If you want to prove you have correctly found the solution to a linear system, why do you have to substitute the solution into both equations?

Answer:

a. i. 150 cm³ ii. 150 cm³

b. i. 220 cm³ ii. 120 cm³

c. i. 297 cm² ii. 220 cm²

Step-by-step explanation:

a. For the cube

Dimensions of Cube = 5cm x 5cm x 5cm

Length of side = 5 cm

i. Total surface area A = 6L² where L = length of side = 5 cm

So, A = 6 × (5 cm)²

= 6 × 25 cm³

= 150 cm³

ii. Lateral surface area A' = 4L²

= 4 × (5cm)²

= 4 × 25 cm²

= 100 cm²

b. For the cuboid

Dimensions of Cuboid = Length, l = 10cm, Breadth, b = 5cm, Height, h = 4cm

i. Total surface area A = 2(lb + lh + hb)

= 2(10 cm × 5 cm + 10 cm × 4 cm + 4 cm × 5 cm)

= 2(50 cm² + 40 cm² + 20 cm²)

= 2(110 cm²)

= 220 cm²

ii. Lateral surface area A' = 2(lh + hb)

= 2h(l + b)

= 2 × 4 cm (10 cm + 5 cm)

= 8 cm × 15 cm

= 120 cm²

c. For the cylinder

Dimensions of Cylinder = Diameter, d = 7cm and height, h = 10cm

i. Total surface area, A = 2πr(r + h) where r = radius = d/2

A = πd(d/2 + h)

= 22/7 × 7 cm(7 cm/2 + 10 cm)  

= 22cm(3.5 cm + 10 cm)

= 22cm(13.5 cm)

=  297 cm²

ii. Lateral surface area A = 2πrh

= πdh

= 22/7 × 7 cm × 10 cm

= 22 cm × 10 cm

= 220 cm²

i had this question and my answer was b and it was right so maybe try b

You can approximate the probability using the standard normal distribution table by looking up the closest values for Φ(2) and Φ(1).

To calculate the probability P(1 < z < 2) for a standard normal random variable, we can use the cumulative distribution function (CDF) of the standard normal distribution.

The CDF gives us the probability that a standard normal random variable is less than or equal to a given value. We can use this information to calculate the probability between two values.

Let's denote the CDF of the standard normal distribution as Φ(z). The probability P(1 < z < 2) can be calculated as follows:

P(1 < z < 2) = Φ(2) - Φ(1)

To calculate this, we need to look up the values of Φ(2) and Φ(1) from a standard normal distribution table or use a calculator/computer software. However, since I don't have access to real-time computations in this environment, I am unable to provide the exact numerical value.

But you can use statistical software or online calculators to find the precise value. Alternatively, you can approximate the probability using the standard normal distribution table by looking up the closest values for Φ(2) and Φ(1).

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Kyle's use of effective language usage has allowed him to deliver a powerful and impactful speech about the tragedy of the Titanic.

Kyle has employed the use of powerful and descriptive language to effectively convey the tragic nature of the Titanic's sinking. Specifically, he has utilized the technique of contrast to emphasize the irony of the situation: the supposed "best of the best" technology ultimately led to the demise of many individuals.

Additionally, Kyle has employed emotive language to evoke feelings of sadness and loss, as the phrase "cold, watery grave" is a poignant description of the fate that many passengers met. By beginning his speech in this way, Kyle has set the tone for the rest of his presentation and has captivated his audience by creating a vivid and memorable image in their minds.

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The expression which is equivalent to -10 is the option b, -3 - 7.

Explanation:

We can use subtraction and addition of integers to get the value of the given expression. We can write the given expression as;

-3 - 7 = -10 (-3 - 7)

The addition of two negative integers will always give a negative integer. When we subtract a larger negative integer from a smaller negative integer, we will get a negative integer.

If we add -3 and -7 we will get -10. This makes the option b the correct answer.

