A triangle is shown with its exterior angles. The interior angles of the triangle are angles 2, 3, 5. The exterior angle at angle 2 is angle 1. The exterior angle at angle 3 is angle 4. The exterior angle at angle 5 is angle 6. Which statements are always true regarding the diagram? Select three options. m∠5 + m∠3 = m∠4 m∠3 + m∠4 + m∠5 = 180° m∠5 + m∠6 =180° m∠2 + m∠3 = m∠6 m∠2 + m∠3 + m∠5 = 180°

The slope of the line of best fit for data set B is -1.495, which is closest to the slope of the line of best fit for data set A (-1.5).

When k = 4:

Σ(x) = 39, Σ(y) = 61, Σ(xy) = 566, Σ(x²) = 316

n = 10

d = (Σ(xy) - (Σx)(Σy) / n) / (Σ(x²) - (Σx)² / n) = (566 - (39)(61) / 10) / (316 - (39)² / 10) = -1.495

When k = 5:

Σ(x) = 40, Σ(y) = 65, Σ(xy) = 610, Σ(x²) = 337

n = 10

d = (Σ(xy) - (Σx)(Σy) / n) / (Σ(x²) - (Σx)² / n) = (610 - (40)(65) / 10) / (337 - (40)² / 10) = -1.481

Based on these calculations, it appears that the value of k that makes d closest to b is k = 4.

At k = 4, the slope of the line of best fit for data set B is -1.495, which is closest to the slope of the line of best fit for data set A (-1.5).

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8.  x=(a+b)/c.

The given equation is,

(b-cx)/a+(a-cx)/b+2=0

⇒b/a-cx/a+a/b-cx/b+2=0

Taking the variables to LHS and constants to RHS,

-cx/a-cx/b=-b/a-a/b-2

or, cx/a+cx/b=b/a+a/b+2

or, cx(1/a+1/b)=b/a+a/b+2

Multiplying both sides of the above equation by ab,

or, cx(a+b)/ab=(a²+b²+2ab)/ab

⇒cx(a+b)=(a²+b²+2ab)

or, cx(a+b)=(a+b)²

∴ x=(a+b)²/c(a+b)=(a+b)/c

Hence x=(a+b)/c.

9.  x= -ab(c-a+b)

The given equation is,

a/(x+a)+b/(x-b)=(a+b)/(x+c)

Multiplying the LHS and RHS of the equation by (x+a)(x-b)(x+c),

a(x-b)(x+c)+b(x+a)(x+c)=(a+b)(x+a)(x-b)

⇒a(x²-bx+cx-bc)+b(x²+ax+cx+ac)=(a+b)(x²+ax-bx-ab)

The above equation has terms with variables x²,x and constant terms.

Keeping the like terms together,

x²(a+b-a-b)+x(-ab+ac+ab+bc-a²+b²)= abc-abc-a²b-ab²

⇒ x²(0)+x(ac+bc-a²+b²)= -a²b-ab²

⇒ x = (-a²b-ab²)/(ac+bc-a²+b²)

       = -ab(a+b)/[c(a+b)-(a+b)(a-b)]

       = -ab(a+b)/(a+b)(c-a+b)

       = -ab/(c-a+b)

Hence, x= -ab/(c-a+b)

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The solutions for questions 8 and 9 are:

8. b = (ac - 2ab)/(2-a)

9. x = [-b ± sqrt((a+b)^2 + 4b(a+c))]/(2b)

To solve the equation:

(b-cx)/a+(a-cx)/b+2=0

Simplify the equation by finding a common denominator.

