helpppp please with answer and explanation thank you!!!! ​

Answer:

48 oz container

Step-by-step explanation:

Price per oz if she buys the 32 oz container : $3.84/32=0.12

Price per oz if she buys the 48 oz container :  $5.28/48=0.11

If she buys the 48 oz container, she is only paying $0.11 per oz versus $0.12 per oz for the 32 oz container.

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the total cost for birch plywood needed to cover a room with a width of 23 feet and a length of 25 feet would be $432.

Now, For the total cost of birch plywood needed for the room, we first need to determine the area of the room.

Hence, We get;

A = 23 x 25

A = 575 square feet.

Assuming that the birch plywood comes in 4 x 8 sheets, we can calculate how many sheets we need by dividing the total area by the area of a single sheet as;

= 575 sq. ft. / (4 ft. x 8 ft.)

= 18 sheets

So, you will need 18 sheets of birch plywood to cover the entire room.

Now, The total cost of the plywood is,

18 sheets x $24 per sheet = $432

Therefore, the total cost for birch plywood needed to cover a room with a width of 23 feet and a length of 25 feet would be $432.

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R is an equivalence relation because it is reflexive, symmetric, and transitive. It is not a partial order because it is not anti-symmetric.

1) R is reflexive because for any integer a, 3a - 5a = -2a,

which is even. Therefore, aRa for all integers a.
2) R is not symmetric because if aRb, then 3a - 5b is even, meaning 3b - 5a is odd.

Thus, bRa does not hold in general. For example, if a = 1 and b = 2, then aRb but not bRa since 3(1) - 5(2) = -7 is odd.
3) R is also not anti-symmetric because if aRb and bRa, 3a - 5b is even, and 3b - 5a is even. Adding these two equations, we get 2a - 2b = 2(a - b), which is even.

Therefore, a - b is even, which means that aRb. For example, if a = 3 and b = 2, then aRb and bRa since 3(3) - 5(2) = 1 and 3(2) - 5(3) = -1 are both odd.
4) R is transitive because if aRb and bRc, 3a - 5b and 3b - 5c are both even. Adding these two equations, we get 3a - 5c = 3a - 5b + 3b - 5c, which is even. Therefore, aRc. For example, if a = 2, b = 1, and c = 0, then aRb and bRc since 3(2) - 5(1) = 1 and 3(1) - 5(0) = 3 are both odd, and aRc since 3(2) - 5(0) = 6 is even.
5) R is an equivalence relation because it is reflexive, symmetric, and transitive. It is not a partial order because it is not anti-symmetric.

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There are 127 three-digit numbers that leave a remainder of 4 when divided by 7.

To find the number of 3-digit numbers that leave a remainder of 4 when divided by 7, we need to consider the possible values of the hundreds, tens, and units place digits.

First, let's examine the remainder pattern when dividing numbers by 7:

0 ÷ 7 = 0 remainder 0

1 ÷ 7 = 0 remainder 1

2 ÷ 7 = 0 remainder 2

3 ÷ 7 = 0 remainder 3

4 ÷ 7 = 0 remainder 4

5 ÷ 7 = 0 remainder 5

6 ÷ 7 = 0 remainder 6

7 ÷ 7 = 1 remainder 0

8 ÷ 7 = 1 remainder 1

9 ÷ 7 = 1 remainder 2

...and so on.

From this pattern, we can observe that the remainder repeats every 7 numbers. Therefore, to find the numbers that leave a remainder of 4 when divided by 7, we can start with the first number that satisfies this condition, which is 4, and then add multiples of 7.

The smallest 3-digit number that leaves a remainder of 4 when divided by 7 is 104. The largest 3-digit number is 997. To find the count of numbers in this range that satisfy the condition, we can subtract the first number from the last number and divide by 7:

(997 - 104) / 7 = 893 / 7 = 127

Therefore, there are 127 three-digit numbers that leave a remainder of 4 when divided by 7.

In summary, by examining the remainder pattern and considering the range of 3-digit numbers, we can determine that there are 127 numbers that satisfy the condition of leaving a remainder of 4 when divided by 7.

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Answer:

The area of the shaded region is the difference between the area of the entire polygon and the area of the unshaded part inside the polygon. The area of the shaded part can occur in two ways in polygons.

Step-by-step explanation:

Observe that the area of the unshaded region is equal to the area of the shaded region subtracted from the area of the rectangle, i.e. . ar(Unshaded) = ar(ABCD) – ar(Shaded).

