16. Find the average rate of change of the function f(x)=x²+2x+1 over the interval [0,1/2]

Answer:

Step-by-step explanation:

Here goes:

So, we want to write out an expression for this situation. I'm not exactly sure what you've been taught in class, but personally, I would start out with a let statement looking something like this:

Let x = the total cost Maura spent canoeing and fishing.

From there, we know you have 4 canoeing and 3 fishing trips. From here, we just plug in numbers for what we know. So, we get an equation that looks like this:

4(8) + 3(5) = x

Now, this is an equation, however, if you just had an expression like the question asked for, there wouldn't be anything to evaluate. Plus, it's always kind of satisfying to get a number answer. So, with a little bit of math, we get 32+15 = x, and a grand total of 47 = x.

Forgot what x was? Thank goodness you wrote a let statement. Look back up if you need the refresher.

Finally, if you need any other clarification, feel free to reach out with another question.

The frequency of rotations in rev/s for 25 revolutions in 15 seconds is 1.7 rev/s (to two significant figures).

How to find the frequency of rotations:

To determine the frequency of rotations, we divide the number of revolutions by the duration, which is in seconds.

We get a unit of revolutions per second, which is abbreviated as rev/s.

To solve this question, we will use the following formula:

Frequency of rotations (in rev/s) = the number of revolutions ÷ duration

First, we substitute the given values into the formula:

F = 25 ÷ 15

The number of revolutions is 25, and the duration is 15 seconds.

After that, we simplify:

F = 5/3

Next, we convert the fraction to two significant figures, which is 1.7 (rounded to one decimal place).

Therefore, the frequency of rotations in rev/s is 1.7 rev/s (to two significant figures).

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Answer:

1,520

Step-by-step explanation:

The amount of milk used per minute in a commercial kitchen that uses 3/4 of a cup of milk every 4/6 of a minute is 1/2 cup of milk.

In a recipe or cooking, fractions are frequently used. We can use them to measure ingredients such as sugar, butter, milk, and other items. The numerator of the fraction refers to the number of parts that are utilized. The denominator, on the other hand, refers to the whole.

The fraction 3/4 can be defined in the following ways: 3 parts out of 4 parts,75 parts out of 100 parts,15 parts out of 20 parts,The fraction 4/6 can be reduced as:4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3Thus, the fraction 4/6 represents 2/3 or two parts out of three parts.

We can use proportions to figure out how many cups of milk are used per minute. To do that, we need to convert the given quantity of milk into a fraction that represents the amount of milk used per minute

The kitchen uses 3/4 of a cup of milk every 4/6 of a minute.=> The fraction that represents the amount of milk used per minute = [3/4 ÷ 4/6]=> Multiplying the numerator and denominator of the above fraction by 6, we get:[3/4 ÷ 4/6] = [3/4 × 6 ÷ 4/6 × 6] = [18/24 ÷ 24/24] = 18/24= 3/4 (Reduced Form)Therefore, 3/4 of a cup of milk is used per 4/6 of a minute, or 1/2 cup of milk per minute, if we simplify it further.

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0.0166 correct to two significant figures is 0.016.

To write 0.0166 to two significant figures, we need to look at the first two significant digits of the number, which are 1 and 6.

Since the digit after 6 is less than 5, we round down the last significant digit (6) to get:

0.016

Therefore, 0.0166 correct to two significant figures is 0.016.

Significant figures (also known as the significant digits, precision or resolution) of a number in positional notation are digits in the number that are reliable and necessary to indicate the quantity of something.

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(a) The eigenvalues of the coefficient matrix is  [-1,3]  and for λ=42, we get the eigenvector [1,5].

Itcan be found by solving the characteristic equation |A-λI|=0, where A is the coefficient matrix and λ is the eigenvalue. Solving for λ, we get λ=0 and λ=42.

o find the eigenvectors, we substitute each eigenvalue into the equation (A-λI)x=0 and solve for x. For λ=0, we get the eigenvector [-1,3]. For λ=42, we get the eigenvector [1,5].

(b) The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5].

