State how many terms are in each algebraic expression:


(a) -112y2 ____________________ [1mark]


(b) 7x2 + 5y – 9xy + 3 __________________ [1mark]

The Surface Area of the sphere is approximately 339.79 square inches.

The surface area of a sphere is given by the formula:

surface area= [tex]4\pi r^{2}[/tex]

where r is the radius of the sphere.

The diameter of the sphere measures 10.4 inches, hence the radius can be calculated as:

r=10.4/2=5.2inches

Hence, the surface area can be calculated as by substituting r=5.2 inches

Therefore, surface area of the sphere is:

Surface Area = [tex]4\pi (5.2)^{2}[/tex]=[tex]4\pi (27.04)[/tex]= 108.16[tex]\pi[/tex] square inches.

So, the Surface Area of the sphere is approximately 339.79 square inches(if we use [tex]\pi[/tex]=3.14 as an approximation)

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Answer:

An integer to describe 6 meters above the sea level would be  meters.

As per the question statement, We are supposed to use an integer to describe the following situation "6 meters above sea level".

We assume that sea level is the datum line and anything above that would be positive and below that would be negative.

So 6 meters above the sea level can be described as .

Integers: Set of whole number containing both positive and negative values of it.

PLS MARK BRAINLIEST

Step-by-step explanation:

The height of the cone in terms of y is h = y / 4.

The cylinder and the cone have the same volume. The cylinder has radius x and height y. The cone has radius 2x.

Therefore,

volume of a cylinder = πr²h

where

Volume of a cone = 1 / 3 πr²h

where

Therefore,

πr²h = 1 / 3 πr²h

πx²y = 1 / 3 π (2x)²h

πx²y = 1 / 3 π 4x² h

multiply both sides by 3

πx²y = π 4x² h

divide both sides by  π 4x²

Hence,

h = πx²y  / π 4x²

h = y / 4

Therefore, the height of the cone is h = y / 4.

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The function in vertex form is f(x) = 4(x - 6)² - 26.

The function in vertex form is f(x) = 4(x - 6)² - 26.

To get to this form, the first step is to factor out the coefficient of x², which is 4:

f(x) = 4(x² - 12x) + 10

Next, complete the square by adding and subtracting (12/2)² = 36 inside the parenthesis:

f(x) = 4(x² - 12x + 36 - 36) + 10

Simplify the expression inside the parenthesis and combine like terms:

f(x) = 4((x - 6)² - 36) + 10

f(x) = 4(x - 6)² - 134

Therefore, the function in vertex form is f(x) = 4(x - 6)² - 26.

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Answer:

Step-by-step explanation:

There are 12 possible cards you can choose.

There are 3 possible results for the spinner.

There are 2 possible results for the coin toss.

When doing all 3 events:

      Total possibilities [tex]=12\times 3 \times 2=72.[/tex]

SOLUTION:   72

we need to find two numbers whose product is 110. Possible combinations include L = 10 feet and W = 11 feet or L = 11 feet and W = 10 feet. Therefore, the dimensions of the brick wall can be either 10 feet by 11 feet or 11 feet by 10 feet.

A brick wall can be shaped like a rectangular prism, and in this case, the wall needs to be 3 feet tall. With the builder having enough bricks for the wall to have a volume of 330 cubic feet, we can calculate the area of the base of the wall.

To find the base area, we can use the formula for the volume of a rectangular prism: Volume = Base Area × Height. In this situation, we know the volume (330 cubic feet) and the height (3 feet), so we can solve for the base area.

330 cubic feet = Base Area × 3 feet
Dividing both sides of the equation by 3, we get:
Base Area = 110 square feet

So, the base area of the brick wall that is shaped like a rectangular prism with a height of 3 feet and a volume of 330 cubic feet will be 110 square feet.

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The ratio of the perimeters is 14/5, which simplifies to 70/25, which reduces to 14/5. So the answer is (d) 5/7.

