please show all steps :)
For the following system: Determine how, if at all, the planes intersect. If they do, determine the intersection. [2T/3A] 2x + 2y + z - 10 = 0 5x + 4y - 4z = 13 3x – 2z + 5y - 6 = 0

From comparing three investment accounts offering different rates, Account A will give Grady at least a 5% annual yield. Therefore, the correct option is option 1.

To determine which investment account will give Grady at least a 5% annual yield, we will need to calculate the Annual Percentage Yield (APY) for each account and compare them. Here are the given terms for each account:

Account A: APR of 4.95%, compounding monthly

Account B: APR of 4.85%, compounding quarterly

Account C: APR of 4.75%, compounding daily

1: Use the APY formula:

APY = (1 + r/n)^(nt) - 1

where r is the annual interest rate (as a decimal), n is the number of compounding periods per year, and t is the number of years.

2: Calculate APY for each account.

Account A:
APY = (1 + 0.0495/12)^(12*1) - 1
APY ≈ 0.0507 or 5.07%

Account B:
APY = (1 + 0.0485/4)^(4*1) - 1
APY ≈ 0.0495 or 4.95%

Account C:
APY = (1 + 0.0475/365)^(365*1) - 1
APY ≈ 0.0493 or 4.93%

3: Compare the APYs to determine which account(s) meet the 5% annual yield requirement.

Based on the calculations, Account A has an APY of 5.07%, which is greater than the 5% annual yield requirement. Therefore, Account A will give Grady at least a 5% annual yield.

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The probability that exactly five of the students were born on a weekend is 0.1514.

Assuming that the probability of being born on a weekend is the same for all students,

we can model the number of students born on a weekend as a binomial random variable with parameters n = 7 (number of trials) and p = 2/7 (probability of success, i.e., being born on a weekend).

The probability of exactly five students being born on a weekend can be calculated using the binomial probability formula:

P(X = 5) = (7 choose 5) * (2/7)^5 * (5/7)^2

where (7 choose 5) = 7! / (5! * 2!) is the number of ways to choose 5 out of 7 students.

Evaluating this expression gives:

P(X = 5) = (7 choose 5) * (2/7)^5 * (5/7)^2

= 21 * (0.0408) * (0.1837)

= 0.1514 (rounded to four decimal places)

Therefore, the probability that exactly five of the seven students were born on a weekend is approximately 0.1514.

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The limit of (69²-24) as x approaches infinity is equal to infinity.

As x approaches infinity, the value of (69²-24) becomes very large, and it goes to infinity. Therefore, the limit of (69²-24) as x approaches infinity is infinity.

B) The limit of (4a+2)/(2-a) as a approaches 2 from the left is equal to -6 and as a approaches 2 from the right is equal to 6.

As a approaches 2 from the left, the denominator (2-a) approaches zero from the negative side, and the numerator (4a+2) approaches -6. Therefore, the limit of (4a+2)/(2-a) as a approaches 2 from the left is -6.

As a approaches 2 from the right, the denominator (2-a) approaches zero from the positive side, and the numerator (4a+2) approaches 6. Therefore, the limit of (4a+2)/(2-a) as a approaches 2 from the right is 6.

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An equation that models this relationship include the following: C. t = 5d.

In Mathematics, a proportional relationship produces equivalent ratios and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) for the data points contained in the table as follows:

Constant of proportionality, k = y/x = t/d

Constant of proportionality, k = 5/1

Constant of proportionality, k = 5.

Therefore, the required equation is given by;

t = kd

t = 5d

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The derivative of y=f(x) = x⁴ - 7x + 5 is dy/dx = 4x³ - 7. For x = 5 and Δx=0.2, dy = 1.986 and Δy = -54.5504.

Given the function y = f(x) = x⁴ - 7x + 5, we can find its derivative with respect to x using the power rule of differentiation:

dy/dx = d/dx(x⁴) - d/dx(7x) + d/dx(5) = 4x³ - 7

Now, we can use the given value of x = 5 and Δx = 0.2 to find the values of dy and Δy:

dy = (4x³ - 7) dx, evaluated at x = 5 and Δx = 0.2

dy = (4(5)³ - 7) (0.2) = 198.6 × 10^(-2)

This means that a small change of 0.2 in x results in a change of about 1.986 in y.

