Two bottled waters and an order of cheese costs $5.50. Three bottled waters and two orders of cheese nachos costs $9.50.

Answer:

g(x) = x² - 4x + 7

216 cm2 is the size of the rectangle border.

the translation of the question is

A painting including its frame is 25 cm long and 10 cm wide, what is the area of ​​the frame if it is 4 cm wide?

What is a rectangle's area?

When the dimensions of a rectangle with length and width are multiplied, the area of the rectangle is determined as follows:

A = lw.

The total area is therefore given by:

A = 25 x 10 = 250 cm².

The white region's size is shown by:

A = (25 - 2 x 4) x (10 - 2 x 4) is equal to 17x 2 and 34 cm2.

Hence, the border's area is as follows:

216 cm2 = 250 cm2 - 34 cm2.

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Given, [tex]$$(\sec n - \tan n) = \frac{1}{4}[/tex], so, using Trigonometry we can obtain [tex]$$\sec n + \tan n = 0$$[/tex].

Trigonometry is a branch of mathematics that deals with the study of relationships between the sides and angles of triangles. It involves the study of trigonometric functions such as sine, cosine, and tangent, and their applications to various fields such as engineering, physics, and navigation. Trigonometry helps in solving problems related to triangles, circles, and periodic phenomena such as waves and oscillations.

To find sec n + tan n using the given equation, we can use the following identity:

[tex]$$\sec^2 n - \tan^2 n = 1$$[/tex]

Multiplying both sides of the given equation by sec n + tan n, we get:

[tex]$$(\sec n - \tan n)(\sec n + \tan n) = \frac{1}{4}(\sec n + \tan n)$$[/tex]

Using the identity above, we can simplify the left-hand side of the equation as:

[tex]$$\sec^2 n - \tan^2 n = 1$$[/tex]

Therefore, we can substitute 1 for [tex]sec^2 n - tan^2[/tex] n in the equation above to get:

[tex]$$(\sec n - \tan n)(\sec n + \tan n) = \frac{1}{4}(\sec n + \tan n)$$[/tex]

[tex]$$1(\sec n + \tan n) = \frac{1}{4}(\sec n + \tan n)$$[/tex]

Simplifying further, we get:

[tex]\frac{3}{4} * $$(\sec n + \tan n) = 0[/tex]

Therefore, we can solve for sec n + tan n as:

[tex]$$\sec n + \tan n = \frac{0}{\frac{3}{4}}$$[/tex]

[tex]$$\sec n + \tan n = 0$$[/tex]

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Answer:

576 books.

Step-by-step explanation:

134+254+188=576 books in total.

Hopefully this helps!

Answer:

We can use the formula for compound interest to solve the problem:

A = P(1 + r/n)^(nt)

where A is the final amount, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this case, we know that P = $2,000, A = $6,883.55, n = 4 (quarterly compounding), and t = 16. We can solve for r by rearranging the formula as follows:

r = n[(A/P)^(1/nt) - 1]

Substituting the values, we get:

r = 4[(6,883.55/2,000)^(1/(4*16)) - 1] = 0.0522 or 5.22%

Therefore, the annual interest rate is approximately 5.22%

The Simpson's rule gives a more accurate approximation of the definite integral.

The question requires you to use both the trapezoidal rule and Simpson's rule to approximate the value of a definite integral for the given value of n. Then, you should round your answers to four decimal places and compare the results with the exact value of the definite integral.Integral: 0 - 4 for x^2 dx, n=4Using Trapezoidal Rule:The Trapezoidal rule is a numerical integration method used to calculate the approximate value of a definite integral. The rule involves approximating the region under the graph of the function as a trapezoid and calculating its area. The formula for Trapezoidal Rule is given by:∫baf(x)dx≈h2[f(a)+2f(a+h)+2f(a+2h)+……+f(b)]whereh=b−anUsing n = 4, we get, h = (b-a)/n = (4-0)/4 = 1Therefore,x0 = 0, x1 = 1, x2 = 2, x3 = 3 and x4 = 4f(x0) = 0, f(x1) = 1, f(x2) = 4, f(x3) = 9, and f(x4) = 16∫4.0x^2 dx = (1/2)[f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4)](1/2)[0 + 2(1) + 2(4) + 2(9) + 16] = 37

