Trina is 4 years less than triple Duane's age. If Duane is 6 years older than his brother who just turned 14, how old is Trina?

Answer:

(x, y) = (cos(2π/5), sin(2π/5)) = ((1/4)(-1 + √5), (1/4)(2√5 + 2))

Step-by-step explanation:

In standard position, the initial side of an angle lies on the positive x-axis, and the terminal side rotates counterclockwise from the initial side.

Since the angle measures 2π/5 radians, we need to find the point on the unit circle that is π/5 radians past the positive x-axis.

Let (x, y) be the coordinates of the point on the unit circle where the terminal side intersects. We can use the following trigonometric identities to find the values of x and y:

cos(2π/5) = x

sin(2π/5) = y

These values can be determined using either a calculator or the unit circle. Alternatively, we can use the fact that the angle 2π/5 is a special angle and can be expressed exactly in terms of square roots:

cos(2π/5) = (1/4)(-1 + √5)

sin(2π/5) = (1/4)(2√5 + 2)

Therefore, the exact coordinates of the point where the terminal side intersects the unit circle are

(x, y) = (cos(2π/5), sin(2π/5)) = ((1/4)(-1 + √5), (1/4)(2√5 + 2))

Answer:

1) x= 2

Step-by-step explanation:

Answer:

d ≈ 5.4

Step-by-step explanation:

calculate the distance d using the distance formula

d = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex]

with (x₁, y₁ ) = (2, 2 ) and (x₂, y₂ ) = (4, 7 )

d = [tex]\sqrt{(4-2)^2+(7-2)^2}[/tex]

   = [tex]\sqrt{2^2+5^2}[/tex]

  = [tex]\sqrt{4+25}[/tex]

  = [tex]\sqrt{29}[/tex]

 ≈ 5.4 ( to the nearest tenth )

The answer choice that is the most appropriate is B. A one-sample z-interval for estimating a population proportion

This is the most appropriate method because you are trying to estimate the proportion of all registered voters who have a favorable impression of Candidate A based on the sample data.

A one-sample z-interval is used when you want to estimate a population proportion using a sample proportion.

To calculate the one-sample z-interval for estimating a population proportion, you can use the following formula:

Confidence Interval = p ± Z * sqrt((p * (1 - p)) / n)

where:

Using this method, you will calculate a confidence interval that estimates the true proportion of registered voters who have a favorable impression of Candidate A, based on the sample data.

This is the most appropriate method because it focuses on estimating a single population proportion and incorporates the sample data and sample size into the calculation.

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Answer:

The median and mean are two different measures of central tendency that provide different information about a data set. The median is the middle value in a data set, where half of the values are above and half are below. The mean, on the other hand, is the sum of all values divided by the total number of values.

In this case, the median number of reviews is much lower than the mean number of reviews, indicating that there are likely a few very high review counts that are skewing the data towards the higher end. This is because the mean is more affected by extreme values, while the median is not. It is possible that there are a few apps with an extremely large number of reviews, which are driving up the mean.

The mean number of reviews would be significantly higher than the median.

The difference between the median and mean number of reviews can be explained by the fact that some apps have very large numbers of reviews, whereas others have very few. The median is the middle value of the dataset, and does not take into account the outliers, which can have a large effect on the mean. For example, if there is one app with 1 million reviews, the mean number of reviews would be significantly higher than the median.

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The amount in the account after compounded quarterly is found as: $1103.8.

Compounding quarterly is the term used to describe the amount of interest that is earned on a quarterly basis on a savings account or investment at which interest is also reinvested.

Although most banks charge interest income here on deposits, which compounds quarterly, this information is useful in computing the fixed deposit income. It can also be used to figure out any revenue from money market instruments or other financial products that pay quarterly income.

The formula for quarterly compounding:

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

P​ $1000​, r​ = 10%, t ​= 2  years;

compounded quarterly : n = 4

A = 1000 * [tex](1+0.10/4)^{4*1}[/tex]

A = 1000 * 1.1038

A = 1103.8

Thus, the amount in the account after compounded quarterly is found as: $1103.8.

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Complete question:

Find the amount in the account for the given​ principal, interest​ rate, time, and compounding period.

P​ $1000​, r​ = 10%, t ​= 2 years; compounded quarterly

If the probability that a player makes a free throw is 0.4, the probability that they do not make a free throw is 1 – 0.4 = 0.6.