To find Δy, we use the formula:

Δy = f(x + Δx) - f(x)

Substituting x = 5 and Δx = 0.2, we get:

Δy = ((5 + 0.2)⁴ - 7(5 + 0.2) + 5) - (5⁴ - 7(5) + 5)

Simplifying this expression gives:

Δy = (122.4496 - 177) = -54.5504

This means that a small change of 0.2 in x results in a change of about -54.5504 in y.

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A model car is drawn at a scale of 21 to 1. If the model car is 9. 2in.  The length of the actual car in feet is approximately 0.7665 feet.

Find out the length of the actual car in feet, we need to first convert the length of the model car from inches to feet.
9.2 inches = 0.767 feet (divide by 12 since there are 12 inches in a foot)
Now, we can use the scale of 21 to 1 to find the length of the actual car in feet.
21 units on the model car = 1 unit on the actual car
So,
1 unit on the actual car = 0.767 feet / 21 = 0.0365 feet
Find the length of the actual car, we can multiply the scale ratio by the length of the model car in units:
21 units x 0.0365 feet per unit = 0.7665 feet
Therefore, the length of the actual car in feet is approximately 0.7665 feet.

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The actual car is  0.7665 feet long.

First, we need to convert the length of the model car from inches to feet:

9.2 in. = 9.2/12 ft. = 0.7667 ft.

Next, we can use the scale to find the length of the actual car:

21 units on the drawing = 1 unit in real life

So, we have:

1 unit in real life = length of actual car

21 units on the drawing = length of model car

Substituting the values we have:

1 unit in real life = (0.7667 ft.)/21 = 0.0365 ft.

Therefore, the length of the actual car is:

1 unit in real life x 21 = 0.0365 ft. x 21 = 0.7665 ft.

So, the actual car is approximately 0.7665 feet long.

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Answer:

Step-by-step explanation:

Let's call the two numbers x and y. We know that:

x + y = 30 (since the sum of the two numbers is 30)

We want to find the values of x and y that maximize their product, which is given by:

P = xy

To solve for x and y, we can use the fact that the sum of the two numbers is 30, so we can rewrite one of the numbers in terms of the other:

y = 30 - x

Substituting this into the equation for the product, we get:

P = x(30 - x)

Expanding this expression, we get:

P = 30x - x^2

To find the maximum value of P, we can take the derivative of this expression with respect to x and set it equal to zero:

dP/dx = 30 - 2x = 0

Solving for x, we get:

x = 15

So one of the numbers is x = 15, and the other is y = 30 - x = 15.

To confirm that this gives the maximum product, we can take the second derivative of P with respect to x:

d2P/dx2 = -2

Since the second derivative is negative, this means that the function P = 30x - x^2 has a maximum at x = 15.

Therefore, the two numbers are 15 and 15, and their product is maximized at P = 15 * 15 = 225.

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Answer:

The pattern is you are subtracting by 8. The subsequent three terms are 51, 43, and 35.

Step-by-step explanation:

First, you can subtract the first value from the second to find the common difference. Then you continue on this pattern. Simple!

the probability of choosing H or P in either selection is 0.84

Two random letters are selected from the word Happy, and we want to find the probability of choosing H or P in either selection.

There are 5 letters, 1 is an H, 2 are P's.

Then the probability of selecting one of these 3 in the first selection is:

p = 3/5 = 0.6

And if we don't chose any of these in the first selection we had the probability:

q = 2/5 = 0.4 (choosing one of the a's)

the probability of choosing one of the p's or the H in the second is again:

q' = 3/5 = 0.6

The joint probability is:

Q = q*q' = 0.4*0.6 = 0.24

Then the total probability is:

p + Q = 0.6 + 0.24 = 0.84

The correct option is the second one.

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The measure of angle U to the nearest tenth is 39.6°

The cosine Rule says that the square of the length of any side of a given triangle is equal to the sum of the squares of the length of the other sides minus twice the product of the other two sides multiplied by the cosine of angle included between them.

C² = a²+b²-2abcosC

450² = 360²+110²+2(110)(360)cosU

202500 = 129600+ 12100+ 79200cosU

202500 = 141700+79200cosU

79200cosU = 202500-141700

79200cosU = 60800

cos U = 60800/79200

cos U = 0.77

U = 39.6°( nearest tenth)

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The area of ΔDEF, to the nearest 10th of a square inch, is approximately 1439.1 square inches.