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The volume the frustum having  right circular cone with height 20, lower base radius 22 and top radius 7 is 4580π or 14,388.5 cubic units.

To find the volume of a frustum of a right circular cone, we use the formula:

V = (1/3)πh(R² + r²2 + Rr)

where h is the height of the frustum, R is the radius of the lower base, and r is the radius of the top base.

In this case, h = 20, R = 22, and r = 7. Plugging these values into the formula, we get:

V = (1/3)π(20)(22² + 7²+ 22*7)
V = (1/3)π(20)(484 + 49 + 154)
V = (1/3)π(20)(687)
V = (1/3)(20π)(687)
V = 4580π or 14,388.5 cubic units.

Therefore, the volume of the frustum of the right circular cone is approximately 4566.67π cubic units.

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As per the given degrees, the temperature recorded by Mari is c.-5.4 °C

In the given question, it is required to determine the temperature that is less than both -3.4 degree Celsius and 1.6 degree Celsius in order to solve this issue. Since both of these temperatures are lower than -5.4 degrees Celsius, they can be visualised as a number line. Since -5.4 is visible to the left of both -3.4 and 1.6, it is clear that this temperature is lower than both of Mari's recorded readings.

Since value of -2.6 is closer to value of -3.4 than it is to 1.6, thus it is not the solution. Furthermore, 3.9 is not the solution because it is bigger than both -3.4 and 1.6. Since 0 is not less than either of the two temperatures that Mari had reported, it is in range of -3.4 and 1.6. The value of -5.4 is the solution since it is smaller than both other values of -3.4 and 1.6.

Complete Question:

Mari used a thermometer to record temperatures of −3. 4° Celsius and 1. 6° Celsius. Which temperature in degrees Celsius is less than both of the temperatures Mari recorded?

a. 2.6 °C

b. 3.9 °C

c.-5.4 °C

d. 0 °C

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By answering the presented question, we may conclude that As a result, exponential growth after 9 years, we may expect  [tex]768[/tex] banana plants to be infected.

The exponential function formula is f(x)=abx, where a and b are positive real values. Draw exponential functions for various values of a and b using the tools provided below.

We may use the exponential growth formula to address this problem:

[tex]A = P(1 + r)^t[/tex]

where A denotes the total number of infected banana plants after t years

P denotes the initial number of infected banana plants.

r denotes the yearly growth rate in decimal form.

t denotes the number of years

In this instance, we have:

[tex]P = 476 \sr = 0.05 \st = 9[/tex]

When we enter these values, we get:

[tex]A = 476(1 + 0.05)^9 \sA \approx 768.44[/tex]

When we round this up to the next full number, we get:

[tex]A \approx 768[/tex]

Therefore, after 9 years, we may expect 768 banana plants to be infected.

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After calculating the surface area of cylindrical box  of height 8 inches and radius 9 inches we came to know we need approximately 960.84 square inches of wrapping paper to cover the top, bottom, and all the way around the cylindrical box.

Surface area is the total area that the surface of an object occupies. It includes all the faces, sides, and tops of the object. It is measured in square units and is used to calculate the amount of material needed to cover the object.

Radius is the distance from the center of a circle to any point on its circumference, which is half the diameter.

Height refers to the measurement of how tall an object or person is from its base to its highest point.

According to the given information :

To find the amount of wrapping paper needed to wrap a cylindrical box with a height of 8 inches and radius of 9 inches, we will use the formula for the surface area of a cylinder:

A = 2πr² + 2πrh

Where A is the surface area, r is the radius of the circular base of the cylinder, and h is the height of the cylinder.

Plugging in the given values, we get:

A = 2π(9)² + 2π(9)(8)

A = 2π(81) + 2π(72)

A = 162π + 144π

A = 306π

Therefore, the surface area of the cylindrical box is 306π square inches. If we use the approximation of π as 3.14, we get:

A ≈ 306(3.14)

A ≈ 960.84

So, we need approximately 960.84 square inches of wrapping paper to cover the top, bottom, and all the way around the cylindrical box.

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Answer:

Step-by-step explanation:

1.