The probability that a player makes three consecutive free throws is 0.4 × 0.4 × 0.4 = 0.064. If a player attempts three free throws, there are three possible outcomes: make all three, miss one and make two, or miss two and make one.

The probability of making all three is 0.064, the probability of missing one and making two is 0.4 × 0.4 × 0.6 + 0.4 × 0.6 × 0.4 + 0.6 × 0.4 × 0.4 = 0.288, and the probability of missing two and making one is 0.6 × 0.6 × 0.4 + 0.6 × 0.4 × 0.6 + 0.4 × 0.6 × 0.6 = 0.432.

Therefore, the probability of leaving the court in three attempts is 0.064 + 0.288 + 0.432 = 0.784. If a player misses all three attempts, they will have to try three more times, and the probability of leaving the court in six attempts is 0.784 + 0.064 × 0.288 × 0.432 = 0.861.

If a player misses all six attempts, they will have to try three more times, and the probability of leaving the court in nine attempts is 0.861 + 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 = 0.926.

If a player misses all nine attempts, they will have to try three more times, and the probability of leaving the court in twelve attempts is 0.926 + 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 × 0.064 × 0.288 × 0.432 = 0.974.

Since the probability of leaving the court in twelve attempts is greater than 0.9, the probability of leaving the court in ten or fewer attempts is greater than 0.9. Therefore, the probability that a player can leave the court in 10 or fewer attempts is greater than 0.9.

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The expected number of patients who will report symptomatic relief is 20.9. and the probability of exactly 20 patients reporting relief being 0.1977, and the probability of at least 20 patients reporting relief being 0.3832.

(a) The expected number of patients who will report symptomatic relief is 20.9.

The probability of a patient reporting symptomatic relief is 0.95, so the probability of a patient not reporting relief is 0.05. Using the binomial distribution formula, the expected value of the number of patients who will report relief is:

E(X) = n * p = 22 * 0.95 = 20.9

(b) The probability that exactly 20 patients will report symptomatic relief is 0.1977.

Using the binomial distribution formula, we can calculate the probability of exactly 20 patients reporting relief:

P(X = 20) = (22 choose 20) * 0.95^20 * 0.05^2 = 0.1977

(c) The probability that at least 20 patients will report symptomatic relief is 0.3832.

Using the binomial distribution formula, we can calculate the probability of 20 or more patients reporting relief:

P(X >= 20) = P(X = 20) + P(X = 21) + ... + P(X = 22)

P(X >= 20) = 1 - P(X < 20)

P(X >= 20) = 1 - (P(X = 0) + P(X = 1) + ... + P(X = 19))

P(X >= 20) = 1 - 0.6168

P(X >= 20) = 0.3832

Therefore, the probability that at least 20 patients will report symptomatic relief is 0.3832.

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The weight of each box of cereal must be between 21 and 23 ounces for the case to pass the quality assurance test. We can write this as the compound inequality: 21 < b < 23.

A quality assurance test is a process that is used to evaluate the quality of a product or service. It involves checking whether the product or service meets certain standards or specifications, and identifying any defects or issues that need to be addressed.

Let's start by finding the weight of a single box of cereal. We know that the weight of the entire case is the weight of 11 boxes plus the weight of the cardboard box, or:

Weight of case = 11b + 33

where b is the weight of a single box of cereal.

To pass the quality assurance test, the weight of the case must be greater than 264 ounces and less than 286 ounces. Therefore, we can write a compound inequality for b as follows:

264 < 11b + 33 < 286

Next, we can simplify this compound inequality by subtracting 33 from each term:

231 < 11b < 253

Finally, we can divide each term by 11 to solve for b:

21 < b < 23

Therefore, the weight of each box of cereal must be between 21 and 23 ounces for the case to pass the quality assurance test. We can write this as the compound inequality:

21 < b < 23 (in ounces)

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The surface area of the playhouse is 564 square feet.

Frank is building a playhouse for his daughter.

The playhouse is a composite figure with a floor and no windows.

The playhouse is a rectangular prism and a triangular prism.

The formula for the surface area of a rectangular prism is 2lw + 2lh + 2wh,

where l is the length,

w is the width,

and h is the height.

Therefore, the surface area of the rectangular prism portion of the playhouse can be calculated by substituting the given values into the formula.

Surface area of the rectangular prism = 2lw + 2lh + 2wh

For the rectangular prism, l = 10 feet, w = 8 feet, and h = 6 feet.