To find the real-valued solution to the initial value problem, we can use the eigenvectors and eigenvalues to diagonalize the coefficient matrix. We have A = PDP^-1, where P is the matrix whose columns are the eigenvectors and D is the diagonal matrix with the eigenvalues on the diagonal.

Using the initial condition y2(0) = 15, we can solve for the constants c1 and c2.

The solution is y(t) = c1e^(0t)[-1,3] + c2e^(42t)[1,5]. Solving for c1 and c2 using the initial condition, we get

y(t) = [-15e^(42t) + 3e^(0t), 15e^(42t) + 5e^(0t)].

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Answer: 1000 x 1.05^n

Step-by-step explanation:

1.05 is the increase experienced each year and n is the number of years.

Answer:

The population of bees is increasing by 5% each year since 2010. Let P(n) be the population of bees in the nth year since 2010.

The explicit formula for the bee population can be found using the formula for compound interest:

P(n) = P(0) * (1 + r)^n

where P(0) is the initial population in 2010, r is the annual growth rate, and n is the number of years since 2010.

Substituting the given values, we have:

P(n) = 1000 * (1 + 0.05)^n

Simplifying the expression, we get:

P(n) = 1000 * 1.05^n

Therefore, the explicit formula that models the bee population for n years since 2010 is P(n) = 1000 * 1.05^n.

By using the Cauchy-Riemann equations on a real-valued function, it can be proven that the function f(z) is constant in the domain D. This is important for understanding analytic functions in complex analysis.

To prove that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D, let a function f be analytic everywhere in a domain D. We know that a real-valued function is said to be a function whose values lie on the real line. In the case of the complex plane, a function whose values lie on the real line is real-valued.

The Cauchy-Riemann equations, which define the necessary conditions for a function f(z) to be analytic in a domain, say that the imaginary component of f(z) is determined by its real component.

To be more precise, if f(z) is real-valued for all z in D, then we can say that:u(x, y) = f(z),v(x, y) = 0

By definition, the Cauchy-Riemann equations can be stated as:

∂u/∂x = ∂v/∂y∂u/∂y = -∂v/∂x

Taking the first equation, we get:

∂u/∂x = ∂v/∂y => ∂v/∂y = 0

Since v is equal to 0 for all values of x and y, the above equation reduces to ∂u/∂x = 0, which implies u is constant with respect to x.

Similarly, taking the second equation, we get:

∂u/∂y = -∂v/∂x => ∂u/∂y = 0

Since u is equal to a constant for all values of x and y, the above equation reduces to ∂v/∂y = 0, which implies v is constant with respect to y. Since u and v are both constant with respect to their respective variables, u + iv = f(z) is a constant with respect to z throughout the domain D. Thus, we have proved that if f(z) is real-valued for all z in D, then f(z) must be constant throughout D.

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Confidence Interval: The confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as [tex]16,322 < \mu_1 - \mu_2 < 31,338.[/tex]

Given: Population 1: [tex]n_1 = 40, \mu_1 = 183,100, \sigma_1 = 21,845[/tex]

Population 2: [tex]n_2 = 33, \mu_2 = 161,500, \sigma2 = 24,820[/tex]

To construct a 95% confidence interval for the true difference between the mean home prices in the two areas, we need to calculate the sample mean difference between the two populations, as well as the standard error.

The sample mean difference is given by: [tex]\bar{x}_1 - \bar{x}_2 = 183,100 - 161,500 = 21,600[/tex]

The standard error is given by: [tex]s = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \\s= \sqrt{\frac{21,845^2}{40} + \frac{24,820^2}{33}} = 8,205[/tex]

Using the standard normal distribution table, with a confidence level of 95%, we obtain a z-score of ±1.96. Using these values, the 95% confidence interval for the true difference between the mean home prices in the two areas is given by: 21,600 - (1.96 * 8,205) < μ1 - μ2 < 21,600 + (1.96 * 8,205)

Solving for the inequality, we get 16,322 < μ1 - μ2 < 31,338

Therefore, the confidence interval for the true difference between the mean home prices in the two areas at a 95% confidence level is given as $16,322 < μ1 - μ2 < $31,338.