In the diagram, "h" represents the height of the tower, "d" represents the original distance between the man and the tower before he walked 42m, and "x" represents the distance the man still has to walk to get to the base of the tower.Using trigonometry, we can write two equations based on the two angles of elevation:tan(40°) = h / (d + 42)tan(50°) = h / dWe want to solve for x, so we need to eliminate "h" from these equations. To do that, we can isolate "h" in each equation:h = (d + 42) tan(40°)h = d tan(50°)Now we can set these two expressions equal to each other:(d + 42) tan(40°) = d tan(50°)Simplifying and solving for "d", we get:d = 42 / (tan(50°) - tan(40°))Now that we know "d", we can find "x" by subtracting 42 from it:x = d - 42Plugging in the values and using a calculator, we get:d = 78.39x = 78.39 - 42 = 36.39Therefore, the man has to walk an additional 36.39 meters to get to the base of the tower.

p.s if you want the others seperate them, or find someone else.

Step-by-step explanation:

five loaves of bread ands three tins of sardines cost N350.00 while two loaves of bread with

two tins of sardine cost N180.00 what is the cost of three loaves of bread and three tins of sardine

The amount Jack charges for mowing and raking leaves is $18 and $8 respectively.

Let

charge of mowing = x

charge of raking = y

4x + 10y = 152

6x + 8y = 172

Multiply (1) by 6 and (2) by 4

24x + 60y = 912

24x + 32y = 688

Subtract to eliminate x

60y - 32y = 912 - 688

28y = 224

divide both sides by 28

y = 224/28

y = 8

Substitute into (1)

4x + 10y = 152

4x + 10(8) = 152

4x + 80 = 152

4x = 152 - 80

4x = 72

divide both sides by 4

x = 72/4

x = 18

Therefore, $18 is charged for mowing and $8 is charged for raking.

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Answer:

False

Step-by-step explanation:

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Answer:

False

Step-by-step explanation:

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By the Integral Test, the series sum_(n=1)^infinity n e⁻³ⁿ also diverges.

Using the Integral Test, we can evaluate the convergence of the series sum_(n=1)^infinity n e⁻³ⁿ.

We can set up the integral as ∫(x=1 to infinity) xe⁻³ˣ dx. Using integration by parts, we can solve the integral as [(-x/3) - (1/9)e⁻³ˣ] from 1 to infinity. Plugging in infinity, we get (-∞/3) - (1/9)e⁻infinity, which is -∞. Therefore, the integral diverges and by the Integral Test, the series sum_(n=1)^infinity n e⁻³ⁿ also diverges.

The Integral Test is a method used to evaluate the convergence of an infinite series by comparing it to an improper integral. The basic idea is that if the integral of the function used to define the series converges, then the series also converges. If the integral diverges, then the series also diverges.

In this case, we set up the integral as ∫(x=1 to infinity) xe⁻³ˣ dx and solved it using integration by parts. When we plugged in infinity, we got -∞, which means the integral diverges.

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Both T-statistics and F-statistics are formed using ratios that involve the difference between two means or sums of squares and their respective variances or mean squares. The top part of the equation for both T-statistics and F-statistics represents the difference between the sample means or the sum of squares due to the factor or regression, while the bottom part of the equation contains the standard error or mean square error, which represents the variability or error in the data. The T-statistic is used to test the significance of the difference between two means, while the F-statistic is used to test the overall significance of a linear model or the equality of variances between two or more groups. Therefore, both ratios provide information about the relative magnitude of the difference between groups or the explanatory power of the model, compared to the variability or error in the data.
T-statistics and F-statistics are both formed using ratios, with each ratio serving to compare different sources of variation in the data.

Similarities in form and meaning:
1. Both are used for hypothesis testing.
2. Both ratios have a numerator (top part) and a denominator (bottom part).
3. The resulting values for both are compared to a critical value, which is determined based on a chosen significance level.

For T-statistics, the ratio is formed as follows:
T = (Sample mean - Population mean) / (Sample standard deviation / sqrt(Sample size))

For F-statistics, the ratio is formed as follows:
F = (Between-group variance) / (Within-group variance)

In both cases, the numerator represents the effect or difference of interest, while the denominator represents an estimate of variability or error. In the T-statistic, the top part contains the difference between the sample mean and the population mean, while the bottom part contains the standard error of the mean. In the F-statistic, the top part contains the between-group variance, which represents the variability between different groups, while the bottom part contains the within-group variance, representing the variability within each group.

In summary, T-statistics and F-statistics both use ratios to compare sources of variation in data, with the top part of the equation representing the effect or difference of interest, and the bottom part representing an estimate of variability or error.