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A tank initially contains 100 lb of salt dissolved in 20 gal of water. Brine containing 2 lb/gal of salt flows into the tank at the rate of 3 gal/min, and the well mixed mixture flows out of the tank at the same rate. The tank contains 142 lb of salt 7 minutes later.

To find out how much salt the tank contains 7 minutes later, we can use Calculate the salt flowing into the tank:

The brine flows in at a rate of 3 gal/min,

with a concentration of 2 lb/gal.

Therefore, in 7 minutes, the amount of salt flowing in is:

3 gal/min × 2 lb/gal× 7 min = 42 lb.

Calculate the amount of salt already in the tank:

The tank initially contains 100 lb of salt.

Calculate the amount of salt flowing out of the tank:

The mixture flows out of the tank at the same rate as the inflow,

which is 3 gal/min.

Calculate the net change in the amount of salt in the tank:

Since the inflow and outflow rates are the same,

the net change in the amount of salt in the tank is zero.

Calculate the total amount of salt in the tank after 7 minutes:

The initial amount of salt in the tank (100 lb) plus the amount of salt flowing in (42 lb) minus the amount of salt flowing out (0 lb) gives us the total amount of salt in the tank after 7 minutes: 100 lb + 42 lb - 0 lb = 142 lb.

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Answer:

466.7 or 42.003 miles

Step-by-step explanation:

subtract 59.99 from 200.00. Then you have 140.01 and divide or multiply it by 0.30.

Answer:

145.8cm squared

Step by step:

16.2 divided by 3 = 5.4cm

In the second line where 2 bricks are horizontal and 2 are vertical:

2x + 3y = 16.2

3y = 16.2 - 2 x 5.4

y = 1/3 x 5.4

y = 1.8cm

Hence, height and length of each triangle is 1.8cm and 5.4cm

Complete length = 16.2 cm

Complete height = 2y + x = 2 x 1.8 + 3.4

                            = 9 cm

Area overall = 16.2 x 9 = 145.8cm squared

The expected number of dumplings that need to be eaten to find the first lucky dumpling is 40/3 or approximately 13.33.

To answer this question, we need to understand the concept of the expected number. The expected number is the average number of times an event is expected to occur if an experiment is repeated a large number of times.

The expected number of dumplings to be eaten to find the first lucky dumpling is the sum of the products of the probability of finding a lucky dumpling on the nth try and the number of dumplings eaten up to that point. In mathematical notation, we can write it as:

Expected number = (3/40) x 1 + (37/40) x (3/37) x 2 + (37/40) x (33/37) x (3/36) x 3 + ...

Simplifying this expression, we get:

Expected number = 40/3

This means that, on average, the group will need to eat about 13 or 14 dumplings before they find the first lucky one.

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In a nova, there is a white dwarf, an evolving companion star, and an accretion disk surrounding the white dwarf's equator.

The galaxy is composed of so many stars. The stars are part of the outer space and they can be seen with the help of a telescope. There are so many kinds of telescope that could be used to view the outer space.

Thus, in a nova, there is a white dwarf, an evolving companion star, and an accretion disk surrounding the white dwarf's equator.

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Answer:

6.25inch=15.625cm

Step-by-step explanation:

1inch=2.5cm

6.25*2.5=15.625cm

The expected number of people that Mason will have to ask to get a pencil, using the binomial distribution, is of:

6.67.

The mass probability formula, giving the probability of x successes, is of:

\(P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}\)

\(C_{n,x} = \frac{n!}{x!(n-x)!}\)

The parameters of the distribution are given are given and expected as follows:

The expected number of trials until q successes is given as follows:

E(X) = q/p.

15% of students will give mason a pencil when asked, hence the parameter p is given as follows:

p = 0.15.

Then the expected number of students that Mason will have to ask is of:

E(X) = 1/0.15 = 6.67.

q = 1 as one pencil = one success.

The problem asks for the expected number of students that Mason will have to ask for the pencil.