Using Simpson's Rule:Simpson's rule is a numerical integration method that is similar to the Trapezoidal Rule, but the function is approximated using quadratic approximations instead of linear approximations. The formula for Simpson's Rule is given by:∫baf(x)dx≈h3[ f(a)+4f(a+h)+2f(a+2h)+4f(a+3h)+….+f(b)]whereh=b−an, and n is even.Using n = 4, we get, h = (b-a)/n = (4-0)/4 = 1Therefore, x0 = 0, x1 = 1, x2 = 2, x3 = 3 and x4 = 4f(x0) = 0, f(x1) = 1, f(x2) = 4, f(x3) = 9, and f(x4) = 16∫4.0x^2 dx = (1/3)[f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4)](1/3)[0 + 4(1) + 2(4) + 4(9) + 16] = 20Comparing the results with the exact value of the definite integral, we have:Integral 0 - 4 for x^2 dx = ∫4.0x^2 dx = [x^3/3]4.0 - [x^3/3]0 = 64/3 ≈ 21.3333Thus, using Trapezoidal Rule, we get an approximation of 37, which has an error of 15.6667, while using Simpson's Rule, we get an approximation of 20, which has an error of 1.3333. Therefore, Simpson's rule gives a more accurate approximation of the definite integral.

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Answer:

4[tex]\sqrt{2\\}[/tex]

Step-by-step explanation:

This is a 45 45 90 Triangle meaning that x would be 4 squareroot 2

so the formula goes lets say 4= a and the other side is b

a=b and x=a[tex]\sqrt{2[/tex]

the volume of the rectangular prism with a width of 10 cm, height of 3 cm, and a depth of 7 cm is 210 cubic cm.

The volume of a rectangular prism is calculated by multiplying its length, width, and height. In this case, the width is 10 cm, the height is 3 cm, and the depth is 7 cm. Therefore, the volume of the rectangular prism can be calculated as follows:

Volume = Length x Width x Height

Since the length of the rectangular prism is not given, we cannot calculate the exact volume. However, we can provide a formula that can be used to calculate the volume of any rectangular prism with the given dimensions.

Formula for the volume of a rectangular prism:

Volume = Width x Height x Depth

Substituting the given values, we get:

Volume = 10 cm x 3 cm x 7 cm

Volume = 210 cubic cm

Therefore, the volume of the rectangular prism with a width of 10 cm, height of 3 cm, and a depth of 7 cm is 210 cubic cm.

It is important to note that the unit of measurement used for the dimensions should be the same for all three dimensions in order to obtain the correct volume. In this case, the unit of measurement used is centimeters (cm), and the volume is expressed in cubic centimeters (cm³).

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Using the Poisson Approximation the probability are:

a)  0.6321

b)  0.3679

c) 0.2642

The Poisson distribution is utilized to compute the likelihood of a particular amount of occurrences happening over a set period. The Poisson approximation will be used to answer the given question, and it is a form of a probability distribution that can be used to approximate the probability of particular events that occur infrequently, and it is suitable for both continuous and discrete variables.

a) The probability of having at least one student with the same birthday as Brad using the Poisson Approximation.

Let the number of other students with the same birthday as Brad be represented by x. Here, x is a discrete variable with a Poisson distribution that follows a Poisson distribution with an average of λ, which is equal to 1:

λ = average number of students having the same birthday as Brad = 1.

Using the Poisson distribution formula, the probability of having at least one student with the same birthday as Brad is given by:

P(X >= 1) = 1 - P(X = 0)

= 1 - e ^ (-λ)P(X = 0)

= (e^(-λ))(λ^0) / 0!

= e^(-λ)

= e^(-1)

= 0.3679

Therefore, the probability of having at least one student with the same birthday as Brad is:

P(X >= 1) = 1 - P(X = 0)

= 1 - 0.3679

= 0.6321

b) The probability of having exactly one student with the same birthday as Brad using Poisson Approximation

P(X = 1) = (e^(-λ))(λ^1) / 1!