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The two power series solutions of the given differential equation about the ordinary point x=0 are:

y1 = 1 - (1/2)x^2 + (1/24)x^4 - (1/720)x^6 + ...
y2 = x - (1/2)x^2 - (1/16)x^4 - (1/240)x^6 - ...

To use the procedure in Example 8 in Section 6.2 to find two power series solutions of the given differential equation about the ordinary point x=0, we first need to find the coefficients of the power series solutions y1 and y2.

For y1, we have:

y1 = 1 + (1/2)x^2 + (1/6)x^3 + ...

To find the coefficients of y1, we differentiate the power series term by term and substitute into the differential equation:

y'' + exy' - y = 0
2(1/2)(1) + ex(2/2)x + (1/2)(1/2)x^2 + (1/6)x^3 + ... - (1 + (1/2)x^2 + (1/6)x^3 + ...) = 0

Simplifying and collecting like terms, we get:

ex + (1/2)x^2 + (1/6)x^3 + ... = 0

Since ex is an exponential function that cannot be expressed as a power series, we can ignore it in this case. Therefore, we get:

(1/2)x^2 + (1/6)x^3 + ... = 0

Solving for the coefficients, we get:

a1 = 0
a2 = -1/2
a3 = 0
a4 = 1/24
a5 = 0
a6 = -1/720
...

Therefore, y1 = 1 - (1/2)x^2 + (1/24)x^4 - (1/720)x^6 + ...

For y2, we have:

y2 = x + (1/2)x^2 + (1/6)x^3 + (1/24)x^4 + ...

To find the coefficients of y2, we differentiate the power series term by term and substitute into the differential equation:

y'' + exy' - y = 0
2(1/2)x + ex(1 + x) + (1/2)x^2 + (1/6)x^3 + (1/24)x^4 + ... - (x + (1/2)x^2 + (1/6)x^3 + (1/24)x^4 + ...) = 0

Simplifying and collecting like terms, we get:

ex + x^2 + (1/6)x^3 + ... = 0

Since ex is an exponential function that cannot be expressed as a power series, we can ignore it in this case. Therefore, we get:

x^2 + (1/6)x^3 + ... = 0

Solving for the coefficients, we get:

b1 = 0
b2 = -1/2
b3 = 0
b4 = -1/16
b5 = 0
b6 = -1/240
...

Therefore, y2 = x - (1/2)x^2 - (1/16)x^4 - (1/240)x^6 - ...

Thus, the two power series solutions of the differential equation about the ordinary point x=0 are:

y1 = 1 - (1/2)x^2 + (1/24)x^4 - (1/720)x^6 + ...
y2 = x - (1/2)x^2 - (1/16)x^4 - (1/240)x^6 - ...

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Sure, I can walk you through the steps to calculate the final price with tax on those shorts.

Here are the steps:

1. The original price of the shorts is $58.

2. BC charges Provincial Sales Tax (PST) at a rate of 7%. 7% of $58 is $4.06.

3. The PST amount is $4.06

4. The price after PST is $58 + $4.06 = $62.06

5. You also need to add Federal Goods and Services Tax (GST) of 5%. 5% of $62.06 is $3.10.

6. The final price with GST added is $62.06 + $3.10 = $65.16

So the final price of the $58 shorts with 7% PST and 5% GST in BC will be $65.16

Let me know if you have any other questions! I'm happy to help explain the steps.

Answer:

Step-by-step explanation:

The final price would be calculated as follows:

- First, calculate the total tax rate by adding the PST and GST: 7% + 5% = 12%

- Next, calculate the amount of tax to be paid on the shorts by multiplying the original price by the tax rate: $58.00 x 12% = $6.96

- Finally, add the tax amount to the original price to get the final price: $58.00 + $6.96 = $64.96

Therefore, the final price for a $58.00 pair of shorts sold in BC with 7% PST and 5% GST would be $64.96.

Confidence in our estimation is important for building confidence in our findings and ultimately our confidence in ourselves.

To find the critical value t* for each situation, we need to look at Table D or use a t-distribution applet.

A) For a 90% confidence interval with n=9, we would look at the row with 8 degrees of freedom (df) in Table D and find the column that contains the closest value to 0.05 (half of the 10% level). The critical value is 1.833.

B) For a 90% confidence interval with n=36, we would look at the row with 35 df and find the column that contains the closest value to 0.05. The critical value is 1.690.

C) For a 90% confidence interval with a sample size of 36 (without knowing the population standard deviation), we would use the same critical value as in part B (1.690) because we would estimate the standard deviation using the sample standard deviation.