To find the area of ΔDEF with given values e = 67 inches, ∠F = 37°, and ∠D = 70°, follow these steps:

Find ∠E using the Triangle Sum Theorem (the sum of the angles in a triangle is always 180°).
∠E = 180° - (∠F + ∠D) = 180° - (37° + 70°) = 180° - 107° = 73°

Use the Law of Sines to find side d.
(sin ∠F) / d = (sin ∠E) / e
(sin 37°) / d = (sin 73°) / 67 inches

Solve for side d.
d = (67 inches * sin 37°) / sin 73°
d ≈ 44.7 inches

Use the formula for the area of a triangle with two sides and the included angle.
Area = 0.5 * d * e * sin ∠D
Area = 0.5 * 44.7 inches * 67 inches * sin 70°
Area ≈ 1439.1 square inches

Thus, the area of ΔDEF, to the nearest 10th of a square inch, is approximately 1439.1 square inches.

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The correlation coefficient (r) for the given data points is approximately 0.9859, indicating a strong positive relationship between the x and y values.

1. First, let's find the mean of the x-values and the y-values. To do this, add all the x-values together and divide by the total number of points (9). Repeat this for the y-values.
Mean of x = (65 + 71 + 79 + 80 + 86 + 86 + 91 + 95 + 100) / 9 ≈ 83.67
Mean of y = (102 + 133 + 144 + 161 + 191 + 207 + 235 + 237 + 243) / 9 ≈ 183.89

2. Next, calculate the deviations of each point from the mean for both x and y.
For example, for the first point (65,102), the deviations are:
x-deviation = 65 - 83.67 ≈ -18.67
y-deviation = 102 - 183.89 ≈ -81.89

3. Then, multiply the x and y deviations for each point and sum the results. Also, square the deviations for both x and y and sum them separately.
Sum of x*y deviations ≈ 47598.73
Sum of squared x deviations ≈ 2678.89
Sum of squared y deviations ≈ 105426.56

4. Finally, calculate the correlation coefficient (r) by dividing the sum of x*y deviations by the square root of the product of the sum of squared x and y deviations.
r = (47598.73) / √(2678.89 * 105426.56) ≈ 0.9859

The correlation coefficient (r) for the given data points is approximately 0.9859, indicating a strong positive relationship between the x and y values.

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The time taken for $400 to amount to $650 at 6% compound interest annually is 8.33 years.

Compound interest is expressed as below:

[tex]A = P(1+\frac{r}{n})^{nt[/tex]

where A is the amount

P is principal

r is the rate of interest

n is the frequency with which interest is compounded per year

t is the time

A = $650

P = $400

r = 0.06

n = 1 because the interest is compounded annually. Thus the frequency of interest compounded per year is 1

650 = 400 [tex](1+0.06)^t[/tex]

1.625 = [tex]1.06^t[/tex]

t = 8.33 years

Thus, it takes 8.33 years for $400 to convert to $650 at 6% compound interest annually.

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To determine if a table or a set of ordered pairs can be modeled by a quadratic function, you should look for the following characteristics:

1. Consistent differences: Examine the differences between consecutive y-values. If there's a constant second difference (i.e., the differences between consecutive first differences remain the same), it's likely that the data can be modeled by a quadratic function.

2. Parabolic shape: Graph the ordered pairs. If the graph resembles a parabola (a U-shaped or inverted U-shaped curve), it indicates that the data can be modeled by a quadratic function.

By analyzing the ordered pairs and their differences, as well as examining the shape of the graph, you can determine if a quadratic function is the best fit for the data.

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The center of the circle lies on the x-axis, the standard form of the equation is (x – 1)² + y² = 3, and the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.

Explanation:

We can rewrite the given equation as (x - 1)² + y² = 9 using completing the square method.

(x² - 2x + 1) + y² - 1 - 8 = 0

(x - 1)² + y² = 9

This is the standard form of the equation of a circle with center (1,0) and radius 3. Therefore, the center lies on the x-axis, and the radius is 3 units.

The circle whose equation is x² + y² = 9 is the equation of a circle with center (0,0) and radius 3, which has the same radius as the given circle.

The shape formed by a solid intersecting with a plane, so the At level X = 3, the area of the vertical cross-section A is 108 square units.

