2

f(5) = 5 - √(5-1) = 3

f(1.25) = 1.25 - √(1.25-1) ≈ 0.354

Therefore, the absolute maximum value of f(t) on the interval [-1,5] is 3, which occurs at t=5, and the absolute minimum value of f(t) on the interval is approximately 0.354, which occurs at t = 1.25.

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From the F-table the critical value of [tex]\frac{\alpha }{2}[/tex] at the degrees of freedom of :

[tex]F_\frac{\alpha }{2}, _d_f_1,_d_f_2=3.91[/tex]

The decision rule

=> Fail to reject the null hypothesis

The first sample size [tex]n_1=12[/tex]

The first sample standard deviation is [tex]s_1=0.038[/tex]

The second sample size is [tex]n_2=10[/tex]

The first sample standard deviation is [tex]s_2=0.042[/tex]

The significance level is [tex]\alpha =0.05[/tex]

The null hypothesis is [tex]H_0:\sigma^2_1=\sigma^2_2[/tex]

The alternative hypothesis is [tex]H_0:\sigma^2_1\neq \sigma^2_2[/tex]

The test statistics is mathematically represented as:

[tex]F_c_a_l=\frac{s^2_1}{s^2_2}[/tex]

[tex]F_c_a_l=\frac{0.038^2}{0.042^2}[/tex]

[tex]F_c_a_l=0.81859[/tex]

The first degree of freedom is :

[tex]df_1=n_1-1=12-1=11[/tex]

The second degree of freedom is :

[tex]df_2=n_2-1=10-1=9[/tex]

From the F-table the critical  value of [tex]\frac{\alpha }{2}[/tex] at the degrees of freedom of :

[tex]df_1=11[/tex] and [tex]df_2=9[/tex]

[tex]F_\frac{\alpha }{2}, _d_f_1,_d_f_2=3.91[/tex]

The decision rule

Fail to reject the null hypothesis

The conclusion

This no sufficient evidence to conclude that there is a difference between the two variance.

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The value of f(t) is tan(t)  + sec(t)  + c.

What is integration?

Calculating areas, volumes, and their extensions requires the use of integrals, which are the continuous equivalent of sums. One of the two basic operations in calculus, along with differentiation, is integration, which is the process of computing an integral.

Here, we have

Given: f'(t) = sec(t) (sec(t) + tan(t))....(1)

We have to find the value f(t).

We take the integral of equation(1) and we get

∫f'(t) = ∫sec²(t)dt + ∫sec(t)tan(t)dt

We let

u = sec(t)

sec(t)tan(t)dt = du

∵ ∫sec²(x)dx = tan(x) + c

∫f'(t) = tan(t) + ∫1 du

f(t)  = tan(t)  + u + c

We substitute the value of u =  sec(t)

f(t) =  tan(t)  + sec(t)  + c

Hence, the value of f(t) is tan(t)  + sec(t)  + c.

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To find the time t when the instantaneous velocity of the ball equals its average velocity, we need to use the formula for average velocity:
average velocity = (change in position) / (change in time)
We can also find the instantaneous velocity by taking the derivative of the position function s(t):
instantaneous velocity = s'(t)
Let's find the average velocity over a certain time interval. Let's say we want to find the average velocity over the interval from t = 0 to t = 5 seconds. Then the change in position would be:
change in position = s(5) - s(0) = (-10 + 250) - (-10 + 250) = 0
And the change in time would be:
change in time = 5 - 0 = 5 seconds
So the average velocity over this time interval is:
average velocity = 0 / 5 = 0 ft/s
Now let's find the instantaneous velocity at time t. Taking the derivative of s(t), we get:
s'(t) = -10
So the instantaneous velocity is a constant -10 ft/s, regardless of the time t.
To find the time t when the instantaneous velocity equals the average velocity, we set these two equal to each other:
s'(t) = average velocity
-10 = 0
This equation has no solution, which means the instantaneous velocity never equals the average velocity. Therefore, there is no time t when this occurs.

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According to the reaction, the value of A is 143.

The given nuclear fission reaction is as follows:

235/92 U + 1/0 n → 90/36 Kr + A/56 Ba + 3 1/0 n

In this reaction, 235/92 U (Uranium-235) and 1/0 n (neutron) are the reactants, and 90/36 Kr (Krypton-90), A/56 Ba (Barium) and 3 1/0 n (neutrons) are the products.

The mass number (A) is the sum of the number of protons and neutrons in the nucleus of an atom. As the mass is conserved during any chemical or nuclear reaction, the mass number of the reactants must be equal to the mass number of the products.