= e^(-1)(1) / 1!

= e^(-1)

= 0.3679

Therefore, the probability of having exactly one student with the same birthday as Brad is:

P(X = 1)

= e^(-1)

= 0.3679

c) The probability of having at least two students with the same birthday as Brad using Poisson Approximation

P(X >= 2) = 1 - P(X < 2)

= 1 - [P(X = 0) + P(X = 1)]

= 1 - [e^(-λ)(λ^0) / 0! + e^(-λ)(λ^1) / 1!]

= 1 - [e^(-1) + e^(-1)(1) / 1!]

= 1 - [e^(-1) + e^(-1)]

= 1 - 2e^(-1)

= 0.2642

Exact probability: 1 - (364/365)^180 = 0.4406

Poisson Approximation Probability: 0.6321

The exact probability is 0.4406, which is less than the Poisson approximation probability, which is 0.6321.

This result indicates that the Poisson approximation formula overestimates the likelihood of having at least one student with the same birthday as Brad.

Exact probability: (364/365)^179(1/365) = 0.3775

Poisson Approximation Probability: 0.3679

The exact probability is 0.3775, which is quite similar to the Poisson approximation probability, which is 0.3679.

This result indicates that the Poisson approximation formula provides a reasonably precise estimate of the likelihood of having exactly one student with the same birthday as Brad.

Exact probability: 1 - [1 + 364/365 + ... + (364!/347!)/365^34] = 0.1827

Poisson Approximation Probability: 0.2642

The exact probability is 0.1827, which is less than the Poisson approximation probability, which is 0.2642.

This result indicates that the Poisson approximation formula overestimates the likelihood of having at least two students with the same birthday as Brad.

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Answer:

Mary runs 600 meters per day, so in one week, she runs:

600 meters/day x 7 days/week = 4200 meters/week

To convert meters to kilometers, we need to divide by 1000:

4200 meters/week ÷ 1000 meters/kilometer = 4.2 kilometers/week

Therefore, Mary runs 4.2 kilometers in one week.

Answer: 4.2 km per week

Step-by-step explanation:

The measure of the angle is x = 25°.

What is angle?

An angle is a geometric figure formed by two rays that share a common endpoint called the vertex. The two rays are called the sides or arms of the angle. Angles are typically measured in degrees, with a full rotation around a point being 360 degrees.

Let the angle be x.

The supplement of the angle is 180° - x.

The complement of the angle is 90° - x.

According to the problem, we have:

180° - x = 2(90° - x) + 25°

Simplifying and solving for x, we get:

180° - x = 180° - 2x + 25°

x = 25°

Therefore, the measure of the angle is x = 25°.

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This inequality will intersect x axis at -2 and y axis at 100.

An inequality in the algebra is called mathematical statement that employs the inequality symbol to represent how two expressions relate to one another. The data on each side of an inequality symbol are nonequal. Relation  between two algebraic expressions that are represented by the inequality symbols are known as literal inequalities.

"A limk is referred an inequality if two real numbers  are connected by the symbols ">," "," "," or "."

Example: 3≤x<8 ( x is greater than or equal to 3 and less than 8)

y<50x+100

For x=0;

y<100

For y=0;

x>-2

The graph of the inequality is attached below:

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After closely analyzing the line graph, it can be seen that the number of pygmy marmosets weighing more than 1/2 ounce are 8.

To create a line plot, we plot each weight of the each pygmy marmosets on the y-axis and the pygmy marmoset number on the x-axis, which will be ultimately being connected using the points with a line.

After ploting the each mentioned weights (in ounces ) and taking a close study of the plotted line graph,  we can see that there are total 8 pygmy marmosets which are weighing more than 1/2 ounce.

Therefore, the total number of pygmy marmosets with weight more than 1/2 ounces are 8.

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The complete question is :

The table shows the weights of 10 newborn pygmy marmosets make a line plot to display the data. How many pygmy marmosets weigh more than 1/2 ounce ?