Substituting the values in the formula.

Surface area of the rectangular prism

= 2lw + 2lh + 2wh

= 2 (10 feet × 8 feet) + 2 (10 feet × 6 feet) + 2 (8 feet × 6 feet)

= 160 feet2 + 120 feet2 + 96 feet2= 376 feet2

The surface area of the rectangular prism of the playhouse is 376 square feet.

The formula for the surface area of a triangular prism is 2lw + lh + ph,

where l is the length, w is the width, h is the height, and p is the slant height.

Therefore, the surface area of the triangular prism portion of the playhouse can be calculated by substituting the given values into the formula.

Surface area of the triangular prism = 2lw + lh + ph

For the triangular prism, l = 8 feet, w = 5 feet, h = 6 feet, and p = 10 feet.

Substituting the values in the formula.

Surface area of the triangular prism = 2lw + lh + ph= 2 (8 feet × 5 feet) + (8 feet × 6 feet) + (10 feet × 6 feet)= 80 feet2 + 48 feet2 + 60 feet2= 188 feet2

The surface area of the triangular prism of the playhouse is 188 square feet.

The total surface area of the playhouse can be found by adding the surface area of the rectangular prism and the surface area of the triangular prism.

Surface area of the playhouse = surface area of rectangular prism + surface area of triangular prism.

= 376 square feet + 188 square feet

= 564 square feet.

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Answer:

The answer is 60

Step-by-step explanation:

2x5x6

rearrange by the commutative property

(2x5)x6

Calculate

10x6

calculate the product or quotient to 60

hence the answer is 60

Hoped this helped:D

Alison has five different hats, six different bracelets, and seven cats. She wants to take a selfie with a hat, a bracelet, and two cats. So she can take 630 different selfies.

         She wants to take selfies with what she has, and the question asks how many different selfies can she take.

    Let's solve the question below:

         The total number of selfies Alison can take is 5 × 6 × ( ₇C₂ )

          As Alison wants to take a selfie with a hat, a bracelet, and two cats, there are 5 hats and 6 bracelets she can select from, and she needs to select 2 cats out of 7 available cats.

          We can use the combination formula to calculate the total number of ways:

            ₇C₂ = [tex](7 * 6)/2[/tex]

                   = 21

  Hence, Alison can take [tex]5 * 6 * 21[/tex] = 630 different selfies with a hat, a bracelet, and two cats.

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Diana needs 60 grams of coffee beans to make 96 fluid ounces of coffee for her guests.

Determine whether two ratios or fractions are equivalent using the proportion formula. Dividing the provided values will reveal the value that is missing. As an example of the percentage formula: A and d are the extreme terms, while b and c are the mean terms. Hence, b::c: d = a/b = c/d.

We can resolve the issue by using proportions.

The ratio of coffee beans to liquid ounces in Diana's ordinary coffee must first be determined.

48 fluid ounces / 30 grammes

By reducing the numerator and denominator by 6, we may simplify the ratio:

Eight fluid ounces/5 grammes

We can now apply this proportion to determine how many grammes of coffee beans Diana needs to prepare 96 ounces of coffee:

5 grams/8 fluid ounces = 96 fluid ounces x grammes

After cross-multiplying and finding x, we obtain:

x = (96 ounces * 5 grammes) / 8 ounces

x=60 grammes.

In order to prepare 96 fluid ounces of coffee for her visitors, Diana needs 60 grammes of coffee beans.

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Total distance of 7m, which is the shortest distance between point P and point R.

Distance is the physical separation between two objects or points. It is usually measured in units such as meters, kilometers, feet, miles, or light-years. Distance is a key concept in physics, mathematics, engineering, and other scientific disciplines. It plays an important role in our everyday life, such as when we measure the distance between two cities, two buildings, or two places.

To determine the shortest distance between point P and point R, we need to calculate the distances between each point and then find the shortest path.

First, we need to calculate the distances between each point. Starting with the points closest to point R, we can find the distance between point R and point Q to be 4m. The distance between point Q and point P is 3m, from the given information. To find the distance between point R and point U, we need to calculate the distance between point U and point V which is 10m, and then add 8m to account for the 8m between point R and point V. This gives us a distance of 18m. The distance between point R and point S is 8m, and the distance between point S and point T is 8m.