To find the area of the vertical cross section A at the level X = 3, we need to find the equation of the circle when it is intersected by the plane X = 3.
First, let's find the value of y when X = 3 using the equation of the circle x^2 + y^2 = 36:

(3)^2 + y^2 = 36
9 + y^2 = 36
y^2 = 27
y = ±√27

Since we are dealing with a circle, there are two points on the circle at X = 3, which are (3, √27) and (3, -√27).

The distance between these two points will be the base of the triangle, which is also equal to its height (as given in the problem).

Base and height of the triangle: 2 * √27

Now we can find the area A of the vertical cross-section, which is a triangle with equal base and height:

A = 1/2 * base * height
A = 1/2 * (2 * √27) * (2 * √27)
A = 4 * 27
A = 108

So, the area of the vertical cross-section A at the level X = 3 is 108 square units.

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the length, in inches, of the other leg of his triangle is 9. 8inches

Using the Pythagorean theorem which states that the square of the longest leg or side of a given triangle is equal to the sum of the squares of the other two sides of the triangle.

From the information given, we have that;

a² + b² = c² represents the relationship between the three sides of a right triangle

Also,

Hypotenuse side = 11 inches

One of the other side = 5 inches

Substitute the values, we have;

11² = 5² + c²

collect like terms

c² = 121 - 25

Subtract the values

c = √96

c = 9. 8 inches

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Answer:

To determine the best measure of variability for the data, we need to consider the type of data we are dealing with. In this case, the data is numerical and discrete, so the best measure of variability would be the range or the interquartile range (IQR).

The range is the difference between the maximum and minimum values in a dataset, while the IQR is the range of the middle 50% of the data. The IQR is less sensitive to outliers than the range, so it is often a better measure of variability.

To calculate the range and IQR for each player, we first need to order the data:

Player A: 1, 2, 2, 2, 3, 3, 3, 4, 8

Player B: 1, 1, 2, 2, 2, 3, 4, 4, 6

Player A has a range of 8 - 1 = 7, and an IQR of Q3 - Q1 = 4 - 2.5 = 1.5.

Player B has a range of 6 - 1 = 5, and an IQR of Q3 - Q1 = 4 - 1.5 = 2.5.

Therefore, Player B has a higher range and a higher IQR, indicating more variability in their performance. Player A has a lower range and a lower IQR, indicating greater consistency in their performance. Therefore, the answer is: Player A is the most consistent.

a. The critical number of f is undefine

b. The open interval(s) on  f is increasing on  (e,∞) and the open interval(s) on which f is decreasing on  (0,1) and (1,e).

c. The local minimum value(s) is 0 and there's no local maximum value.

d. Concave downward on (0, e^1/2) and concave upward on (e^1/2, ∞).

e. The inflection point(s) of the graph of f is (e^1/2, e(ln e^1/2 - 1)^2).

(a) To find the critical numbers of f, we need to find where the derivative of f is zero or undefined.

f'(x) = 2x ln x + x - 2x = 2x (ln x - 1) = 0

This gives us x = 1 or x = e. However, f'(x) is undefined at x = 0, so we also need to check this point.

(b) To determine the intervals of increase and decrease, we need to test the sign of f'(x) on each interval.

When x < 1, ln x < 0, so ln x - 1 < -1, and f'(x) < 0.

When 1 < x < e, ln x > 0, so ln x - 1 < 0, and f'(x) < 0.

When x > e, ln x > 1, so ln x - 1 > 0, and f'(x) > 0.

Therefore, f is decreasing on (0,1) and (1,e), and increasing on (e,∞).

(c) To find the local minimum and maximum values, we need to check the critical points and the endpoints of the intervals.

f(1) = 0 is a local minimum.

f(e) = e^2 (ln e - 1) = e^2 (1 - 1) = 0 is also a local minimum.

(d) To find the intervals of concavity, we need to test the sign of f''(x) on each interval.

f''(x) = 2 ln x - 1

When x < e^1/2, ln x < 1/2, so f''(x) < 0, and f is concave downward on (0, e^1/2).

When x > e^1/2, ln x > 1/2, so f''(x) > 0, and f is concave upward on (e^1/2, ∞).

(e) To find the inflection points, we need to find where the concavity changes.

f''(x) = 0 when ln x = 1/2, or x = e^1/2.