Pygmy Marmoset      Weight (ounces)

1                                 0.4

2                                 0.5

3                                 0.6

4                                 0.7

5                                 0.8

6                                 0.9

7                                 1.0

8                                 1.1

9                                 1.2

10                                 1.3

A higher confidence level is associated with a wider confidence interval.

a. The point estimate of µ = (14 + 20 + 21 + 16 + 18 + 19) / 6 = 108 / 6 = 18.

b. For 80% confidence interval for µ, the confidence coefficient is 1 - α = 0.8, so α = 0.2 / 2 = 0.1. From the z-table, we can find the corresponding value of z to be 1.28. The confidence interval can be calculated as follows:

Upper Bound: µ + z*σ / √n = 18 + (1.28)(2.3) / √6 = 21.35

Lower Bound: µ - z*σ / √n = 18 - (1.28)(2.3) / √6 = 14.65

The 80% confidence interval is (14.65, 21.35). For 98% confidence interval for µ, the confidence coefficient is 1 - α = 0.98, so α = 0.01. From the z-table, we can find the corresponding value of z to be 2.33. The confidence interval can be calculated as follows:

Upper Bound: µ + z*σ / √n = 18 + (2.33)(2.3) / √6 = 23.88

Lower Bound: µ - z*σ / √n = 18 - (2.33)(2.3) / √6 = 12.12

The 98% confidence interval is (12.12, 23.88). The 80% and 98% confidence intervals are different because as we move to higher confidence levels, the z-values become larger, which in turn causes the confidence intervals to become wider. Therefore, a higher confidence level is associated with a wider confidence interval.

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Answer:

Step-by-step explanation:

[tex]\frac{2}{x} -3=\frac{1}{2}[/tex]

      [tex]\frac{2}{x}=\frac{1}{2}+3[/tex]      (-3 both sides)

      [tex]\frac{2}{x}=\frac{7}{2}[/tex]            (added fraction)

      [tex]2=\frac{7x}{2}[/tex]           (×[tex]x[/tex] both sides)

      [tex]4=7x[/tex]           (×2 both sides)

      [tex]x=\frac{4}{7}[/tex]             (÷7 both sides)

A triangle has sides with lengths of 3 feet, 4 feet, and 5 feet which is a right triangle.

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, while the other two sides are called the legs.

The converse of the Pythagorean theorem states that if a triangle has sides of lengths a, b, and c, and a² + b² = c², then the triangle is a right triangle.

In this case, the triangle has sides of lengths 3 feet, 4 feet, and 5 feet. To determine whether it is a right triangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse  is the same as the total of the squares formed by the other two sides' lengths.

If the triangle is a right triangle, then the longest side (the hypotenuse) must satisfy this equation. Thus, we have:

5² = 3² + 4²

25 = 9 + 16

25 = 25

Since this equation is true, we can conclude that the triangle is a right triangle.

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The slopes are,

1) 7/6

2)7/2

3) -1

4) -2

5) 10/9

What is slope?

Calculated using the slope of a line formula, the ratio of "vertical change" to "horizontal change" between two different locations on a line is determined. The difference between the line's y and x coordinate changes is known as the slope of the line.Any two distinct places along the line can be used to determine the slope of any line.

1) The given points , [tex](x_1,y_1) =(0,1)[/tex] and [tex](x_2,y_2) = (6,8)[/tex] then,

=> slope  = [tex]\frac{y_2-y_1}{x_2-x_1}[/tex] = [tex]\frac{8-1}{6-0} = \frac{7}{6}[/tex]

2) The given points [tex](x_1,y_1) =(-1,10)[/tex] and [tex](x_2,y_2) = (-5,-4)[/tex] then,

=> Slope = [tex]\frac{-4-10}{-5+1} = \frac{-14}{-4}=\frac{7}{2}[/tex]

3)  The given points [tex](x_1,y_1) =(-10,2)[/tex] and [tex](x_2,y_2) = (-3,-5)[/tex] then,

=> slope = [tex]\frac{-5-2}{-3+10} = \frac{-7}{7}=-1[/tex]