Let's use the fact that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding side lengths. If we let the corresponding side lengths be a and b, then we have:

(area of first triangle)/(area of second triangle) = (a^2)/(b^2)

We are given that the areas of the two triangles are 50 cm^2 and 98 cm^2, respectively. Let's call the side lengths of the first triangle a1, a2, and a3, and the side lengths of the second triangle b1, b2, and b3. Then we have:

(50)/(98) = (a1a2)/(b1b2)

We don't know the actual values of a1, a2, b1, and b2, but we can find the ratio of their perimeters by adding up the side lengths of each triangle. Let's call the perimeters of the first and second triangles P1 and P2, respectively. Then we have:

P1 = a1 + a2 + a3

P2 = b1 + b2 + b3

Dividing P1 by P2, we get:

P1/P2 = (a1 + a2 + a3)/(b1 + b2 + b3)

We can rewrite the ratios of the side lengths using the equation we found earlier:

P1/P2 = [(a1a2)/(b1b2)]*[(a1 + a2 + a3)/(a1 + a2 + a3)]

P1/P2 = [(a1a2)/(b1b2)]*1

P1/P2 = (a1a2)/(b1b2)

We still don't know the values of a1, a2, b1, and b2, but we can eliminate them by using the equation we found earlier:

(50)/(98) = (a1a2)/(b1b2)

Simplifying this expression, we get:

(a1/a2) = sqrt((98/50)*(b1/b2))

We can use this to substitute for one of the ratios of side lengths in the equation for P1/P2:

P1/P2 = [(a1a2)/(b1b2)]*[(a1 + a2 + a3)/(a1 + a2 + a3)]

P1/P2 = [sqrt((98/50)(b1/b2))][(a1 + a2 + a3)/(a1 + a2 + a3)]

P1/P2 = sqrt((98/50)*(b1/b2))

Now we can substitute the given values to get:

P1/P2 = sqrt((98/50)(b1/b2)) = sqrt((98/50)(2/1)) = sqrt(196/50) = 14/5

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The  local time in Sydney when the plane arrives will be 18:58, or 6:58 PM on a 24-hour clock.

The time difference between Speed and Sydney is 2 hours, and the plane will take some time to fly from Singapore to Sydney at an Speed of 757.2 km/h over a distance of 6310 km.

The time it takes the plane to fly from Speed to Sydney can be calculated as:

time = distance / Speed= 6310 km / 757.2 km/h = 8.33Speed

So the plane will take 8.33 hours to fly from Singapore toSpeed

Now we need to add the 2-hour time difference between Singapore and Sydney to determine the local time in Sydney when the plane arrives.

07:45 (Singapore time) + 8.33 hours (flight time) + 2 hours (time difference) = 18:58

Therefore, the local time in Sydney when the plane arrives will be 18:58, or 6:58 PM on a 24-hour clock.

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The average monthly exchange rate between the us dollar and the Australian dollar for 2018

A. The components of a time series of average monthly exchange rates include trend, seasonality, cyclical fluctuations, and random noise. The trend represents the long-term movement of the exchange rate, seasonality represents repeating patterns within a fixed period, cyclical fluctuations are changes due to economic cycles, and random noise consists of unpredictable fluctuations.

B. To smooth out the patterns that include everything the model learned, you can apply an exponential smoothing method with a chosen smoothing coefficient (alpha). A suitable coefficient could be 0.2, representing a balance between giving weight to recent data and considering the historical pattern. The decision on the coefficient depends on the specific characteristics of the data and the desired degree of smoothing.

C. To forecast the monthly exchange rate for 2020, you can apply various techniques, such as moving average, exponential smoothing, autoregressive integrated moving average (ARIMA), and machine learning-based methods. Each method has its advantages and limitations, and it's important to analyze the performance of each technique on historical data to choose the most appropriate method for forecasting.

D. Comparing the forecasting results of different techniques applied in part (C) requires measuring their accuracy using metrics like mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE). The technique with the lowest error values would be considered more accurate in predicting the monthly exchange rates. It is crucial to consider the data characteristics and the goals of the forecast when deciding on the most suitable technique.

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The area of the regular pentagon is approximately 392.2 square centimeters.

The area of a regular polygon can be calculated using the formula:

A = (1/2) * apothem * perimeter

where perimeter = number of sides * side length.