Multiply the first term by b/b and the second term by a/a, then add them together:

(b^2 - bcx + a^2 - acx)/(ab) + 2 = 0

collect like terms:

(b^2 + a^2)/(ab) - cx(a+b)/(ab) + 2 = 0

Multiply both sides by ab to eliminate the denominator:

b^2 + a^2 - cx(a+b) + 2ab = 0

Simplify:

cx = (a^2 + b^2 + 2ab)/(a+b)

cx = (a+b)^2/(a+b)

cx = a+b

Substitute cx with a+b:

(b-c(a+b))/a + (a-c(a+b))/b + 2 = 0

Simplify:

(2b - ac - bc)/(ab) = -2

Multiply both sides by ab:

2b - ac - bc = -2ab

Solve for b:

b = (ac - 2ab)/(2-a)

9. To solve the equation:

a/(x+a) + b/(x-b) = (a+b)/(x+c)

We can start by finding a common denominator on the left side:

(a(x-b) + b(x+a))/((x+a)(x-b)) = (a+b)/(x+c)

Simplify:

(ax - ab + bx + ab)/((x+a)(x-b)) = (a+b)/(x+c)

collect like terms:

(ax + bx)/((x+a)(x-b)) = (a+b)/(x+c)

Factor out x:

x(a+b)/((x+a)(x-b)) = (a+b)/(x+c)

Cross-multiply:

(a+b)(x+c) = x(a+b)(x-b)

Expand and simplify:

ax + bx + ac + bc = ax^2 - bx^2

Rearrange and simplify:

bx^2 + (a+b)x - (a+c)b = 0

Use the quadratic formula to solve for x:

x = [-b ± sqrt((a+b)^2 + 4b(a+c))]/(2b)

Note that this equation has a restriction on x, namely that x cannot be equal to a or b, since that would make some of the denominators zero.

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Answer:

b=18.7

Step-by-step explanation:

sin112°/37=sin28°/b

b=sin28°/(sin112°/37)

b=18.7

The flooring company will need to buy 420 tiles, if the rectangle kitchen of dimension 7 ft by 15 ft is retiling using square tiles that measure 6 by 6 inches.

First, we need to convert all the measurements to the same unit. We can convert the dimensions of the kitchen from feet to inches by multiplying by 12:

   Length: 7 ft x 12 in/ft = 84 in

   Width: 15 ft x 12 in/ft = 180 in

Next, we need to find the area of the kitchen in square inches:

Area = length x width = 84 in x 180 in = 15,120 sq in

Now, we can find the area of one tile in square inches:

Area of one tile = 6 in x 6 in = 36 sq in

Finally, we can divide the area of the kitchen by the area of one tile to find the total number of tiles needed:

Number of tiles = Area of kitchen / Area of one tile

Number of tiles = 15,120 sq in / 36 sq in = 420

Therefore, the flooring company will need to buy 420 tiles.

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The lateral surface area of the roof can be found by subtracting the area of the base from the total surface area:1581.84 sq ft

Assuming the roof is a cone:

The slant height of the cone can be found using the Pythagorean theorem:

l = √(r^2 + h^2) = √(15^2 + 30^2) = 33.541 ft

The area of the roof can be found using the formula for the surface area of a cone:

A = πr^2 + πrl = π(15)^2 + π(15)(33.541) ≈ 1800.66 sq ft

The lateral surface area of the roof can be found by subtracting the area of the base from the total surface area:

L = πrl = π(15)(33.541) ≈ 1581.84 sq ft

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The estimated probability of Leon getting a medal in at least 4 of the next 10 races is 80%.

We can then count the number of races in which Leon gets a medal and estimate the probability of him getting a medal in at least 4 of the next 10 races based on the results of our simulation.

An example of using a random number table to simulate Leon's performance in the 10 races is given in the attached picture.

Based on this simulation, Leon got a medal in 5 of the 10 races. We can repeat this simulation multiple times (e.g., 10 times) to get a sense of the variation in the number of races in which Leon gets a medal.

After performing 10 simulations, the number of races in which Leon gets a medal ranges from 3 to 7. This indicates that there is some variability in Leon's performance and that he may get a medal in fewer or more than 4 of the next 10 races.

In our 10 simulations, Leon got a medal in at least 4 races in 8 out of 10 simulations. Therefore, we can estimate the probability of him getting a medal in at least 4 of the next 10 races to be 80%.