4)  The given points [tex](x_1,y_1) =(-3,-4)[/tex] and [tex](x_2,y_2) = (-1,-8)[/tex] then,

=> slope = [tex]\frac{-8+4}{-1+3} = \frac{-4}{2}=-2[/tex]

5)The given points [tex](x_1,y_1) =(0,1)[/tex] and [tex](x_2,y_2) = (-9,-9)[/tex] then,

=> slope = [tex]\frac{-9-1}{-9+0} = \frac{-10}{-9}=\frac{10}{9}[/tex]

Hence the slopes are,

1) 7/6

2)7/2

3) -1

4) -2

5) 10/9

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Answer:

a) 44 children can safely play in the playground of area 154 m^2.

b) The smallest playground area in which 24 children can play is 84 m^2.

Step-by-step explanation:

We have the ratio 210m^2 : 60.

a) 154/210 is 11/15. Multiplying this scale factor gives the ratio 154 m^2 : 44.

44 is found by multiplying 11/15 by 60.

44 children can safely play in the playground of area 154 m^2.

b) 24/60 is 2/5. Multiplying this scale factor gives the ratio 84 m^2 : 24

84 is found by multiplying 2/5 by 210.

The smallest playground area in which 24 children can play is 84 m^2.

Hope this helps!

As we proved that the commutator  [cA+dB. C]=c[A. C]+d[B. C] is bilinear

Let us start with the left-hand side of the equation: [cA + dB, C]. By definition, this is equal to (cA + dB)C - C(cA + dB).

Using the distributive property of multiplication, we can expand this expression to get

=> cAC + dBC - cCA - dBC.

Now let us consider the right-hand side of the equation:

=> c[A,C] + d[B,C].

By definition,

=> [A,C] = AC - CA and [B,C] = BC - CB.

Substituting these expressions into the right-hand side of the equation, we get

=> c(AC - CA) + d(BC - CB).

Using the distributive property of multiplication, we can expand this expression to get

=> cAC - cCA + dBC - dCB.

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the blue bucket holds 5 x 4 = 20 cups of water.

what is a probability?

In mathematics, probability is a measure of the likelihood or chance of an event occurring. It is represented as a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain.

Since 1 quart of water is equivalent to 2 pints of water and 1 pint of water is equivalent to 2 cups of water, then 1 quart of water is equivalent to 4 cups of water (2 pints x 2 cups per pint = 4 cups).

To determine which bucket holds more water, we need to know the amount of water in the other bucket. Without that information, we cannot compare the two buckets.

Assuming that Janelle filled the other bucket with water as well, we would need to know how many cups of water it holds in order to compare the two buckets.

Therefore, the blue bucket holds 5 x 4 = 20 cups of water.

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The given series in the sigma notation can be written as [tex]\sum_{n = 1} ^ 5 10n - 8[/tex].

An arithmetic series is a set of integers where each term is made up of the common difference, a fixed amount, and the sum of the terms before it. In other words, the terms of the series may be represented as follows if the first term of an arithmetic series is a and the common difference is d:

a, a + d, a + 2d, a + 3d, ...

The given series is 2 + 12 + 22 + 32 + 42.

The total number of terms are 5.

The first term is 2, and the common difference is:

d = 12 - 2 = 10

Now, using the nth term of sequence we have:

an = 2 + (n - 1) 10

= 10n - 8

= [tex]\sum_{n = 1} ^ 5 10n - 8[/tex]

Hence, the given series in the sigma notation can be written as [tex]\sum_{n = 1} ^ 5 10n - 8[/tex].

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Answer:

9. 6.32

10. 4.12

Step-by-step explanation:

9. c^2 = 2^2 + 6^2 = 4 + 36 = 40

c = √40 = 6.32

10. c^2 = 1^2 + 4^2 = 17

c = √17 = 4.12

The fireworks display occurred due north of the person at point A. This can be determined by calculating the direction and speed of sound. Assuming the speed of sound is approximately 343 meters per second, the fireworks display must have occurred approximately 0.58 seconds away from point A, which is approximately 343 meters due north of point A. This means that the fireworks display occurred somewhere between the two points.