For a pentagon with side length s = 15.1 cm, the perimeter is:

perimeter = 5 * 15.1 = 75.5 cm

The apothem is a = 10.4 cm.

Using the formula, we get:

A = (1/2) * 10.4 * 75.5

A = 392.2 cm^2

Therefore, the area of the regular pentagon is approximately 392.2 square centimeters.

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If Juan does not read any books before day 4 and he starts reading at the same rate as Patti for the rest of the month, he would have read 6 books by day 12. The correct option is A.

If Juan does not read any books before day 4, it means he has missed out on the opportunity to read for the first three days. Assuming Patti and Juan have been reading at the same rate since day 4, we can calculate the total number of books they would have read by day 12.

Patti reads one book per day, so by day 12, she would have read a total of 9 books (from day 4 to day 12). If Juan starts reading at the same rate as Patti from day 4, he would also have read 9 books by day 12.

However, we have to account for the fact that Juan did not read any books before day 4. This means that he missed out on the opportunity to read 3 books (one book per day for the first three days). Therefore, by day 12, Juan would have read a total of 6 books (3 books missed + 3 books read from day 4 to day 12).

Therefore, the answer is A. Juan would have read 5 books less than Patti by day 12, since Patti would have read a total of 9 books and Juan would have only read 6.

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First, we need to convert the mixed number 2 3/8 to an improper fraction:

2 3/8 = (2 x 8 + 3) / 8 = 19/8

To find out how many times Craig needs to fill a 1/3 cup measuring cup, we can divide 19/8 by 1/3:

(19/8) / (1/3) = (19/8) x (3/1) = 57/8

Therefore, Craig needs to fill a 1/3 cup measuring cup 57/8 times to measure 2 3/8 cups.

The fraction is half of the sweaters cost $ 50 or less.

After placing the given set of numbers in order, the median is the value that falls in the middle of the group.

From the given box plot,

The value of the median is $ 50

We can clearly see that the median is $ 50 this means, 50% of the clothes are below $ 50 and 50% of the clothes are above $ 50.

The percentage of $ 50 or less sweaters is now 50%.

So, Required fraction = 50 / 100

= 5 / 10

= 1 / 2

Therefore, the fraction is half of the sweaters cost $ 50 or less.

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Given question is incomplete, complete question is below

Bella works at a popular clothing store. last winter ,her manager asked her to track the sweater sales. this box plot show the results.

question: what fraction of the sweater cost $50 or less?

In Year 6 at Woodland Junior School, there are 90 children split into three classes of 14, 24, and 12 on the basis of there selection of sports. In 6M, 14 not chose hockey, and the rest of the class was split equally between football and netball. In 6P, 24 not chose netball, 16 chose football, and 8 choose hockey  In 6T, the ratio of football to netball to hockey was 1:2:3. The completed table is shown.

To complete the table, we need to distribute the remaining children who did not choose their sport in each class. Here's how we can calculate it

In 6M, the number of children who did not choose hockey is 27 - 13 = 14.

Since the rest of the class was split equally between football and netball, each of these two sports will have 14/2 = 7 children.

In 6P, the number of children who did not choose netball is 33 - 9 = 24.

Since twice as many children chose football as chose hockey, we can write the number of footballers as 2x, and the number of hockey players as x. Then we have 2x + x + 9 = 33, which gives x = 8. Therefore, we have 16 children who chose football, and 8 children who chose hockey. The number of children who did not choose any of these two sports is 33 - 16 - 8 - 9 = 0.

In 6T, the ratio of children who chose football to netball to hockey was 1:2:3. Let's call the number of children who chose netball as 2x, and the number of children who chose hockey as 3x. Then we have x + 2x + 3x = 30, which gives x = 6. Therefore, we have 6 children who chose football, 12 children who chose netball, and 18 children who chose hockey.

Thus, the completed table is shown.

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The price of the old model is given by $22.289 and dimensions of the rectangle by 16units and 32 units.