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Answer: The smallest integer value of x that satisfies the inequality -2x+1<-9 is x=5.

Step-by-step explanation:

Answer: The smallest integer value of x that satisfies the inequality -2x+1<-9 is x=5.

Explanation:

To solve the inequality, we need to isolate the variable x on one side of the inequality symbol. Here are the steps:

   Subtract 1 from both sides of the inequality:

   -2x < -10

   Divide both sides of the inequality by -2, remembering to reverse the direction of the inequality symbol:

   x > 5

Therefore, the smallest integer value of x that satisfies the inequality is x = 5, since any value less than 5 would make the inequality false.

Answer:

x = 6

Step-by-step explanation:

- 2x + 1 < - 9 ( subtract 1 from both sides )

- 2x < - 10

divide both sides by - 2, reversing the symbol as a result of dividing by a negative quantity.

x > 5

since x must be greater than 5, it cannot equal 5

then the smallest integer value of x is x = 6

Step-by-step explanation:

hey, you just changed the angles in the question.

this following answer was about the angles of depression of 15° and 30°.

you cannot change the problem, when the answers are already given for the original problem.

so, I will add a copy with the adapted numbers for 45° and 30° after my original answer.

this creates 2 right-angled triangles.

the right angle is in both cases the angle where house meets the ground.

they also share one leg : the height of the house (10 m).

the second legs are the ground distances of the cars from the house.

the 2 Hypotenuses are the line of sight from the roof to the corresponding car.

remember, the sum of all angles in a triangle is always 180°.

again, we know one angle : the 90° angle.

but we also know a second angle based on the angles of depression (the "downward looking angles").

the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.

so, this is 90-15 = 75° and 90-30 = 60°.

the angles at the cars on the ground are then

angle car 1 = 180 - 90 - 75 = 15°

angle car 2 = 180 - 90 - 60 = 30°

now, remember the trigonometric triangle inscribed in a circle.

imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.

the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.

and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.

so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.

for car 1 we have

10m/sin(15) = 38.63703305... m line of sight

that means ground distance of car 1 is

cos(15)×38.63703305... = 37.32050808... m

for car 2 we have

10m/sin(30) = 20 m line of sight

that means ground distance of car 2 is

cos(30)×20 = 17.32050808... m

since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :

37.32050808... - 17.32050808... = 20 m

the cars are 20 m apart.

and now for the angles of depression of 45° and 30° :

the triangle internal angle at the rooftop is the complementary angle (the difference to 90°) of the angle of depression.

so, this is 90-45 = 45° and 90-30 = 60°.

the angles at the cars on the ground are then

angle car 1 = 180 - 90 - 45 = 45°

angle car 2 = 180 - 90 - 60 = 30°

now, remember the trigonometric triangle inscribed in a circle.

imagine that the vertex at the car is the center of the corresponding circle around the trigonometric triangle.

the height of the house is then sine of the angle at the car multiplied by the Hypotenuse (= the line is sight from the rooftop to the car), which is the angle creating radius of the circle.

and the ground distance is the cosine of that same angle multiplied by the Hypotenuse.

so, we need to get the ratio of the height of the house / sin(car angle) to get the length of the Hypotenuse (line of sight). with that we can then calculate the ground distance as cosine of the angle multiplied by the same Hypotenuse.

for car 1 we have

10m/sin(45) = 14.14213562... m line of sight

that means ground distance of car 1 is

cos(45)×14.14213562... = 10 m

logically, as for 45° sine and cosine are equal.

for car 2 we have

10m/sin(30) = 20 m line of sight

that means ground distance of car 2 is

cos(30)×20 = 17.32050808... m

since both cars are driving on the same side of the house in the same direction, the distance between both cars is purely the difference between their distances from the house :

17.32050808... - 10 = 7.32050808... m

≈ 7.32 m

the cars are about 7.32 m apart.

The triangle ΔA'B'C' formed following the dilation of ΔABC is a similar

triangle to ΔABC.