Next, we need to calculate the shortest path between point P and point R. From point P, the shortest route would be to travel 3m south to point Q, then 4m east to point R. This gives us a total distance of 7m, which is the shortest distance between point P and point R.

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Complete questions as follows-
Point U is 8m east of point R. Point P is 3m north of point Q. Point U is 10m north of point V. Point W is 8m west of point V. Point R is exactly between point S and point T. Point R is 4m east of point Q. Point T is in 8m south of point S.
What is the shortest distance between point P and point R?

Answer: 5/2


fraction of 2.5 is 5/2

Answer:

2 1/2 or 5/2 as improper fraction

2 is a whole number and you have .5 left over

To convert to fraction .5 is the same as 1/2

So it gives you 2 1/2

By the way thanks for the Brainly points :)

The water surface in the conical tank is rising at a rate of approximately 0.16 feet per minute when the water depth is 5 feet.

To solve for this, we can use related rates. Since the tank is being filled with water at a constant rate of 2 ft³/min , we know that the rate of change of the volume of water in the tank is constant. We can use the formula for the volume of a cone to relate the volume of water in the tank to the height of the water in the tank:

V = (1/3) * pi * r² * h

Taking the derivative of both sides with respect to time gives:

dV/dt = (1/3) * pi * r² * dh/dt

We know that dV/dt = 2 ft³/min, r = 2 ft, and h = 5 ft. Solving for dh/dt:

dh/dt = (3 / pi * r²) * dV/dt

dh/dt = (3 / 4 * pi) * 2

dh/dt = 0.16 ft/min

Therefore, the water surface is rising at a rate of approximately 0.16 feet per minute when the water depth is 5 feet.

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Between 4 and 4.5

sqrt(20)

Between 4.5 and 5

sqrt(22)

sqrt(23.1)

Between 5 and 5.5

sqrt(30)

sqrt(25.9)

sqrt(27.3)

Between 5.5 and 6

sqrt(31)

sqrt(34.9)

Therefore , the solution of the given problem of function comes out to be  f(g(4)) = 66.

The midterm examination will have questions on each subject, mathematics, variable design, and both real and imagined locations. a list of the relationships between different elements that collaborate to create the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input. Each postbox also has a distinct address, which may be an enclave.

Here,

a) We must multiply f(x) and g(x) in order to obtain f(x)  g(x):

=>  f(x) · g(x) = (-2x+ 4)

The formula is

=> (-6x - 7) = 12x² + 14x - 24x - 28 = 12x² - 10x - 28.

As a result,

=>  12x² - 10x - 28 is what f(x)  g(x) equals.

b) We must first determine g(4) in order to obtain f(g(4)):

=> g(4) = -6(4) - 7

=>    -31

We can now change g(4) to f(x) as follows:

=> f(g(4)) = f(-31)

=>   -2(-31) + 4

=>  62 + 4

Consequently, f(g(4)) = 66.

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Step-by-step explanation:

Blank 1 = 82

Blank 2 = 82

Blank 3 = 98

Answer:

Step-by-step explanation:

We can start by simplifying the expression using the laws of exponents:

343^(2x) * 49^(-x/2) = 49^(x/2)

We can then simplify further by expressing everything in terms of 7 (since 7 is the common factor of both 343 and 49):

(7^3)^(2x) * (7^2)^(-x/2) = (7^2)^(x/2)

Applying the laws of exponents again, we get:

7^(6x) * 7^(-x) = 7^x

7^(6x - x) = 7^x

7^(5x) = 7^x

Now we can solve for x by equating the exponents on both sides:

5x = x

4x = 0

x = 0

Therefore, the solution to the equation is x = 0.

Answer:

x = 0

Step-by-step explanation:

343^(2x) * 49^(-x/2) = 49^(x/2)

(7^3)^(2x) * (7^2)^(-x/2) = (7^2)^(x/2)

7^(6x) * 7^(-x) = 7^(x)

Now, we can simplify the equation further by combining the like terms on both sides:

7^(6x - x) = 7^(x)

7^(5x) = 7^(x)

We can solve for x by equating the exponents on both sides of the equation:

5x = x

4x = 0

x = 0

Therefore, the solution to the equation is x = 0.

a) mv = Mv.

b) The columns of M are dependent where [v1, v2, ..., vn]T is a vector combination

If a matrix M is multiplied by a vector v, the result will be a linear combination of the columns of the matrix. That is, if M is an m×n matrix and v is a vector with n entries, then the product Mv is a linear combination of the columns of M with coefficients from v. Thus, mv = Mv.