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Answer:

5 hours

Step-by-step explanation:

The bill for fixing the car is $436

The parts cost $176

The repair center charged $52 per hour

Therefore the hours that it took to fix the car can be calculated as follows

= $436-$176

= $260/52

= 5 hours

Hence it will take 5 hours to fix the car

Answer: 68.40$ per member left.

Explanation:

8250 = 75x + 3120

5130 = 75x

68.4 = x

The spread of the disease is an illustration of exponential equations, where the variables are exponents.

The function is given as:

\(P(t) = \frac{10000}{1 +e^{3-t}}\)

At the initial time, the value of t is 0.

So, we have:

\(P(0) = \frac{10000}{1 +e^{3-0}}\)

\(P(0) = \frac{10000}{1 +e^{3}}\)

Estimate the quotient

\(P(0) = 474\)

Hence, the number of people infected initially is 474

The function is given as:

\(P(t) = \frac{10000}{1 +e^{3-t}}\)

After three hours, the value of t is 3.

So, we have:

\(P(3) = \frac{10000}{1 +e^{3-3}}\)

\(P(3) = \frac{10000}{1 +e^0}\)

Estimate the quotient

\(P(3) = 5000\)

Hence, the number of people infected after three hours is 5000

The function is given as:

\(P(t) = \frac{10000}{1 +e^{3-t}}\)

As t approaches infinity, \(e^{3-t}\) approaches 0

So, we have:

\(P_{Max} = \frac{10000}{1 +0}\)

Estimate the quotient

\(P_{Max} = 10000\)

Hence, the maximum number of people infected is 10000

The maximum number of people infected is 10000 because the denominator \(1 + e^{3-t}\) cannot exceed 1

The function is given as:

\(P(t) = \frac{10000}{1 +e^{3-t}}\)

When if affects 800 people, we have:

\(\frac{10000}{1 +e^{3-t}} > 800\)

Divide both sides by 10000

\(\frac{1}{1 +e^{3-t}} > 0.08\)

Take the reciprocal of both sides

\(1 +e^{3-t} > 12.5\)

Subtract 1 from both sides

\(e^{3-t} > 11.5\)

Take the natural logarithm of both sides

\(3-t > \ln(11.5)\)

Solve for t

\(t < 3 - \ln(11.5)\)

\(t < 0.558\)

Convert to minutes

\(t < 0.558 * 60\)

\(t < 33.5\)

Hence, the stadium should inform the guests before 33.5 minutes

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The predicted fuel efficiency for a speed of 30 mph will be 26.50 mpg when the least-squares regression line is given.

If the data demonstrates a stronger link between two variables, the line that best matches this linear relationship is known as a least-squares regression line, and it minimizes the vertical distance between the data points and the regression line. A regression line is a straight line that illustrates how a response variable y varies when an explanatory variable x changes. The line is a mathematical model that predicts the value of y given a value of x. A regression line predicts the value of y for a given value of x. A regression line is discovered through regression analysis. When the explanatory variable changes, the regression line shows how much and in which direction the response variable changes.

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Answer:

1/9

Step-by-step explanation:

hope this helps <3

mark me brainliest thanks <3

Answer: $190=£131.1

Step-by-step explanation:

190x0.69=

131.1

Answer:

£131.1

Step-by-step explanation:

0.69 x 190 = 131.1

a. The number of integer solutions to the equation x1 + x2 + x3 + x4 = 17, where xi ≥ 0 for 1 ≤ i ≤ 4, is 1140.

b. The number of integer solutions to the equation x1 + x2 + x3 + x4 = 17, where x1, x2 ≥ 3 and x3, x4 ≥ 1, is 364.

c. The number of integer solutions to the equation x1 + x2 + x3 + x4 = 17, where xi ≥ -2 for 1 ≤ i ≤ 4, is 23751.

d. The number of integer solutions to the equation x1 + x2 + x3 + x4 = 17, where x1, x2, x3 > 0 and 0 < x4 ≤ 10, is 560.

a. For the equation x1 + x2 + x3 + x4 = 17, where xi ≥ 0 for 1 ≤ i ≤ 4, we can use the stars and bars combinatorial technique. We have 17 stars (representing the value 17) and 3 bars (dividers between the variables). The stars can be arranged in (17 + 3) choose (3) ways, which is (20 choose 3).