Two dimensions make up a rectangle: the length and, perpendicular to that, the breadth. A triangle's or an oval's interior likewise has two dimensions. Despite the fact that we don't consider them to have "length" or "height," they do span a territory that is expansive in more than one way.

A circle can be measured in any direction. Why do we just consider it to be two dimensional? Because only one direction—the direction perpendicular to the first measurement—can be used to make a second measurement, for a total of two directions.

Let us assume that, price of the old model is Px .

so,

→ Price of 32" LED television = P(2x - 15.500)

A/q,

→ (2x - 15.500) = 29.078

→ 2x = 29.078 + 15.500

→ 2x = 44.578

→ x = $22.289

Therefore, price of the old model is $22.289.

Let us assume that, width of the rectangle is x unit.

so,

→ Length = twice of width = 2x = 2x unit .

then,

→ Perimeter = 2(Length + width)

A/q,

→ 2(2x + x) = 96

→ 3x = 48

→ x = 16 unit .

therefore,

Width of rectangle = x = 16 units .

Length of rectangle = 2x = 32 units.

Hence, the dimensions of the rectangle are 16units and 32 units.

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The price of the old televison is P22,289

The dimensions of the rectangle are 16 and 32

We have to read the problem carefully so as to be able to know how to translate the problem effectively and that is what we are going to do below.

We know that;

Let the price of the old 32'' LED television be x

Now;

29,078. 00 = 2x - 15,500

29,078. 00 + 15,500 = 2x

x = 29,078. 00 + 15,500 /2

x = P22,289

ii) Given that;

l = 2w

Perimeter = 2(l +w)

P = 2(2w + w)

P = 2(3w)

P = 6w

w = 96/6

w = 16

Then l = 2(w) = 32

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lets assume.

Algebraic expression: c - 2.5. The variable is c, which represents Julie's initial weight in kilograms.

Algebraic equation: Jhon's height = 2m + 5. The variables are m, which represents Peter's height in centimeters, and the height of Jhon, which is represented by the equation.

Algebraic equation: Ador's age = 3(Emy's age - d). The variables are Ador's age and Emy's age, which is d years less than 9.

Algebraic expression: n + 7. The variable is n, which represents Jupiter's current age in years.

Algebraic equation: Anna's age = 3p + 4. The variables are p, which represents Anna's sister's age in years, and Anna's age, which is represented by the equation.

SO ANNA  current age is P=3+4

and p=7

Anna's age = 3p + 4, where p is the age of Anna's sister in years, and Anna is 4 years older than thrice the age of her sister.

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The length of CB is 3 unit.

In the given figure ;

By SAS property of similar of triangles,

ΔCED and ΔCAB are similar.

Therefore,

CE/CB = DE/AB = DC/AC

⇒ CE/CB = DC/AC

⇒ 2/CB = 2.4/3.6

⇒ CB = (3.6/2.4)X2 = 3

Hence CB = 3

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Using the combine clues method:

[tex]R_1=7\\\\R_2=9\\\\R_3=8\\\\S_1=2\\\\S_2=4\\\\S_3=5\\\\T_1=10\\\\T_2=14[/tex]

Certain items that must be worn, used, or accessed in order to access particular clues. The maximum skill level needed to solve each master clue is shown.

Clue 1: [tex]R_1 + R_2 + R_3 = 24[/tex]

Clue 2: [tex]R_1 + R_2 =16[/tex]

Since, [tex]R_1 + R_2 + R_3 = 24[/tex] (from clue 1)

[tex]R_1 + R_2 + R_3 = 24\\\\16 + R_3 = 24[/tex]

Subtract both the sides by 16,

[tex]R_3=8[/tex]

Clue 3:

We need to find the blank [tex]R_1 +[/tex] __ [tex]+S_2=20[/tex] and [tex]S_2=4[/tex]

Since, [tex]R_1 + R_2 = 16[/tex] (from clue 2)

So,

[tex]R_1 + R_2 + S_2 = 20\\\\16 + S_2 = 20[/tex]

Subtract 16 from both sides.