To double-check the calculations, we can look at the two points and the direction in which the sound traveled. Point A is due north of the fireworks display, and point B is due west. This means that the sound traveled both north and west, which is consistent with the calculations.


Therefore, we can conclude that the fireworks display occurred due north of point A.

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For (a), the value for x is 11, as P(X>x) = 0.5 when the mean is 7 and the standard deviation is 4. This can be found using the standard normal table.

For (b), the value for x is 19, as P(X>x) = 0.95 when the mean is 7 and the standard deviation is 4. This can also be found using the standard normal table.

For (c), the value for x is 9, as P(x< X<9) = 0.2 when the mean is 7 and the standard deviation is 4. This is found by subtracting the z-scores of x and 9 from each other, and then finding the area of the z-score between those two numbers using the standard normal table.

For (d), the value for x is 4, as P(3< X) = 0.2 when the mean is 7 and the standard deviation is 4. This is found by subtracting the z-score of 3 from the mean and then finding the area of the z-score to the left of that number using the standard normal table.

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Answer:

[tex]=20x^{2} +2x-6[/tex]

Step-by-step explanation:

First step: you're going to multiply each variable from the first bracket to the first and second variable from the second bracket:

=4x(5x)+4x(-2)

=[tex]20x^{2} -8x[/tex]

Second step: now you're going to do the same thing but for the second variable:

=2(5x)+3(-2)

=10x-6

we're going to add the answers that we got from the first two steps to get the final answer:

[tex]=(20x^{2} -8x)+(10x-6)\\[/tex]

[tex]=20x^{2} +(-8x+10x)+(-6)\\=20x^{2} +2x-6[/tex]

Using the digits 9, 8, 7, 6, 5, 4, 3, 2, and 1 in that order to create 100 by using the mathematical operations, (9 + 8 + 7) x (6 - 5) x (4 + 3 + 2) - 1 = 100

It is not possible to create 100 using the digits 9, 8, 7, 6, 5, 4, 3, 2, and 1 in that order without using any mathematical operations.

However, it is possible to create 100 by using mathematical operations like addition, subtraction, multiplication, division, and parentheses. One possible way to do this is:

(9 + 8 + 7) x (6 - 5) x (4 + 3 + 2) - 1 = 100

Here, we first add the first three digits (9 + 8 + 7 = 24), subtract the fourth and fifth digits (6 - 5 = 1), add the next three digits (4 + 3 + 2 = 9), and finally multiply all three results and subtract the last digit (24 x 1 x 9 - 1 = 215 - 1 = 100).

So, using this sequence of digits and mathematical operations, we can create 100.

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The proportionate relationship is used to determine that:

Y=0.5 when x = 1.

Y Equals 1.25 when x = 2.5.

Y = Y = 1.75 when x = 3.5.

What does "proportional relationship" mean?

In a proportional connection, the output variable is determined by the input variable multiplied by a proportionality constant, as in the equation: y = kx.

where k is the proportionality constant.

The relationship in this issue is provided by:

y = x/2

Hence, y = 1/2 = 0.5 when x = 1.

Y = 2.5/2 = 1.25 when x = 2.5.

When x = 3.5:

y = 3.5/2 = 1.75

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Answer:

The answer is 657 and 365.

Step-by-step explanation:

Let the two numbers be x and y respectively

In first case,

x+y=1022

x=1022-y----------- eqn i

In second case

x-y=292

1022-y-y=292 [From eqn i]

1022-2y=292

1022-292=2y

730=2y

730/2=y

y=365

Substituting the value of y in eqn i

x=1022-y

x=1022-365

x=657

Hence two numbers are 657 and 365.

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The area of a triangle is given by the formula A = 1/2 * b * h, where b is the base and h is the height.

The sides of an isosceles triangle are two equal sides and one unequal side. Therefore, to maximize the area of the triangle, we can set the two equal sides equal to x and the remaining side equal to 2x.

The area as a function of x is then:
A(x) = 1/2 * x * 2x = x2.
Therefore, the length of the base of the triangle is x, and the length of the remaining sides is 2x.

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