Therefore, the inflection point is (e^1/2, f(e^1/2)) = (e^1/2, e(ln e^1/2 - 1)^2).

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the sequence 1, 4, 16, ... is a geometric sequence with a common ratio of 4.

what is geometric sequence ?

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed number called the common ratio (r).

In the given question,

The sequence 1, 4, 16, ... is geometric.

To determine the common ratio, we divide any term by the previous term. For example:

The ratio between 4 and 1 is 4/1 = 4.

The ratio between 16 and 4 is 16/4 = 4.

Since the ratio is the same for any two consecutive terms, we can conclude that the common ratio is 4.

We can also verify this by using the general formula for a geometric sequence:

aₙ= a₁ * r⁽ⁿ⁻¹⁾

where aₙ is the nth term, a_1 is the first term, r is the common ratio, and n is the term number.

Using the given sequence, we have:

a₁ = 1 (the first term)

a₂ = 4 (the second term)

a₃ = 16 (the third term)

We can use these values to solve for the common ratio:

a₂ / a₁ = r

4 / 1 = r

r = 4

Therefore, the sequence 1, 4, 16, ... is a geometric sequence with a common ratio of 4.

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To calculate d²y/dx², we first need to find the first derivative of y, which is dy/dx. For y = 0.5x^-0.2, we can use the power rule of differentiation, which states that the derivative of x^n is n*x^(n-1). Therefore,

dy/dx = -0.1x^-1.2

To find the second derivative, d²y/dx², we need to differentiate dy/dx again. Using the power rule again, we get:

d²y/dx² = 0.12x^-2.2

This is the second derivative of y with respect to x.

In calculus, a derivative is a measure of how a function changes as its input changes. The second derivative is a measure of how the rate of change of the function itself changes as its input changes. It tells us about the curvature of the function at any given point.

In this case, we have calculated the second derivative of y, which gives us information about the rate of change of the slope of the function. If the second derivative is positive, the function is concave up (curving upward), and if it is negative, the function is concave down (curving downward). If the second derivative is zero, the function has an inflection point (a point where the curvature changes direction).

Overall, the second derivative is a powerful tool in calculus that helps us understand the behavior of functions in more detail.

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Producing 0 washing machines is not a practical solution for a company.

To maximize profit, we need to find the difference between revenue and cost functions, which gives us the profit function P(x):

P(x) = R(x) - C(x) = (0.3x²) - (10,000 + 0.7x²)

Simplify the profit function:

P(x) = -0.4x² + 10,000

Now, to maximize profit, we'll find the critical points by taking the first derivative of P(x) with respect to x:

P'(x) = dP(x)/dx = -0.8x

Set P'(x) to zero and solve for x:

-0.8x = 0
x = 0

Since the profit function P(x) is a quadratic with a negative leading coefficient, the maximum value will occur at the critical point x = 0. However, producing 0 washing machines is not a practical solution for a company.

To maximize profit while producing washing machines, the company should consider other factors beyond the given cost and revenue functions, such as market demand and production capacity.

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Amount after 11 hours: 3,477,373 grams; Amount after 50 hours: 33,320 grams.

The Radioactive decay formula provided in the question for the amount A(t) of a sample of uranium-240 remaining after t hours is:

A(t) = 5600100(3[tex])^(-11/14)[/tex]

To find the amount of the sample remaining after 11 hours, we substitute t = 11 in the formula and calculate:

A(11) = 5600100(3[tex])^(-11/14)[/tex] ≈ 3477373 grams

Therefore, the amount of the sample remaining after 11 hours is approximately 3,477,373 grams (rounded to the nearest gram).

Similarly, to find the amount of the sample remaining after 50 hours, we substitute t = 50 in the formula and calculate:

A(50) = 5600100(3[tex])^(-50/14)[/tex] ≈ 33320 grams

Therefore, the amount of the sample remaining after 50 hours is approximately 33,320 grams (rounded to the nearest gram).

The exponential formula for radioactive decay describes the behavior of a radioactive substance, where the amount of the substance decreases over time as it decays. In this case, uranium-240 has a half-life of 14 hours, which means that half of the initial amount of the substance will decay in 14 hours. After another 14 hours, half of the remaining amount will decay, and so on.