If the columns of M are linearly dependent, it implies that they are multiples of each other, i.e., one column is equal to a scalar multiple of another column, which can be written as an equation of the form, ci = aj where c and a are scalar multiples of jth and ith columns of M, respectively.

Hence, when we compute Mv, the linear combination of the columns of M will depend on the scalar multiples c and a.

For instance, let us assume that column j and i of M are linearly dependent. We have;

ci = aj or

M(:,j) = a*M(:,i)

where M(:,j) and M(:,i) represent jth and ith columns of M, respectively. Then, we can express M as;

M = [M(:,1), ..., M(:,j-1), M(:,i), M(:,j+1), ..., M(:,n)] = [M(:,1), ..., M(:,j-1), a*M(:,i), M(:,j+1), ..., M(:,n)]

Thus, we can rewrite the product Mv as;

Mv = [M(:,1), ..., M(:,j-1), a*M(:,i), M(:,j+1), ..., M(:,n)][v1, v2, ..., vn]T

where [v1, v2, ..., vn]T is a vector combination of the columns of M.

Therefore, if a linear combination of the columns of M results in the zero vector, it implies that the columns of M are dependent.

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Answer:

Step-by-step explanation:

To find the volume of a pyramid, we can use the formula:

V = (1/3) * base area * height

First, we need to find the height of the pyramid. To do this, we can draw a diagram of the pyramid and create a right triangle using half a side of the base, the height of the pyramid, and one of the segments connecting the apex to a midpoint of a side of the base.

Using the Pythagorean theorem, we can solve for the height:

h^2 + (14/2)^2 = 25^2

h^2 + 49 = 625

h^2 = 576

h = 24

Now we can find the base area:

Base area = 14^2

Base area = 196

And finally, we can use the formula to find the volume:

V = (1/3) * 196 * 24

V = 1568 cubic units

Therefore, the volume of the pyramid is 1568 cubic units.

the surface area of the given triangular prism with sides of 4in, 5in, 6in, 6in, 8in, and 7in is 146 square inches.

To calculate the surface area of a triangular prism, you need to find the area of each of the faces and add them up.

First, let's find the area of the two triangular faces. To do this, we need to find the base and height of each triangle. Since the prism is isosceles, the base of each triangle is 6 inches (the length of one of the sides of the equilateral triangle). The height of each triangle can be found using the Pythagorean theorem. We have two sides of the triangle: 4 inches and 5 inches. Using the Pythagorean theorem, we can find the height:

[tex]h^2 = 5^2 - 4^2\\h^2 = 25 - 16\\h^2 = 9\\h = 3[/tex]

So the height of each triangular face is 3 inches. Now we can find the area of each triangular face:

Area of one triangular face = (1/2) x base x height

= (1/2) x 6 x 3

= 9 square inches

Since there are two triangular faces, the total area of the triangular faces is:

Total area of triangular faces = 2 x 9 = 18 square inches

Next, let's find the area of the three rectangular faces. We have two rectangles with sides of 6 inches by 8 inches, and one rectangle with sides of 4 inches by 8 inches. The area of each rectangular face is:

Area of rectangular face = length x width

So the area of the rectangular faces are:

Area of rectangular face 1 = 6 x 8 = 48 square inches

Area of rectangular face 2 = 6 x 8 = 48 square inches

Area of rectangular face 3 = 4 x 8 = 32 square inches

Therefore, the total surface area of the triangular prism is:

Total surface area = 18 + 48 + 48 + 32 = 146 square inches

So the surface area of the given triangular prism with sides of 4in, 5in, 6in, 6in, 8in, and 7in is 146 square inches.

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Answer:

1.15 dollars

Step-by-step explanation:

To calculate the cost of 1 pound of apples, we can divide the total cost of 4 pounds of apples by 4.

The total cost of 4 pounds of apples is 4.60 dollars.

So, the cost of 1 pound of apples would be 4.60 / 4 = 1.15 dollars.

The cost in dollars of 1 pound apples is $1.15

Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers.

The process of dividing a number up into equal parts, and finding how many equal parts can be made. It is represented by the symbol "÷" or sometimes "/"

Division is breaking a number up into an equal number of parts.

Given that, a shopper buys 4 pounds of apples for 4.60, we need to find the cost in dollars of 1 pound of apples,

To calculate the cost of 1 pound of apples, we can divide the total cost of 4 pounds of apples by 4.