Therefore, we can write the mass number balance equation for the given nuclear fission reaction as:

235 + 1 = 90 + A + (3 × 1)

Simplifying the above equation, we get:

236 = 90 + A + 3

A = 236 - 90 - 3

A = 143

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Answer:

The mean of the following scores is the sum of the numbers divided by the amount of terms,

The set is equal to 47, divided by the amount of numbers (10) = 4.7

Answer = 4.7

The probability distribution for the proportion of the 206 polled voters that support a new tax on fast food can be modeled by a normal distribution with a mean of 0.40 and a standard deviation of 0.0341 (rounded to four decimal places).

To answer your question, we need to find the mean and standard deviation for the normal probability distribution representing the proportion of polled voters that support a new tax on fast food.

1. Calculate the mean (µ):
The mean of the proportion can be found using the formula µ = p, where p is the proportion of voters that support the tax. In this case, p = 0.40. So, µ = 0.40.

2. Calculate the standard deviation (σ):
The standard deviation for a proportion can be calculated using the formula σ = √[p(1-p)/n], where n is the number of voters polled. In this case, n = 206.
σ = √[0.40(1-0.40)/206] = √[0.24/206] = √0.001165 = 0.0341 (rounded to 4 decimal places)

3. Complete the boxes with mean and standard deviation values:
Mean (µ): 0.40
Standard Deviation (σ): 0.0341

The probability distribution for the proportion of the 206 polled voters that support a new tax on fast food can be modeled by a normal distribution with a mean of 0.40 and a standard deviation of 0.0341 (rounded to four decimal places).

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The interval of convergence is:
I = (-1, 1)
So, the radius of convergence is:
R = 1

To find the radius of convergence and interval of convergence of the given series, we'll use the Ratio Test. The given series is:

Σ (from n=1 to infinity) 5(-1)^n nx^n

Let's consider the absolute value of the general term and apply the Ratio Test:

L = lim (n -> infinity) | (5(-1)^(n+1) (n+1)x^(n+1)) / (5(-1)^n nx^n) |

L = lim (n -> infinity) | ((-1)(n+1)x) / n |

Now, let's find the limit:

L = |-x| lim (n -> infinity) | (n+1) / n |

The limit is 1 as n goes to infinity. Therefore:

L = |-x|

For the Ratio Test, if L < 1, the series converges. So:

|-x| < 1

This inequality gives us the interval of convergence:

-1 < x < 1

Thus, the interval of convergence is:

I = (-1, 1)

The radius of convergence (R) is the distance from the center of the interval to either endpoint:

R = (1 - (-1)) / 2 = 2 / 2 = 1

So, the radius of convergence is:

R = 1

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The convergence range is I = [-1/5, 1/5)

We employ the ratio test to determine the radius of convergence:

lim┬(n→∞)⁡|5(-1)nx| = lim(n)|x(n+1)/n| = lim(n)|(n+1)/n| = n (n+1)x(n+1)|/|5(-1)n nx|

As a result, R = 1/5 is the radius of convergence.

Test the endpoints x = -1/5 and x = 1/5 to determine the interval of convergence:      

The series changes to: when x = -1/5

Σ 5(-1)^n n(-1/5)^n = Σ (-1)^n n/5^n

Since n/5n is decreasing and this alternate series has diminishing terms, it converges according to the alternating series test. Therefore, the interval of convergence includes x = -1/5.

The series changes to: when x = 1/5.

Σ 5(-1)^n n(1/5)^n = Σ (n/5)^n

Since this series is positive, we can perform the ratio test:

lim┬(n→∞)⁡|(n+1)/5|^(n+1)/(n/5)"n" = lim(n)(n+1).^{n+1}/n^n/5 = ∞

When x = 1/5, the series diverges, according to the ratio test.

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Therefore, the probability that more than one drug in a batch is defective is 0.347.

To solve this problem, we can use the binomial probability distribution. Let X be the number of defective drugs in a batch of 15. Then, X follows a binomial distribution with parameters n = 15 and p = 0.08.
(i) To determine the probability that none of the drugs in a batch is defective, we need to find P(X = 0). This can be calculated using the binomial probability formula:
P(X = 0) = (15 choose 0) × [tex]0.08^0[/tex] × [tex]0.92^{15}[/tex] = 0.327
Therefore, the probability that none of the drugs in a batch is defective is 0.327.
(ii) To determine the probability that more than one drug in a batch is defective, we need to find P(X > 1). This can be calculated using the binomial probability formula and some algebra:
P(X > 1) = 1 - P(X <= 1)
         = 1 - P(X = 0) - P(X = 1)
         = 1 - [(15 choose 0) × [tex]0.08^0[/tex] × [tex]0.92^{15}[/tex] + (15 choose 1) × [tex]0.08^1[/tex] × [tex]0.92^{14}[/tex]]
         = 0.347

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To Calculate the total within-cluster variation by summing the squared distances for both clusters.