[tex]16 + S_2 - 16 = 20 -16\\\\S_2 = 4[/tex]

So,

[tex]R_1 + R_2 + S_2 = 20[/tex]

Clue 4:

We need to find the blank [tex]R_2[/tex] = __

[tex]S_2 + R_2 = 13[/tex]

Since, [tex]S_2 = 4[/tex]

[tex]S_2 + R_2 = 13\\\\4 + R_2 = 13[/tex]

Subtract 4 from both sides.

[tex]R_2 = 9[/tex]

So, [tex]R_2 = 9[/tex]

Clue 5:

We need to find the blank ___ [tex]+ T_2 = 22[/tex] and [tex]T_2=[/tex] ___

[tex]R_3 + T_2 = 22[/tex]

Since, [tex]R_3 =8[/tex] (from the Clue 2)

[tex]R_3 + T_2 = 22[/tex]

[tex]8 + T_2 = 22[/tex]

Subtract both the sides by 8,

[tex]T_2 = 14[/tex]

So, [tex]R_3 + T_2 = 22[/tex]

Clue 6:

We need to find the blank [tex]T_1 +[/tex] ___ = 24 and ___ = 10

Since, [tex]T_1 + T_2 = 24[/tex] (from the Clue 5)

[tex]T_1 + 14 = 24[/tex]

Subtract both the sides by 14,

[tex]T_1 = 10[/tex]

Clue 7:

We need to find the value of [tex]S_1[/tex],

Given [tex]T_1 + R_3 + S_1 =20[/tex]

Since, [tex]T_1=10[/tex] and [tex]R_3=8[/tex]

[tex]10+ 8 + S_1 = 20\\\\18 + S_1 = 20[/tex]

Subtract both the sides by 18,

so, [tex]S_1=2[/tex]

Combining this with what we learned from Clue 2 and Clue 6, we know

[tex]S_1=2[/tex]

Now, we need to find the blank, [tex]S_1 + S_2 +[/tex] ___ = 11

Combining this with what we learned from Clue 3 and Clue 7, we know:

___ = 5

[tex]S_1 + S_2 + S_3 = 11\\\\2 + 4 + S_3 = 11\\\\6 + S_3 = 11[/tex]

Subtract both the sides by 6,

[tex]S_3 = 5[/tex]

For [tex]R_1[/tex]

Since,

[tex]R_1 + R_2 + R_3 = 24\\\\R_1 + 9 + 8 = 24\\\\R_1 + 17 = 24[/tex]

Subtract both the sides by 17,

[tex]R_1 = 17[/tex]

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Let's assume that the number of shirts that must be sold to cover the costs is x.

The revenue from selling x shirts is 12x dollars, and the total cost is the sum of the setup fee and the cost per shirt multiplied by the number of shirts, which is 65 + 7x dollars.

To break even, the revenue must be equal to the cost:

12x = 65 + 7x

Subtracting 7x from both sides:

5x = 65

Dividing both sides by 5:

x = 13

Therefore, the student council must sell 13 shirts to break even.

To find the total costs, we can substitute x = 13 into the expression for the total cost:

65 + 7x = 65 + 7(13) = 156

So the total costs will be $156.

Answer:

The student council's costs will be $146 and they need to sell 13 shirts to cover their costs.

Step-by-step explanation:

Let's call the number of shirts sold "x".

The revenue from selling x shirts at $12 each is:

The total cost of making x shirts is:

In order to break even (i.e. cover their costs), the revenue must equal the total cost:

[tex]\sf:\implies 12x = 7x + 65[/tex]

Solving for x:

[tex]\sf:\implies 5x = 65[/tex]

[tex]\sf:\implies \boxed{\bold{\:\:x = 13\:\:}}\:\:\:\green{\checkmark} [/tex]

Therefore, the student council must sell 13 shirts to break even.

To find the total costs, we can substitute x = 13 into the total cost equation:

[tex]\sf:\implies Total\: Cost = 7(13) + 65 = \boxed{\bold{\:\:\$146\:\:}}\:\:\:\green{\checkmark} [/tex]

So the student council's costs will be $146 and they need to sell 13 shirts to cover their costs.