D) As we increase the confidence level, the critical value t* increases as well, making the margin of error larger. As we increase the sample size, the critical value t* decreases, making the margin of error smaller. These relationships illustrate that a larger sample size and a higher level of confidence increase our confidence in the accuracy of our estimate, but they also increase the range of values within which the true population mean may lie. Therefore, it is critical to carefully choose the appropriate confidence level and sample size based on the research question and available resources. Confidence in our estimation is important for building confidence in our findings and ultimately our confidence in ourselves.

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Answer:

84 inches

Step-by-step explanation:

7 x 12 = 84  There are 12 inches in a foot.

Helping in the name of Jesus.

Answer: 84 inches.

Step-by-step explanation:

Since 1 foot = 12 inches, we can multiply 7 feet by 12 inches to get 84 inches.

Answer:

148.3° and 31.7°

Step-by-step explanation:

Supplementary angles add up to 180°

Let x represent the angle we are trying to find

Let y represent the supplementary angle of x

We have x + y = 180  [1]

We are given
x - y = 116.6  [2]

Add both equations. [1] + [2]

x + y + x - y = 180 + 116.6

2x = 296.6

x = 296.6 / 2 = 148.3

y = 180 - 148.3 = 31.7

Therefore the two angles measure 148.3° and 31.7°

Check:
148.3 - 31.7 = 116.6

We can observe that ET is the permutation matrix that reverses the permutation of columns performed by E.

Firstly, to generate the permutation matrix E defined in (2), we need to enter the commands provided in the example. This can be done in MATLAB by simply copying and pasting the commands into the command window.

Once we have the permutation matrix E, we can generate a 5 x 5 matrix A with integer entries using the command A = floor(10*rand(5)). This command generates a matrix A with random integers between 0 and 10.

Next, we need to compute the product EA and compare the answer with the matrix A. The product EA is computed in MATLAB by typing E*A. The resulting matrix is related to A by a permutation of its rows. Specifically, the rows of A are rearranged according to the permutation matrix E.

Left multiplication by the permutation matrix E has the effect of permuting the rows of the matrix A. Specifically, the ith row of A is replaced by the row of A corresponding to the ith row of E.

Similarly, we can compute the product AE and compare the answer with the matrix A. The product AE is computed in MATLAB by typing A*E. The resulting matrix is related to A by a permutation of its columns. Specifically, the columns of A are rearranged according to the permutation matrix E.

Right multiplication by the permutation matrix E has the effect of permuting the columns of the matrix A. Specifically, the ith column of A is replaced by the column of A corresponding to the ith column of E.

Moving on to part (b) of the question, we need to compute E-1 and ET. The inverse of the permutation matrix E can be computed in MATLAB using the command inv(E). The transpose of the permutation matrix E can be computed using the command E'.

Observing E-1 and ET, we can see that they are also permutation matrices. This is because the inverse of a permutation matrix is also a permutation matrix, and the transpose of a permutation matrix is also a permutation matrix.

Furthermore, we can observe that E-1 is the permutation matrix that reverses the permutation of rows performed by E. Specifically, the ith row of A is replaced by the row of A corresponding to the ith row of E-1.

Similarly, we can observe that ET is the permutation matrix that reverses the permutation of columns performed by E. Specifically, the ith column of A is replaced by the column of A corresponding to the ith column of ET.

Overall, we can conclude that permutation matrices are a powerful tool in linear algebra, allowing us to manipulate the rows and columns of a matrix in a precise and structured manner.

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1.) complementary angles add up to 90°

2.) supplementary angles add up to 180°

3.)The three angle measures = 32°,28°,30°

The complementary angles are those angles that sums up to 90° while supplementary angles are those angles that sums up to 180°.

For question 3.)

The three angles as

re complementary angles that sums up to 90°

that is;

X+3+X-1+X+1 = 90°

3x +3 = 90

3x = 90-3

3x = 87

X = 87/3

X = 29

The three angle measures = 32°,28°,30°

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Thus, the area of the rectangular playground is found as 1500 sq. ft.

A parallelogram with four opposing, parallel, congruent sides is referred to as a rectangle. The rectangle's corners are at a right angle. The fact that a rectangle's sides are not all equal is the only distinction between it and a square.

A two-dimensional shape's area is the interior blank space. The quantity of space that a shape occupies is another way to define area. When calculating a rectangle's area, we multiply the length by the width of a rectangle.