- Total within-cluster variation: 138 + 54 = 192

The within-cluster variation of the given partitioning is 192.

To calculate the within-cluster variation of the given partitioning, we need to find the sum of squared distances of each point from its respective centroid.

For cluster C1:

- Distance from p1 to C1 = sqrt((6-7)^2 + (3-1)^2) = sqrt(2)

- Distance from p2 to C1 = sqrt((3-7)^2 + (8-1)^2) = sqrt(74)

- Distance from p3 to C1 = sqrt((5-7)^2 + (9-1)^2) = sqrt(68)

Sum of squared distances for cluster C1 = (sqrt(2))^2 + (sqrt(74))^2 + (sqrt(68))^2 = 2 + 74 + 68 = 144

For cluster C2:

- Distance from p4 to C2 = sqrt((10-8)^2 + (7-2)^2) = sqrt(53)

- Distance from p5 to C2 = sqrt((3-8)^2 + (2-2)^2) = sqrt(29)

Now,

Calculate the sum of squared distances within each cluster.

- Sum of squared distances for C1: 5 + 65 + 68 = 138

- Sum of squared distances for C2: 29 + 25 = 54

Now

Calculate the total within-cluster variation by summing the squared distances for both clusters.

- Total within-cluster variation: 138 + 54 = 192

The within-cluster variation of the given partitioning is 192.

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The minimum hourly wage that the airline needs to pay to be in the top 10% of hourly wages is $25.58 per hour (rounded to two decimal places).

Since the hourly wages of maintenance crew members follows a normal distribution, we can use the z-score formula to find the minimum hourly wage needed to be in the top 10% of wages:

z = (x - μ) / σ

where z is the z-score corresponding to the top 10% of wages, μ is the mean hourly wage of $20.50, σ is the standard deviation of $3.50, and x is the minimum hourly wage we need to pay.

To find the z-score for the top 10%, we look up the corresponding z-score from the standard normal distribution table or use a calculator with the inverse normal function. The z-score for the top 10% is approximately 1.28.

When the values are substituted into the formula, we get:

1.28 = (x - 20.50) / 3.50

Solving for x, we get:

x = 1.28 * 3.50 + 20.50

x = 25.58

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The expected value of XY = 2.25. So, the expected value of XY for the given condition is 2.25.

To solve this problem, we need to first find all the possible pairs of positive integers (X, Y) that satisfy X^2 + Y^2 < 13.

We can do this by listing out all the possible values of X and Y that satisfy this inequality:

(X,Y) = (1,1), (1,2), (2,1), (2,2), (1,3), (3,1), (2,3), (3,2)

Now, we can calculate the value of XY for each of these pairs:

(1,1): XY = 1
(1,2): XY = 2
(2,1): XY = 2
(2,2): XY = 4
(1,3): XY = 3
(3,1): XY = 3
(2,3): XY = 6
(3,2): XY = 6

Next, we need to find the probability of choosing each of these pairs. Since X and Y are randomly chosen positive integers, the probability of choosing any particular pair is 1/8 (since there are 8 possible pairs in total).

Now we can find the expected value of XY:

E(XY) = (1/8)(1) + (1/8)(2) + (1/8)(2) + (1/8)(4) + (1/8)(3) + (1/8)(3) + (1/8)(6) + (1/8)(6)
E(XY) = 3

Therefore, the expected value of XY is 3.

Remember to bring your solutions with you to class at the appointed time. Two problems will be graded for correctness and the rest for completeness.

To find the expected value of XY for randomly chosen positive integers X and Y satisfying X^2 + Y^2 < 13, we first need to identify the possible (X,Y) pairs that meet the condition.

The possible pairs are:
(1,1), (1,2), (2,1), and (2,2)

Now, let's calculate the products XY for each pair:
(1*1), (1*2), (2*1), and (2*2) which result in 1, 2, 2, and 4.

To find the expected value of XY, we need to find the average of these products:
(1+2+2+4)/4 = 9/4 = 2.25

So, the expected value of XY for the given condition is 2.25.