As time goes on, the amount of uranium-240 remaining decreases exponentially, and the rate of decay is determined by the half-life of the substance. The formula provided in the question allows us to calculate the amount of uranium-240 remaining after any given amount of time, based on its initial amount and half-life.

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The radioactive substance uranium-240 has a half-life of 14 hours.  The amount of the sample remaining after 11 hours is approximately 2265 grams, and the amount of the sample remaining after 50 hours is approximately 95 grams.

The formula for the amount of uranium-240 remaining after t hours is given by: A(t) = 5600 * (1/2)^(t/14).
Find the amount of the sample remaining after 11 hours, we substitute t = 11 into the formula and evaluate:
A(11) = 5600 * (1/2)^(11/14)
A(11) ≈ 2265 grams (rounded to the nearest gram)
Find the amount of the sample remaining after 50 hours, we substitute t = 50 into the formula and evaluate:
A(50) = 5600 * (1/2)^(50/14)
A(50) ≈ 95 grams (rounded to the nearest gram)
Therefore, the amount of the sample remaining after 11 hours is approximately 2265 grams, and the amount of the sample remaining after 50 hours is approximately 95 grams.

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Answer:

Congruent, impossible, not congruent.

Step-by-step explanation:

a) Congruent because of AAS congruency.

b) Impossible to tell. There is no congruency rule with 1 angle and 1 side.

c) Not congruent. Sides should not be equal.

The sale price of the product would be Birr 105.80.

If the cost of the product is Birr 92 and the company has a policy of 15% mark-up on the cost, then the sale price can be found by adding 15% of the cost to the cost itself.

To calculate this, we can use the formula:

Sale price = Cost + Mark-up

where the mark-up is 15% of the cost.

Mark-up = 15% of Cost = 0.15 * 92 = Birr 13.80

So, the sale price = Cost + Mark-up = 92 + 13.80 = Birr 105.80.

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Answer:

The largest volume that such a box can have is (3/4)²(3/2)²/4 = 1.6875 m³

Step-by-step explanation:

4. Let the sides of the rectangle be 'l' and 'w', where lw = A, the fixed area. The perimeter P is given by P = 2l + 2w. To minimize P, we need to find the values of 'l' and 'w' that make P as small as possible. Solving the equation for 'w' in terms of 'l' from lw = A, we get w = A/l. Substituting this into the equation for P, we get P = 2l + 2(A/l). Taking the derivative of P with respect to 'l' and setting it to zero, we get 2 - 2A/l² = 0, which implies l = √A. Substituting this value into lw = A, we get w = √A. Therefore, a square with sides of length √A has the minimum perimeter among all rectangles with a fixed area of A.

Let the side length of the square base be 'x' and the height of the box be 'h'. Then the volume of the box is V = x²h = 400. We need to minimize the surface area S of the box, which is given by S = x² + 4xh. Solving the equation for 'h' in terms of 'x' from V = x²h, we get h = 400/x². Substituting this into the equation for S, we get S = x²+ 4x(400/x²) = x² + 1600/x. Taking the derivative of S with respect to 'x' and setting it to zero, we get 2x - 1600/x² = 0, which implies x = 10 cm. Therefore, the dimensions of the box that minimizes the amount of material used are 10 cm x 10 cm x 4 cm.

Let the side length of the square cut out from each corner be 'x', and the height of the box be 'h'. Then the volume of the box is V = x²h. The length and width of the base of the box are (3-2x) and (3-2x) respectively. We need to maximize the volume V of the box subject to the constraint that the length and width of the base are positive. Taking the derivative of V with respect to 'x' and setting it to zero, we get h = (3-2x)²/4. Substituting this into the equation for V, we get V = x²(3-2x)²/4. Taking the derivative of V with respect to 'x' and setting it to zero, we get x = 3/4 m. Therefore, the largest volume that such a box can have is (3/4)²(3/2)²/4 = 1.6875 m³

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Given d = ba, we can write:

d11 = b1a1 + b2a3

d12 = b1a2 + b2a4

d21 = b3a1 + b4a3

d22 = b3a2 + b4a4

To find the product of d, we need to find the values of b and a such that d11 = d12 = d21 = d22.