Answer:

The answer to your problem is, 259,644

Step-by-step explanation:

an = 1880 ( it 25% [tex])^{t}[/tex]

= 1880 x (1.25[tex])^{t}[/tex]

ai = 1880, r will equal 1.25

Sn = [tex]\frac{aill-r^{4} }{l = r}[/tex]

[tex]S_{16}[/tex] = [tex]\frac{1880(l-1.25^{16}) }{l - 1.25}[/tex]

= 259,644

Thus the answer to your problem is, 259,644

The function with y and x, representing cost and shipping charges will be y = 3.50x + 9.50.

The express that can be used to write the function is as follows-

Total amount = number of shirts × cost of shirts + shipping fee

Write the function based on the formula -

y = 3.50×x + 9.50

As stated x represents the number of shirts bought online. We know that shipping charges will be calculated once while cost of shirt will vary according to the number of shirts.

Hence the function will be -

y = 3.50x + 9.50

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To find f-1(6), we need to find the value of x that gives f(x) = 6. However, we don't have enough information about the function f to do this. We need to know whether f is a linear function or not.

When the function and its inverse intersect, we have f(x) = f-1(x). Substituting y for both f(x) and f-1(x), we get y = f(y). To find the value of y when this is true, we need to solve for y:
y = f(y)
y = f-1(y)
Substituting y = f(x), we get:
f(x) = f-1(f(x))
f(x) = x

So the function and its inverse intersect when y = x.
If a = 0, then the linear function is f(x) = b, which is a constant function. Constant functions do not have inverses, so the inverse of f(x) = b does not exist.

If a ≠ 0, then the inverse of f(x) = ax + b is given by:
f-'(x) = (x - b) / a
This is also a linear function, so the inverse of a linear function is always a linear function when a ≠ 0.
(a) To find the inverse of a linear function, you need to swap the x and y values. Given that f(4) = 6, the inverse function f^(-1)(6) would yield the value of x when y = 6. Since we know that f(4) = 6, it implies that f^(-1)(6) = 4.
A function and its inverse intersect when the input value (x) is equal to the output value (y). In other words, they intersect when y = x.

(c) Yes, the inverse of a linear function is always a linear function. If f(x) = ax + b, where a ≠ 0, then the inverse function, f^(-1)(x), can be found by swapping x and y values and solving for y. In this case, x = ay + b. Solving for y, we get y = (x - b) / a, which is also a linear function.

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The mean and the median of the data set is given as the median, 4, is greater than the mean, 3. The correct answer is option a.

To find the mean of the data set, we add up all of the numbers and divide by the total number of days. So, the mean is (1+4+5+5+1+5+5)/6 = 3.

To find the median, we first need to put the numbers in order from least to greatest: 1, 1, 4, 5, 5, 5. The median is the middle number, which in this case is 4.

Now we need to determine which value is greater. The options are:

A. The median, 4, is greater than the mean, 3.
B. The mean, 3, is greater than the median, 4.
C. The mean, 4, is greater than the median, 2.
D. The median, 5, is greater than the mean, 0.

We can eliminate options B and D because they don't make sense. The mean cannot be less than 0, and the median cannot be less than the mean.

Option A is correct. The median, 4, is greater than the mean, 3. This makes sense because there are two 1's in the data set, which bring down the mean. However, the median is not affected by outliers like this, so it remains higher.

Option C is incorrect because the mean is not 4, it is 3.

In summary, the mean of the data set is 3 and the median is 4. The median is greater than the mean, which is represented by option A.

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The landscaper needs to buy approximately 85.33 bags of topsoil to cover the whole garden.

The landscaper uses 4 bags of topsoil to cover 3/8 of the garden. To cover the whole garden, he would need:

First, we need to find how many bags of topsoil he needs per 1/8 of the garden:

4 bags / 3/8 of the garden = 32/3 bags of topsoil per 1 garden

Then, we can find the number of bags he needs for the whole garden:

32/3 bags * 8 = 256/3 bags or 85.33 bags (rounded to two decimal places)

Therefore, the landscaper needs to buy approximately 85.33 bags of topsoil to cover the whole garden.

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Answer:
-Reflection across line p followed by 240 clockwise rotation about point T.