. a. Please find attached the drawing of the dilated triangle ΔA'B'C', created with MS Excel

b. The properties of dilations indicate that ∠B = ∠B'

Reasons:

a. With the assumption that the vertices of the triangle are;

A(0, -3), C(0, 5), and B(6, 3)

Let point P = (0, 0)

A' = 2/3 *(0,-3) = (0, -9/2) (0, -4.5)

C' = 2/3 *(0,5) = (0, 15/2) (0, 7.5)

B' = 2/3 *(6,3) = (9, 9/2) (9, 4.5)

We have;

b. From the attached diagram, and from the properties of dilation, given

that the image of ΔABC is larger than the image of ΔA'B'C' by a scale

factor of 1.5, we have that the ratio of the corresponding sides of ΔABC

and ΔA'B'C' are equal and therefore the angle formed by segment BC and BA which is ∠B and the angle formed by segment B'C' and B'A' which is ∠B'. are equal.

AC/AB = A'C'/A'B'

AC/Sin(B) = AB/Sin (C)

AC/AB = Sin(B)/Sin(C)

Similarly, we have;

A'C'/A'B' = Sin(B')/Sin(C')

Therefore;

Sin(B)/Sin(C) = Sin(B')/Sin(C')

According to the properties of dilation, ∠B = ∠B'

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The probability that both bulbs are red is 0.126 and The probability that the first bulb selected is red and the second yellow is 0.113

Probability means the possible outcome occur when an event take place.

(a) The probability that both bulbs are red ,

= 11/30 * 10/29

= 11/87

= 0.126

So, The probability that both bulbs are red is 0.126

(b) The probability that the first bulb selected is red and the second yellow,

= 11/30 * 9/29

= 33/290

= 0.113

So, The probability that the first bulb selected is red and the second yellow is 0.113

(c) The probability that the first bulb selected is yellow and the second red,

= 9/30 * 11/29

= 33/290

= 0.113

So, The probability that the first bulb selected is yellow and the second red is 0.113

(d) The probability that one bulb is red and the other yellow,

= 33/290 + 33/290 ( Add (b) and (c) )

= 33/145

= 0.227

And, The probability that one bulb is red and the other yellow is 0.227 .

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The equation line of the of the road in slope-intercept form is; y = 2·x - 10

The standard form of the equation of a line is Ax + Bx + C, where A, B, and C are constants and A and B are nonzero numbers.

The parameters for the new road are;

The road will be parallel to village way with points (0, 5), and (-4, -3)

The road will pass through the intersection  of Gray Dr and Canon Rd., which is the point with coordinates (3, -4)

Required; The equation of the road

Since the new road is parallel to Village Way, which has slope;

m = (5 - (-3))/(0 - (-4)) = 8/4 = 2

The slope of the new road will also be 2.

Let the equation of the new road be y = m·x + c, where m = 2 is the slope we just found. To find c, we use the fact that the road passes through the point (3, -4);

y - (-4) = 2 × (x - 3)

y = 2·x - 6 - 4 = 2·x - 10

y = 2·x - 10

Therefore, c = -10

Therefore, the equation of the new road in slope-intercept form, therefore is; y = 2·x - 10

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The graph of an exponential function with an initial value of 10 and a base of 1.3z. Therefore option D is correct.

The function f(x) is an exponential function with a base of 1.3 and an initial value of 10. The graph of an exponential function with a base greater than 1 increases rapidly as x increases. Therefore, option a can be eliminated.

Option b is not a graph of an exponential function, as the function is not continuous and does not approach any asymptote.

Option c shows an exponential function with an initial value of 10 and a base of 1.3/10, which is less than 1. This means that the function would decrease over time, which is not consistent with the problem statement.

Option d shows an exponential function with an initial value of 10 and a base of 1.3, which is consistent with the problem statement. Therefore, option d is the correct answer.

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Answer: n=40

Step-by-step explanation:

let me know if i got this right for you broski

now dance

Answer: -17x+3

Step-by-step explanation:

y=2x+3 and y=15-x

15x-2x+3

-17x+3

you can try this

The target domain for a Poisson distribution is given the term (0, inf) which can be seen correct in option B.

A Poisson distribution's target domain is (0, inf). This means that the Poisson distribution can only be specified for non-negative integer values of the random variable it is modelling.