Let's assume that d11 = d12 = d21 = d22 = x. Then, we have:

b1a1 + b2a3 = x

b1a2 + b2a4 = x

b3a1 + b4a3 = x

b3a2 + b4a4 = x

We can solve for b1, b2, b3, and b4 in terms of a1, a2, a3, and a4:

b1 = (x - b2a3)/a1

b2 = (x - b1a1)/a3

b3 = (x - b4a3)/a1

b4 = (x - b3a1)/a3

Substituting these values of b1, b2, b3, and b4 into the equation d = ba, we get:

d11 = x = a1(x - b1a3)/a1 + a3(x - b2a1)/a3

= x - b1a3 + x - b2a1

= 2x - (b1a3 + b2a1)

d12 = x = a2(x - b1a3)/a1 + a4(x - b2a1)/a3

= (a2/a1)x - (b1a2 + b2a4) + (a4/a3)x - (b1a4 + b2a4)

= x - (b1a2 + b2a4)

d21 = x = a1(x - b3a3)/a1 + a3(x - b4a1)/a3

= (a1/a1)x - (b3a3 + b4a3) + (a3/a3)x - (b3a1 + b4a3)

= x - (b3a1 + b4a3)

d22 = x = a2(x - b3a3)/a1 + a4(x - b4a1)/a3

= (a2/a1)x - (b3a2 + b4a4) + (a4/a3)x - (b3a4 + b4a4)

= x - (b3a2 + b4a4)

We can rewrite these equations in matrix form as:

| 2 -a3-a1 0 0 || x | | b1a3 + b2a1 |

| 0 a2 0 -a4 || | = | b1a2 + b2a4 |

| -a3-a1 0 2 -a1 || | | b3a1 + b4a3 |

| 0 -a4 -a1 2 || | | b3a2 + b4a4 |

To solve for x, we need to invert the matrix on the left and multiply it by the vector on the right:

| x | | 2 -a3-a1 0

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The volume of the given sphere having radius of 4 units is 267.94 units³.

Given the radius of the sphere (r) = 4 units

To find the volume of the given sphere, we have to substitute the radius in the below volume formula of the sphere,

the volume of the sphere = 4/3 * π * r³

the volume of the given sphere = 4/3 * 3.14 * (4)³        

[π  is approximately equal to 3.14]

the volume of the given sphere = 267.94 units³

So from the above analysis, we can conclude that the volume of the sphere having 4 units radius is 267.94 units³.

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Given question is not having complete information, the complete question is written below:

What is the volume of the sphere having 4 units radius?

The expression g - 68 shows how many gumdrops Hanson has left after eating 68 out of g.

To find out how many gumdrops Hanson has left after eating 68 out of g, we need to subtract 68 from g. Therefore, the expression that shows how many gumdrops Hanson has left is:

g - 68

This expression represents the remaining gumdrops after Hanson has eaten 68 out of g. For example, if Hanson had 100 gumdrops before eating 68 of them, then the expression would be:

100 - 68 = 32

Therefore, Hanson would have 32 gumdrops left after eating 68 out of 100.

In summary, the expression g - 68 shows how many gumdrops Hanson has left after eating 68 out of g. The value of g represents the total number of gumdrops Hanson had before eating 68.

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The 95% confidence interval is: b) -1.38 to d) 3.44.

The value of the sample mean difference is 1.74, which falls:

b) within.

Here, we have to determine the 95% confidence interval for the difference of sample means and complete the statements, we need to use the sample mean difference provided and the confidence interval limits given as options.

We'll compare the sample mean difference to the interval to see if it falls within or outside the interval.

Given that the sample mean difference is 1.74, let's analyze the options:

Options for the confidence interval limits:

Lower limit options:

a) -1.26

b) -1.38

c) -3.48

d) -3.44

Upper limit options:

a) 1.26

b) 3.48

c) 1.38

d) 3.44

Since the sample mean difference is 1.74, we need to check if it falls within the interval formed by the lower and upper limits.

Looking at the options for the lower limit, the closest value to 1.74 is -1.38, and the closest value to the upper limit is 3.44.

So, the 95% confidence interval would be:

-1.38 to 3.44

Now, completing the statements:

The 95% confidence interval is: b) -1.38 to d) 3.44

The value of the sample mean difference is 1.74, which falls:

b) within

So, the completed statements are:

The 95% confidence interval is -1.38 to 3.44.

The value of the sample mean difference is 1.74, which falls within the 95% confidence interval.

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