-Reflection across line r.

-120 counterclockwise rotation about point T.

Step-by-step explanation:

All these transformations map the triangle onto itself.

The length of the hypotenuse of the right triangle is 15 cm.

The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Using this formula, we can calculate the length of the hypotenuse of a right triangle.

Given that the longer sides of the right triangle are 9 cm and 12 cm long, we can assume that one of these sides is the shorter side and the other is the longer side. Let's assume that the shorter side is 9 cm long and the longer side is 12 cm long.

Using the Pythagorean Theorem, we can calculate the length of the hypotenuse as follows:

Hypotenuse² = Shorter side² + Longer side²

Hypotenuse² = 9² + 12²

Hypotenuse² = 81 + 144

Hypotenuse² = 225

Taking the square root of both sides, we get:

Hypotenuse = √225

Hypotenuse = 15

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Word problem: The ratio of the number of people that attended Michael party to the number of people that attended Joshua's party is 5: 25

First, we need to know that equivalent ratios are those ratios that can usually be simplified to a similar value.

Also, algebraic expressions are defined as expressions that are composed of terms, variables, coefficients, terms, constants and factors.

These expressions are also made up of arithmetic operations such as addition, multiplication, subtraction, bracket, parentheses, etc

From the information given,

The word problem is;

The ratio of the number of people that attended Michael party to the number of people that attended Joshua's party is 5: 25

This is represented as;

5/25

Divide the values

1/5

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The equation given in the solution is X(t) = (cos(4t) + C2sin(4t)). This equation represents the position of the mass at time t after the spring has been released with an initial velocity of 0. The terms "spring" and "stretch" indicate that Hooke's Law is being used to determine the spring constant k.

The term "velocity" is used to describe the initial velocity of the mass, which is given as 0. The position of the mass at time t is determined by the value of X(t), which is a function of time. Therefore, the position of the mass at any given time can be found by plugging in the value of t into the equation X(t) = (cos(4t) + C2sin(4t)).
Hi! Based on the information provided, you have a spring with a mass of 6 kg and a natural length of 0.3 m. A force of 38.4 N is required to stretch it to 0.7 m. The spring constant, k, is determined to be 96. The spring is then stretched to 0.7 m and released with an initial velocity of 0. To find the position of the mass at time t, you can use the equation:
x(t) = C1 * cos(4t) + C2 * sin(4t)

Given the initial condition x(0) = 0.4, we find that C1 = 0.4. The initial velocity x'(0) is 0, leading to the conclusion that C2 = 0. Therefore, the equation for the position of the mass at time t is:
x(t) = 0.4 * cos(4t)

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So the value of the integral ∫ dx/√(1-2x^2) over the interval [-1/√2, 1/√2] is (1/√2)π.
The value of the integral ∫ dx/√(1-2x^2) is given by:
∫ (1/√(1-2x^2)) dx = (1/2) * arcsin(√2 * x) + C
where C is the constant of integration.

The value of the integral ∫ dx/√(1-2x^2) is equal to the integral of the function 1/√(1-2x^2) with respect to x. This is an example of an integral that requires a trigonometric substitution to evaluate. Specifically, we can let x = sin(θ)/√2 and dx = cos(θ)/√2 dθ. Substituting these expressions into the integral yields:

∫ dx/√(1-2x^2) = ∫ (cos(θ)/√2) / √(1-2(sin(θ)/√2)^2) dθ

Simplifying the denominator gives:

√(1-2(sin(θ)/√2)^2) = √(1 - sin^2(θ)) = cos(θ)

Substituting this expression into the integral gives:

∫ dx/√(1-2x^2) = ∫ (cos(θ)/√2) / cos(θ) dθ = ∫ dθ/√2 = (1/√2)θ + C

To find the value of the integral, we need to substitute back in for x and evaluate at the limits of integration. If we are integrating over the interval [-1/√2, 1/√2], then:

(1/√2)θ evaluated from -π/4 to π/4 gives:

(1/√2)(π/4 + π/4) - (1/√2)(-π/4 - π/4) = (1/√2)π

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