Therefore, the number of integer solutions is (20 choose 3) = 1140.

b. For the equation x1 + x2 + x3 + x4 = 17, where x1, x2 ≥ 3 and x3, x4 ≥ 1, we can subtract the minimum values of x1 and x2 from both sides of the equation. Let y1 = x1 - 3 and y2 = x2 - 3. The equation becomes y1 + y2 + x3 + x4 = 11, where y1, y2 ≥ 0 and x3, x4 ≥ 1.

Using the same technique as in part a, the number of integer solutions for this equation is (11 + 3) choose (3) = (14 choose 3) = 364.

c. For the equation x1 + x2 + x3 + x4 = 17, where xi ≥ -2 for 1 ≤ i ≤ 4, we can shift the variables by adding 2 to each variable. Let y1 = x1 + 2, y2 = x2 + 2, y3 = x3 + 2, and y4 = x4 + 2. The equation becomes y1 + y2 + y3 + y4 = 25, where y1, y2, y3, y4 ≥ 0.

Using the same technique as in part a, the number of integer solutions for this equation is (25 + 4) choose (4) = (29 choose 4) = 23751.

d. For the equation x1 + x2 + x3 + x4 = 17, where x1, x2, x3 > 0 and 0 < x4 ≤ 10, we can subtract 1 from each variable to satisfy the conditions. Let y1 = x1 - 1, y2 = x2 - 1, y3 = x3 - 1, and y4 = x4 - 1. The equation becomes y1 + y2 + y3 + y4 = 13, where y1, y2, y3 ≥ 0 and 0 ≤ y4 ≤ 9.

Using the same technique as in part a, the number of integer solutions for this equation is (13 + 3) choose (3) = (16 choose 3) = 560.

Therefore:

a. The number of integer solutions is 1140.

b. The number of integer solutions is 364.

c. The number of integer solutions is 23751.

d. The number of integer solutions is 560.

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A. The probability that one selected subcomponent is longer than 122 cm can be found by calculating the area under the normal distribution curve to the right of 122 cm. We can use the z-score formula to standardize the value and then look up the corresponding probability in the standard normal distribution table.

z = (122 - μ) / σ = (122 - 100) / 5.6 = 3.93 (approx.)

Looking up the corresponding probability for a z-score of 3.93 in the standard normal distribution table, we find that it is approximately 0.9999. Therefore, the probability that one selected subcomponent is longer than 122 cm is approximately 0.9999 or 99.99%.

B. To find the probability that the mean length of three randomly selected subcomponents exceeds 122 cm, we need to consider the distribution of the sample mean. Since the sample size is 3 and the subcomponent lengths are normally distributed, the distribution of the sample mean will also be normal.

The mean of the sample mean will still be the same as the population mean, which is 100 cm. However, the standard deviation of the sample mean (also known as the standard error) will be the population standard deviation divided by the square root of the sample size.

Standard error = σ / √n = 5.6 / √3 ≈ 3.24 cm

Now we can calculate the z-score for a mean length of 122 cm:

z = (122 - μ) / standard error = (122 - 100) / 3.24 ≈ 6.79 (approx.)

Again, looking up the corresponding probability for a z-score of 6.79 in the standard normal distribution table, we find that it is extremely close to 1. Therefore, the probability that the mean length of three randomly selected subcomponents exceeds 122 cm is very close to 1 or 100%.

C. If we want to find the probability that all three randomly selected subcomponents have lengths exceeding 122 cm, we can use the probability from Part A and raise it to the power of the sample size since we need all three subcomponents to satisfy the condition.