The total cost of 4 pounds of apples is 4.60 dollars.

So, the cost of 1 pound of apples would be 4.60 / 4 = 1.15 dollars.

Hence, the cost in dollars of 1 pound apples is $1.15

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a) The linear function that gives the sales in x years after 2011 is of: S(x) = 18x + 191.

b) The slope of the graph of S(x) is of 18, meaning that the sales increase by 18 million a year.

c) The online sales were of 227 billion in the year of 2013.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

We take 2011 as the reference year, when the sales were of 191 billion, hence the parameter b is given as follows:

b = 191.

In 3 years, the sales increased by 54 billion, hence the slope m is given as follows:

m = 54/3

m = 18.

Hence the function modeling the sales in x years after 2011 is given as follows:

y = 18x + 191.

The sales were of 227 billion x years after 2011, for which y = 227, hence:

227 = 18x + 191

18x = 36

x = 2.

2 + 2011 = 2013.

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The number of pushups done by Grace and Kathryn is 7 and 35 respectively

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data,

Let the equation be represented as A

Now , the value of A is

Let the number of pushups by Grace be x

Let the number of pushups by Kathryn be y

Now, Kathryn did 5 times as many pushups as Grace

[tex]y = 5x[/tex] be equation (1)

And, they did a total of 42 pushups together

[tex]x + y = 42[/tex] be equation (2)

Substituting the value of y in the equation, we get

[tex]x + 5x = 42[/tex]

[tex]6x = 42[/tex]

Divide by 6 on both sides, we get

[tex]x = 7[/tex] pushups

And, [tex]y = 35[/tex] pushups

So, the number of pushups by Grace is 7 and by Kathryn is 35 respectively

The value of x using similar triangles theorem is = 13/3.

Triangles that resemble one another but may not be exactly the same size are said to be comparable triangles.

When two objects have the same shape but different sizes, they can be said to be comparable.

This indicates that comparable shapes superimpose one another when amplified or de-magnified.

The term "Similarity" refers to this characteristic of like shapes.

Now in the given figure,

the proportion of the triangles' side is same.

So, 12/4 = 13/x

x = 13/3

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Therefore, the rate of change over the interval [-1, 2] for Function B is 32/3.

In mathematics, a function is a rule that assigns to each input value (also known as the independent variable) a unique output value (also known as the dependent variable). In other words, a function is a relationship between a set of inputs and a set of possible outputs with the property that each input is related to exactly one output. Functions can be represented in various forms such as algebraic expressions, tables, graphs, or even words. Functions play a crucial role in many areas of mathematics, science, and engineering, as they provide a way to model and describe real-world phenomena and relationships between quantities.

Here,

For Function A, the formula is f(x) = 4x + 1. To find the rate of change over the interval [-1, 2], we need to calculate the slope of the line that passes through the points (-1, f(-1)) and (2, f(2)). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the values for Function A, we get:

slope = (f(2) - f(-1)) / (2 - (-1))

slope = (4(2) + 1 - (4(-1) + 1)) / (2 - (-1))

slope = (9 + 5) / 3

slope = 14/3

Therefore, the rate of change over the interval [-1, 2] for Function A is 14/3.

For Function B, the formula is g(x) = 10x + 2. To find the rate of change over the interval [-1, 2], we need to calculate the slope of the line that passes through the points (-1, g(-1)) and (2, g(2)). The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

slope = (y2 - y1) / (x2 - x1)

Substituting the values for Function B, we get:

slope = (g(2) - g(-1)) / (2 - (-1))

slope = (10(2) + 2 - (10(-1) + 2)) / (2 - (-1))

slope = (20 + 12) / 3

slope = 32/3

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b + d ≥ 9 this inequality states that the sum of hours babysitting, b, and hours walking dogs, d, must be greater than or equal to 9 in order for Ryan to meet his weekly work requirement.

What is inequality?

In mathematics, an inequality is a statement that compares two quantities, indicating whether they are equal or not, and in what direction they differ. An inequality is represented by one of the following symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to).

The inequality that represents the possible values for the number of hours babysitting, b, and the number of hours walking dogs, d, that Ryan can work in a given week is:

b + d ≥ 9

This inequality states that the sum of hours babysitting, b, and hours walking dogs, d, must be greater than or equal to 9 in order for Ryan to meet his weekly work requirement.

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