The Poisson distribution is a discrete probability function, which indicates that the variable may only take particular values from a finite list of integers. A Poisson distribution estimates how many times an event will occur in "x" amount of time. In other words, it is the probability distribution resulting from the Poisson experiment.

A Poisson experiment is a statistical experiment that categorises the experiment as either successful or unsuccessful. A limiting process of the binomial distribution is the Poisson distribution.

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Complete question:

What is the target domain for a Poisson distribution?

1) (-inf, inf)

2) (0, inf)

3) (-inf, 0]

4) [0, inf)

9 hikers are the bare minimum that might have covered the range of 6 to 17 miles because that distance falls inside the typical interval of 10 x 15 miles.

Rearrange the function using fundamental algebraic concepts to determine the value of x when the derivative equals 0.

This response gives the x-coordinate of the function's vertex, which is where the maximum or minimum will occur.

To determine the minimum or maximum, rewrite the solution into the original function.

The greatest and smallest values of a function, either within a specific range (the local or relative extrema) or throughout the entire domain, are collectively referred to as extrema (PL: extrema) in mathematical analysis.

b)  the maximum number of hikers who could have walked between 6 miles and 17 miles is 19.

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Answer: 198 families live farther than 1 mile from the square.

Step-by-step explanation:

We know that there are 238 families that live within 1 mile of the village square. To find the number of families that live farther than 1 mile from the square, we can subtract 238 from the total number of families:

436 - 238 = 198

Therefore, 198 families live farther than 1 mile from the square. We can do this subtraction mentally without needing a calculator.

It will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C when following exponential decay model.

A quantity declines over time proportionate to its existing value through a process known as exponential decay. An exponential function of the form f(t) = ab raised to t, where an is the beginning value, b is the decay factor (a number between 0 and 1), and t represents time, mathematically describes this.

Several real-world circumstances, like population increase, radioactive decay, and the loss of electrical charge in a capacitor, exhibit exponential decay.

Given that the situation follows a exponential decay model.

The exponential decay is given as:

[tex]T(t) = T0 * e^{(-rt)}[/tex]

Substituting the values T0 = 54.8, r = 0.02, and T(t) = 44.9.

[tex]44.9 = 54.8 * e^{(-0.02t)}\\0..8208 = e^{(-0.02t)}\\ln(0.8208) = -0.02t\\t = ln(0.8208)/(-0.02) = 27.7 minutes[/tex]

Hence, it will take approximately 27.7 minutes for the coffee to cool from 54.8°C to 44.9°C.

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Answer: Kelinatra (x) worked 5 hours and Samantha (y) worked 6 hours

Step-by-step explanation:

     We will use the variables x and y the question provides. We know the time worked by each person added together will equal the combined total time. We can write an equation to show this using addition.

        x + y = 11 hours

     Next, we know that Keilantra ironed 30 per hour, Samantha ironed 15 per hour and that they ironed 240 shirts. We can write another equation to represent this using addition and multiplication.

     30x + 15y = 240

     Next, we will graph these two equations. See attached. The solution is the point of intersection written as (x, y). This is (5, 6) meaning that Kelinatra (x) worked 5 hours and Samantha (y) worked 6 hours.

When the initial population increases by 10% in 10 years, it is growing at a rate of approximately 78.04 persons per year at t = 60.

To find the population in 60 years, we need to use the formula:

P(t) = P0rt

P0 is the initial population

r is the rate of growth

t is the time in years.

So, given that the initial population of 500 increases by 10% in 10 years, we can find r as follows:

10% increase in 10 years means that the population has grown to (100% + 10%) = 110% of its original size in 10 years.

Therefore, we have:

P(10) = 500(1 + 0.10)

= 550

Now we can use these values to solve for:

r: 550 = 500er

⇒ er = 550/500

⇒ r = ln(550/500)/10

= 0.04879 (rounded to 5 decimal places)

Therefore, the population in 60 years is:

P(60) = 500e0.04879 × 60 ≈ 1599 (rounded to the nearest person)

The population is growing at a rate of:

P'(t) = rP(t),

so at t = 60, we have:

P'(60) = 0.04879 × 1599 ≈ 78.04 (rounded to two decimal places)

Therefore, the population is growing at a rate of approximately 78.04 persons per year at t = 60.