Given that-

P = 2(l + w)

160 = 2 (50 + w)

80 = 50 +w

w = 80 - 50

w = 30 feet

area = length* width

area = 50*30

area = 1500 sq. ft

Thus, the area of the rectangular playground is found as 1500 sq. ft.

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Correct question-

The perimeter of the playground shown is 160 feet. Find the area.

Length is 50 ft.

Answer:  25% gain

Step-by-step explanation:

math :)

Answer:

1. Since ABCD is a rhombus, all sides are congruent.

2. Since △ACB ≅ △DBC, ∠ACB ≅ ∠DBC.

3. Since opposite angles of a parallelogram are congruent, ∠ABC ≅ ∠DCB.

4. Since ∠ACB ≅ ∠DBC and ∠ABC ≅ ∠DCB, then ∠ACB + ∠ABC = ∠DBC + ∠DCB.

5. Since the sum of the angles in a triangle is 180°, then ∠ACB + ∠ABC = 180° and ∠DBC + ∠DCB = 180°.

6. Therefore, ABCD is a rectangle.

7. Since ABCD is both a rhombus and a rectangle, it must be a square.

To find the coordinate matrix of x in Rn relative to the standard basis, we need to express x as a linear combination of the standard basis vectors. In R2, the standard basis vectors are e1 = (1,0) and e2 = (0,1).

We can write x as:

x = 7(1,0) - 6(0,1)

This means that the coordinate matrix of x in R2 relative to the standard basis is:

[x] = [7 -6]

Note that the first column corresponds to the coordinate of x with respect to e1, and the second column corresponds to the coordinate of x with respect to e2.

To find the coordinate matrix of the vector x in R^n relative to the standard basis, you simply need to represent the vector x as a column matrix using its given components. In this case, x = (7, -6), so the coordinate matrix of x relative to the standard basis is:

[ 7 ]
[ -6 ]

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The means and variances of the given quantities are.

a. E(X+Y) = 4, Var(X+Y) = 6

b. E(X-Y) = -2, Var(X-Y) = 3

c. E(3X+2Y) = 9, Var(3X+2Y) = 29

d. E(5Y-2X) = 13, Var(5Y-2X) = 21

We can use the following properties of means and variances of linear combinations of random variables

If a and b are constants and X and Y are random variables, then E(aX+bY) = aE(X) + bE(Y).

If X and Y are independent random variables, then Var(X+Y) = Var(X) + Var(Y).

If X and Y are independent random variables and a and b are constants, then Var(aX+bY) = a^2Var(X) + b^2Var(Y).

Using these properties, we can find the means and variances of the given quantities:

a. X+Y

E(X+Y) = E(X) + E(Y) = 1 + 3 = 4

Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y) = 2^2 + 1^2 + 2(0.5)(2)(1) = 6

b. X-Y

E(X-Y) = E(X) - E(Y) = 1 - 3 = -2

Var(X-Y) = Var(X) + Var(Y) - 2Cov(X,Y) = 2^2 + 1^2 - 2(0.5)(2)(1) = 3

c. 3X+2Y

E(3X+2Y) = 3E(X) + 2E(Y) = 3(1) + 2(3) = 9

Var(3X+2Y) = 3^2Var(X) + 2^2Var(Y) + 2(3)(2)(0.5) = 29

d. 5Y-2X

E(5Y-2X) = 5E(Y) - 2E(X) = 5(3) - 2(1) = 13

Var(5Y-2X) = 5^2Var(Y) + 2^2Var(X) - 2(5)(2)(0.5) = 21

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Therefore, a basis for the column space (image) of P is given by:
{[1; 0; 0], [0; 0; 1]}

The orthogonal projection onto the xz-plane in R3 can be represented by the transformation matrix

[tex]P = [1 0 0; 0 0 0; 0 0 1].[/tex]

To find the kernel and image of this transformation, we can solve for the null space and column space of P.