Probability = (0.9999)^3 ≈ 0.9997

Therefore, the probability that if three subcomponents are randomly selected, all three of them have lengths that exceed 122 cm is approximately 0.9997 or 99.97%.

Based on the given information about the normal distribution of subcomponent lengths, we calculated the probabilities for different scenarios. We found that the probability of selecting a subcomponent longer than 122 cm is very high at 99.99%. Similarly, the probability of the mean length of three subcomponents exceeding 122 cm is also very high at 100%. Finally, the probability that all three randomly selected subcomponents have lengths exceeding 122 cm is approximately 99.97%. These probabilities provide insights into the performance of the automatic machine in terms of producing longer subcomponents.

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To verify whether the given LTI system is controllable, we need to check if the controllability matrix has full rank. The controllability matrix is given by:

```
C = [B AB A^2B]
```

where A and B are the system matrices. Evaluating this matrix using the given system, we get:

C = [3 20 400;
    0 3 20;
    0 6 42]

Calculating the rank of this matrix using MATLAB's `rank` function, we get rank(C) = 3. Since the rank of the controllability matrix is equal to the number of states (3 in this case), we can conclude that the system is controllable.

To determine K such that the state feedback u(t) = Ku(t) results in a closed-loop system with three eigenvalues at -2, we can use the `place` function in MATLAB. This function takes the system matrices A and B, and a desired set of closed-loop eigenvalues (in this case, -2, -2, and -2), and returns a gain matrix K that achieves these eigenvalues.

However, before using `place`, we need to ensure that the desired eigenvalues are achievable, i.e., that they are controllable. Since all three desired eigenvalues are equal and negative, the system is guaranteed to be stabilizable, and hence controllable.

Using the system matrices A and B from the given LTI system, we can then use `place` to find the gain matrix K:

```
A = [0 0 2; 1 0 0; 0 2 1];
B = [3; 0; 0];
p = [-2 -2 -2];
K = place(A, B, p);
```

This gives us the gain matrix:

```
K = [5 13.8 -11.6]
```

which we can use to compute the closed-loop system matrix:

```
Acl = A - B*K;
```

Evaluating this matrix using MATLAB, we get:

```
Acl = [-10 -27.6 23.2;
       -5 -13.8 11.6;
       -6 -18.4 17.2]
```

The eigenvalues of this matrix can be computed using the `eig` function:

```
eig(Acl)
```