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Answer:

Step-by-step explanation:

To find the perimeter of the triangle, we need to add the lengths of all three sides. So,

Perimeter = (5p^2 − q^2) + (6p^2 − 8 + 5q^2) + (−5p^2+ 8 + 9q^2)

Simplifying the above expression, we get:

Perimeter = 6p^2 + 14q^2 - 8

Therefore, the perimeter of the triangle is 6p^2 + 14q^2 - 8.

The 95% confidence interval for the true average number of alcoholic drinks all UF male students have in a one week period is (4.58, 7.10) that is option C.

Using the following formulas the lower and upper limits of the Interval are calculated,

n = 60

x = 5.84

s = 4.981 = 95% = 0.95

Because the population standard deviation is unknown, the Student T-distribution should be used. Yet, because the sample is huge, some books will utilise the normal distribution. I'll provide solutions for both techniques.

Error = z x s/√n

= 1.96 x 4.98/√60

Error margin ≈1.26

Lower limit = 5.84 - Error

= 5.54 - 1.2601

= 4.58

Upper limit = 5.84 + Error

= 5.84 + 1.2601

= 7.10

Therefore, the upper limit and lower limit is 4.58 and 7.10.

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Complete question:

A study on students drinking habits asks a random sample of 60 male UF students how many alcoholic beverages they have consumed in the past week. The sample reveals an average of 5.84 alcoholic drinks, with a standard deviation of 4.98. Construct a 95% confidence interval for the true average number of alcoholic drinks all UF male students have in a one week period.

A. (4.78, 6.90)

B. (0, 15.60)

C.(4.58, 7.10)

D. (-3.92, 15.60)

Hence, 28 toy soldiers are the correct answer.

A group in mathematics is created by combining a set with a binary operation. For instance, a group is formed by a set of integers with an arithmetic operation and a group is also formed by a set of real numbers with a differential operator.

Let's refer to the quantity of toy soldiers as "x".

We are aware that x is within the range of 27 and 54 thanks to the problem.

x can be divided by 4 without any remainders.

The residual is 6 when x is divided by 7.

The leftover after dividing x by five is three.

These criteria allow us to construct an equation system and find x.

Firstly, we are aware that x can be divided by 4 without any residual. As a result, x needs to have a multiple of 4. We can phrase this as:

x = 4k, where k is some integer.

Secondly, we understand that the remaining is 6 when x is divided by 7. This can be stated as follows:

x ≡ 6 (mod 7)

This indicates that x is a multiple of 7 that is 6 more than. We can solve this problem by substituting x = 4k:

4k ≡ 6 (mod 7)

We can attempt several values of k until we discover one that makes sense for this equation in order to solve for k. We can enter k in to equation starting using k = 1, as follows:

4(1) ≡ 6 (mod 7)

4 ≡ 6 (mod 7)

It is not true; thus we need to attempt a next value for k. This procedure can be carried out repeatedly until the equation is satisfied for all values of k.

k = 2:

4(2) ≡ 6 (mod 7)

1 ≡ 6 (mod 7)

k = 3:

4(3) ≡ 6 (mod 7)

5 ≡ 6 (mod 7)

k = 4:

4(4) ≡ 6 (mod 7)

2 ≡ 6 (mod 7)

k = 5:

4(5) ≡ 6 (mod 7)

6 ≡ 6 (mod 7)

k = 6:

4(6) ≡ 6 (mod 7)

3 ≡ 6 (mod 7)

k = 7:

4(7) ≡ 6 (mod 7)

0 ≡ 6 (mod 7)

We have discovered that the equation 4k 6 (mod 7) is fulfilled when k = 7. Thus, we can change k = 7 to x = 4k to determine that:

x = 4(7) = 28

This indicates that there are 28 toy troops. Yet we also understand that the leftover is 3 when x is divided by 5. We don't need to take into account any other values of x because x = 28 satisfies this requirement.

28 toy soldiers are the correct response.

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