Null space (kernel) of P:
To find the null space of P, we need to solve the equation Px = 0. This is equivalent to the system of equations:
x1 = 0
x3 = 0
where [tex]x = [x_1; x_2; x_3][/tex]is a vector in R3. The solutions to this system form the kernel of P. We can see that any vector in the xz-plane will satisfy this system since x2 can take any value. Therefore, a basis for the kernel is given by:
{[0; 1; 0]}

Column space (image) of P:
To find the column space of P, we need to determine the span of its columns. Since the second column of P is zero, we only need to consider the first and third columns. These are the standard basis vectors for R3 in the xz-plane. Therefore, a basis for the column space (image) of P is given by:
{[1; 0; 0], [0; 0; 1]}

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Given;

f(x):3x5 -20x3 in the interval I-1,2]
The critical numbers for the function f(x) = 3x^5 - 20x^3 in the interval [-1, 2] are x = 0. The maximum value of f on the interval is 96, and the minimum value of f on the interval is -23.

To find the critical numbers for the function f(x) = 3x^5 - 20x^3 in the interval [-1, 2], follow these steps:

1. Find the derivative of the function:
f'(x) = 15x^4 - 60x^2

2. Set the derivative equal to zero and solve for x to find critical numbers:
15x^4 - 60x^2 = 0
x^2(15x^2 - 60) = 0
x^2(5x^2 - 20) = 0

Critical numbers are x = 0, x = ±2√2.

3. Check which critical numbers are within the given interval [-1, 2]:
Only x = 0 is within the interval.

4. Evaluate the function at the endpoints and critical numbers:
f(-1) = 3(-1)^5 - 20(-1)^3 = -23
f(0) = 0
f(2) = 3(2)^5 - 20(2)^3 = 96

5. Determine the maximum and minimum values on the interval:
The maximum value of f on the interval is 96, which occurs at x = 2.
The minimum value of f on the interval is -23, which occurs at x = -1.

The critical numbers for the function f(x) = 3x^5 - 20x^3 in the interval [-1, 2] are x = 0. The maximum value of f on the interval is 96, and the minimum value of f on the interval is -23.

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All the correct values are,

1) 3π/2

2) 11π/18

3) 31π/12

4)  150 degree

5)  135 degree

6)  540 degree

7) tan 90° = ∞

8|) sin 7π/6 = - 1/2

We can change the value degree to radian as;

1) 270°

⇒ 270° × π/180

⇒ 3π/2

2) 110°

⇒ 110 × π/180

⇒ 11π/18

3) 315°

⇒ 315 × π/180

⇒ 63π/36

⇒ 31π/12

We can change the value radian to degree as;

4) 5π/6

⇒ 5π/6 × 180 /π

⇒ 5×180 / 6

⇒ 5 × 30

⇒ 150 degree

5) 3π/4

⇒ 3π/4 × 180/π

⇒ 3 × 180 / 4

⇒ 3 × 45

⇒ 135°

6) 3π

⇒ 3π × 180 / π

⇒ 540°

The exact value of trig function are,

7) tan π/2

⇒ tan 90° = ∞

8) sin 7π/6

⇒ sin 210°

⇒ - 1/2

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The general solution to the differential equation is:

[tex]x = c_1 ~s^{(-1 + \sqrt{1 - 4s^2})/2} + c_2 ~s^{(-1 - \sqrt{1 - 4s^2})/2}[/tex]

We have,

Starting from the differential equation for the aging spring:

m d²x/dt² + α dx/dt + kx = 0

We can substitute s = (2/α) x √(k/m) - (α/2)  x t to obtain:

dx/dt = dx/ds x ds/dt = (dx/ds) x (-α/2) x (1/√(k/m))

d²x/dt² = d/dt (dx/dt) = (d/ds) x (dx/dt) x (ds/dt) = (d²x/ds²) x (α²/4km)

Substituting these expressions for dx/dt and d²x/dt² into the original differential equation and simplifying, we obtain:

s² d²x/ds² + s d/ds(x) + s² x = 0

This is the differential equation in terms of the new variable s.

To find the general solution, we assume a solution of the form x = [tex]s^n[/tex].

Substituting this into the differential equation, we obtain:

s² d²/ds² ([tex]s^n[/tex]) + s d/ds ([tex]s^n[/tex]) + s² [tex]s^n[/tex] = 0

Simplifying and dividing through by [tex]s^n[/tex], we get:

n (n - 1) + n + s²  = 0

This is a quadratic equation in n, which has the solutions:

n = (-1 ± √(1 - 4s²))/2

Therefore,

The general solution to the differential equation is:

[tex]x = c_1 ~s^{(-1 + \sqrt{1 - 4s^2})/2} + c_2 ~s^{(-1 - \sqrt{1 - 4s^2})/2}[/tex]

where [tex]c_1 ~and ~c_2[/tex] are constants of integration.

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