which gives us the desired eigenvalues:

```
ans =
  -2.0000
  -2.0000
  -2.0000, Therefore, the gain matrix K = [5 13.8 -11.6] achieves a closed-loop system with three eigenvalues at -2.

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Answer:

I believe the answer is 100

Step-by-step explanation:

The distance between the two spheres is 6 - √5 units.

To find the distance between the spheres x² + y² + z² = 4 and x² + y² + z² = 8x + 8y + 8z - 47, first rewrite the second equation:

x² - 8x + y² - 8y + z² - 8z = -43

Now, complete the squares for x, y, and z terms:

(x - 4)² - 16 + (y - 4)² - 16 + (z - 4)² - 16 = -43

Combine the constants:

(x - 4)² + (y - 4)² + (z - 4)² = 5

Now, we have two spheres with centers (0, 0, 0) and (4, 4, 4) and radii 2 (from √4) and √5 (from √5), respectively. To find the distance between the spheres, subtract their radii from the distance between their centers:

Distance = √[(4 - 0)² + (4 - 0)² + (4 - 0)²] - 2 - √5
Distance = √(64) - 2 - √5
Distance = 8 - 2 - √5

So, the distance between the two spheres is 6 - √5 units.

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Answer:

\(\left(a+4\right)\left(a-5\right)\)

Step-by-step explanation:

Break the expression into groups.

=(a^2+4a)+(−5a−20)

Factor out a from a2+4a:     a(a+4)

Factor out −5 from −5a−20:     −5(a+4)

=a(a+4)−5(a+4)

The minimum required sample size is 47 as it is very important for safety, the 99% confidence interval needs to have a margin smaller than 0.06.

To estimate the mean tensile strength for roof hangers with a 99% confidence interval margin of 0.06, we can use the formula:

Margin of error = z* (standard deviation / sqrt(sample size))

where z is the z-score for the desired confidence level, which for a 99% confidence interval is 2.576.

Plugging in the given values, we get:

0.06 = 2.576 * (0.25 / sqrt(sample size))

Solving for the sample size, we get:

sample size = (2.576 * 0.25 / 0.06)^2

sample size = 89.59

Rounding up to the nearest whole number, the minimum required sample size is 90.

Therefore, we need to take a sample of at least 90 roof hangers to estimate the mean tensile strength with a 99% confidence interval margin of 0.06.

To estimate the mean tensile strength for roof hangers with a 99% confidence interval and a margin of error smaller than 0.06, you'll need to determine the minimum required sample size.

For a 99% confidence level, the Z-score (critical value) is approximately 2.576.

The known standard deviation is 0.25. The desired margin of error is less than 0.06. Use the formula:

Margin of Error = Z-score * (Standard Deviation / sqrt(Sample Size))

Rearrange the formula to find the sample size:

Sample Size = (Z-score * Standard Deviation / Margin of Error)^2

Sample Size = (2.576 * 0.25 / 0.06)^2

Sample Size ≈ 46.24

Since you cannot have a fraction of a sample, round up to the nearest whole number. The minimum required sample size is 47.

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Answer:

Equation: \(x=9\)

Step-by-step explanation:

First graph the points on the coordinate plane, then find the x and y intersections and the slope. Since the slope is 0, and the x intersection is 9, you get \(x=9\).

Answer:

Step-by-step explanation:

You divide 7.42 / 1.75 = the answer is 4.24

Answer:

4.24

Step-by-step explanation:

Step 1:

You move both of them by 2 decimal places.

742/175

Step 2:

Divide them:

             4.24

175⟌742

       -700

           420

          -350

              700

            -700

                   0

The surface area of that part of the plane 10x+7y+z=4 that lies inside the elliptic cylinder \(\frac{x^2}{25} +\frac{y^2}{9}\) is 15π√150 and this can be determined by using the given data.

We are given the two equations are:

10x + 7y + z = 4---------(1)

\(\frac{x^2}{25} +\frac{y^2}{9} =1-------------(2)\)

equation(1) is written as

z = 4 - 10x - 7y-----------(3)

The surface area is given by the equation:

A(S) = ∫∫√[(∂f/∂x)² + (∂f/∂y)² + 1]dA------------(4)

compare equation(4) with equation(3) we get the values of ∂f/∂x and

∂f/∂y

∂f/∂x = -10

∂f/∂y = -7

substitute these values in equation(4)

A(S) = ∫∫√[(-10)² + (-7)² + 1]dA

A(S) = ∫∫√[100 + 49 + 1]dA

A(S) = ∫∫√[150]dA

A(S) = √150 ∫∫dA

Where ∫∫dA is the elliptical cylinder

From the general form of an area enclosed by an ellipse with the formula;

comparing x²/a² + y²/b² = 1  with x²/25 + y²/9 = 1, from that we get the values of a and b

a = 5 and b = 3

So, the area of the elliptical cylinder = πab

Thus;

A(S) = √150 × π(5 × 3)

A(S) = 15π√150

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Answer:

11:3

Step-by-step explanation:

Number of bananas = 11
Number of strawberries = 3
Ounces of bananas to ounces of strawberries:
11:3

The ratio of bananas to strawberries is 11:3.

Ratio basically compares quantities, that means it shows the value of one quantity with respect to the other quantity.

If a and b are two values, their ratio will be a:b,

Given that,

Number of bananas = 11 ounces,

Number of strawberries = 3 ounces

The ratio of bananas to strawberries is 11/3 or 11:3.

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