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The numeric value of the composition of the functions is given as follows:

f ( g ( f ( f ( f ( g ( 27 ) ) ) ) ) ) = 1.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The functions for this problem are given as follows:

We obtain the numeric values from the inside out, hence:

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The only solution that satisfies the given conditions is r = 0, y = 2, and Q = r'y = 0.

To find the two positive numbers r and y that maximize Q=r'y, we need to use the Lagrange multiplier method. Let's define a Lagrangian function L(r, y, λ) as follows:
L(r, y, λ) = r'y + λ(x + y - 2)
where λ is the Lagrange multiplier. We need to find the values of r, y, and λ that maximize L(r, y, λ).
Taking partial derivatives of L with respect to r, y, and λ, we get:
∂L/∂r = y
∂L/∂y = r + λ
∂L/∂λ = x + y - 2
Setting these partial derivatives equal to zero, we get:
y = 0 (this is not a valid solution as we need positive numbers)
r + λ = 0
x + y - 2 = 0
From the second equation, we get r = -λ. Substituting this into the first equation, we get y = 0. Substituting r and y into the third equation, we get x = 2.

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The particular solution of the indicated linear system that satisfies the initial conditions is (1/5) e⁻ᵃ [1 1/5] + (8/5) t e⁻ᵃ [1 1/5] + (4/5) e⁻ᵃ [1 1/5] + (3/5) e²ᵃ [1 1]

The first step in finding a particular solution of a linear system that satisfies given initial conditions is to write the system in matrix form, which is already given as:

X ′ = [ 3 − 1

5 − 3]x

Here, X ′ is the derivative of the vector X with respect to time t, and x is the vector of unknown functions that we want to find. To solve this system, we need to find the eigenvalues and eigenvectors of the matrix [3 -1; 5 -3], which can be done by finding the roots of the characteristic equation det([3 -1; 5 -3] - λI) = 0, where I is the identity matrix and λ is the eigenvalue.

Solving the characteristic equation, we get λ = -1 and λ = -1, which means that we have one repeated eigenvalue. To find the eigenvectors, we need to solve the equation ([3 -1; 5 -3] - (-1)I)x = 0 for each eigenvalue. For λ = -1, we get the equation

[4 -1; 5 -2]x = 0

which has the general solution x = c[1; 1/5], where c is a constant. For a repeated eigenvalue, we also need to find the generalized eigenvectors, which are solutions of the equation ([3 -1; 5 -3] - (-1)I)x = v, where v is a nonzero vector orthogonal to the eigenvector.

For λ = -1, we can choose v = [0; 1] and solve the equation ([3 -1; 5 -3] - (-1)I)x = [0; 1], which gives the solution x = [1/5; 1/25]. Thus, the eigenvector matrix P and the generalized eigenvector matrix Q are

P = [1 1/5; 1 1/5] and Q = [1 1/5; 0 1/25]

respectively. Using these matrices, we can write the general solution of the system as

x = c₁ e⁻ᵃ [1 1/5] + c₂ t e⁻ᵃ [1 1/5] + c₃ e⁻ᵃ [1 1/5] + c4 e^(2t) [1 1]

where c₁, c₂, c₃, and c4 are constants determined by the initial conditions.

Now, we can use the given initial conditions x(0) = [1 1] to find the values of c₁, c₂, c₃, and c4. Substituting t = 0 and x = [1 1] into the general solution, we get

[1 1] = c₁ [1 1/5] + c₂ (0) [1 1/5] + c₃ [1 1/5] + c4 [1 1]

which simplifies to

c₁ + c₃ + c4 = 1

c₁ + (1/5)c₃ + c4 = 1

Using the given initial conditions x'(0) = [2 4], we can also find the values of c₂ and c₃ by differentiating the general solution and substituting t = 0 and x' = [2 4]. This gives us the equations

x'(0) = [-1 0]c₁ + [-1/5 + 1]c₂ + [-1/5]c₃ + [2 2]c4 = [2 4]

Simplifying this equation, we get

c₁ - (1/5)c₃ + 2c4 = 2

c₂ + 2c4 = 4

We now have a system of four equations in four unknowns, which can be solved using algebraic manipulation. Solving for c₁, c₂, c₃, and c4, we get

c₁ = 1/5

c₂ = 8/5

c₃ = 4/5

c4 = 3/5

Substituting these values back into the general solution, we get the particular solution that satisfies the given initial conditions:

x = (1/5) e⁻ᵃ [1 1/5] + (8/5) t e⁻ᵃ [1 1/5] + (4/5) e⁻ᵃ [1 1/5] + (3/5) e²ᵃ [1 1]

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Complete Question:

Find a particular solution of the indicated linear system that satisfies the given initial conditions.

X ′ = [ 3 − 1

          5 − 3]x

x_1 = e^(2t) [1    1]

x_2 = e^(-2t) [ 1    5]

The integral value of S1 -1 t(1-t)²dt using the distributive property of multiplication is ½t² - ⅔t³ + ¼t⁴ + C.

To evaluate the integral S1 -1 t(1-t)²dt, we can start by expanding the integrand using the distributive property of multiplication:

t(1-t)² = t(1-2t+t²) = t - 2t² + t³

Then, we can integrate each term separately:

∫t dt = ½t² + C1

∫2t² dt = ⅔t³ + C2

∫t³ dt = ¼t⁴ + C3

Putting everything together, we get:

S1 -1 t(1-t)²dt = ½t² - ⅔t³ + ¼t⁴ + C

where C = C1 + C2 + C3 is the constant of integration.

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The probability that at least 10 out of 12 flips are heads is 0.5845, or about 58.45%.

The probability of getting a head on a single flip of the biased coin is 0.6, and the probability of getting a tail is 0.4.

To find the probability that at least 10 out of 12 flips are heads, we need to add the probabilities of getting 10, 11, or 12 heads. We can calculate these probabilities using the binomial probability formula:

P(X=k) = (n choose k)[tex]\times[/tex] [tex]p^k[/tex] [tex]\times[/tex] [tex](1-p)^{(n-k)[/tex]

where X is the random variable representing the number of heads in n flips, k is the number of heads we want to calculate the probability for, p is the probability of getting a head on a single flip, and (n choose k) is the binomial coefficient, which represents the number of ways to choose k heads out of n flips.

Using this formula, we can calculate the probability of getting 10 heads in 12 flips as:

P(X=10) = (12 choose 10) [tex]\times 0.6^10 \time 0.4^2 = 0.232[/tex]

The probability of getting 11 heads in 12 flips is:

P(X=11) = (12 choose 11) [tex]\times 0.6^{11} \times0.4^1 = 0.2835[/tex]

The probability of getting 12 heads in 12 flips is:

P(X=12) = (12 choose 12) [tex]\times 0.6^{12} \times 0.4^0 = 0.069[/tex]

Therefore, the probability of getting at least 10 heads in 12 flips is:

P(X>=10) = P(X=10) + P(X=11) + P(X=12) = 0.232 + 0.2835 + 0.069 = 0.5845

So, the probability that at least 10 out of 12 flips are heads is 0.5845, or about 58.45%.

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The dimensions of the box for which the volume is maximized are  2√15 cm and 5√15 cm.

Let's denote the side length of the square base by "x" and the height of the box by "h". Then, the total surface area of the box is:

S = [tex]x^{2}[/tex] + 4xh

We know that S = 300, so we can write:

[tex]x^{2}[/tex] + 4xh = 300

To maximize the volume of the box, we need to find the values of x and h that satisfy this equation and give us the largest possible value for V, the volume of the box.

The volume of the box is given by:

V = [tex]x^{2}[/tex]h

To find the maximum value of V, we can use the method of Lagrange multipliers. We want to maximize V subject to the constraint that S = 300, so we define the Lagrangian function:

L(x, h, λ) = [tex]x^{2}[/tex]h + λ([tex]x^{2}[/tex] + 4xh - 300)

To find the maximum of V, we need to solve the system of equations:

∂L/∂x = 2xh + 2λx + 4λh = 0

∂L/∂h = [tex]x^{2}[/tex] + 4λx = 0

∂L/∂λ = [tex]x^{2}[/tex] + 4xh - 300 = 0

Solving these equations, we get:

h = 5x/2

[tex]x^{2}[/tex] = 60

Substituting h = 5x/2 and [tex]x^{2}[/tex] = 60 into the equation for the volume, we get:

V = [tex]x^{2}[/tex]h = (60)(5x/2) = 150x

So, to maximize the volume, we need to find the value of x that maximizes V. Since [tex]x^{2}[/tex] = 60, we have x = √60 = 2√15. Substituting this value into the equation for h, we get:

h = 5x/2 = 5(2√15)/2 = 5√15

Therefore, the dimensions of the box for which the volume is maximized are:

length of the side of the square base = 2√15 cm

height of the box = 5√15 cm.

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