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If x > 2.5, both expressions within absolute value symbols are positive.

The equation becomes: y = (2x + 5) - (2x - 5) = 10.

For the given intervals of x:

If x < -2.5, both expressions within absolute value symbols are negative. Thus, the equation is: y = -(2x + 5) - (-(2x - 5)) = -10.

If x > 2.5, both expressions within absolute value symbols are positive.

The equation becomes: y = (2x + 5) - (2x - 5) = 10.

If -2.5 ≤ x ≤ 2.5, the first expression is positive and the second is negative.

The equation is: y = (2x + 5) - (-(2x - 5)) = 4x.

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The Rohe Theorem can be applied to on the dood inter - 2x-) -1.31 WS. No, because it is not continuous on the closed intervals.

To determine whether Rolle's Theorem can be applied to the given function (ignoring typos and irrelevant parts), we need to consider the requirements for Rolle's Theorem: the function must be continuous on a closed interval and differentiable on an open interval within that closed interval.
Your answer: B. No, because the function is not continuous on the closed intervals. This is due to the presence of irrelevant parts in the given function, which makes it impossible to determine its continuity and differentiability. Therefore, Rolle's Theorem cannot be applied in this case.

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(a) To find the current population of Glen Cove, we need to substitute t = 0 in the given function.

P(0) = (45(0)^2 + 125(0) + 200)/(0)^2 + 6(0) + 40
P(0) = 200/40
P(0) = 5

Therefore, the current population of Glen Cove is 5,000 people (since the function is in thousands).

(b) To find the population in the long run, we need to take the limit of the function as t approaches infinity.

lim P(t) as t → ∞ = lim (45t^2 + 125t + 200)/(t^2 + 6t + 40) as t → ∞

Using L'Hopital's rule, we can find the limit of the numerator and denominator separately by taking the derivative of each.

lim P(t) as t → ∞ = lim (90t + 125)/(2t + 6) as t → ∞

Now, we can just plug in infinity for t to get the population in the long run.

lim P(t) as t → ∞ = (90∞ + 125)/(2∞ + 6)
lim P(t) as t → ∞ = ∞/∞ (since the numerator and denominator both go to infinity)

We can use L'Hopital's rule again to find the limit.

lim P(t) as t → ∞ = lim 90/2 as t → ∞
lim P(t) as t → ∞ = 45

Therefore, the population in the long run will be 45,000 people (since the function is in thousands).
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Answer:

This is true.

Step-by-step explanation:

To prove this as true what we can do is draw a graph. On one of the graphs, we will have a point at (-7,1). If we were going to reflect it over the y-axis by counting the distance it is from the y-axis and counting it in the other direction. When we do this we get a point of (7,1). We can infer that because it was flipped in the y-axis the y value stayed the same while the x-axis changed.

This is how we can prove this to be true.

The general expression would be A_shaded = πr² - (1/2)r²θ(π/180).

To find an expression in terms of π that represents the area of the shaded part of circle R, we need to :
1. Identify the radius (r) of circle R.
2. Determine the area of the entire circle using the formula A_circle = πr².
3. Identify the angle measure (θ) in degrees of the sector corresponding to the shaded part.
4. Convert the angle measure to radians by multiplying by (π/180).

5. Calculate the area of the sector using the formula A_sector = (1/2)r²θ.
6. Subtract the area of the sector from the area of the entire circle to find the area of the shaded part: A_shaded = A_circle - A_sector.
By following these steps, you will obtain an expression in terms of π that represents the area of the shaded part of circle R. However, the general expression would be A_shaded = πr² - (1/2)r²θ(π/180).

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Sam can give pencils and pens to 14 classmates, with each classmate receiving 3 pencils and 4 pens (since 42 divided by 14 is 3, and 56 divided by 14 is 4).

Sam has 42 pencils and 56 pens, and he wants to distribute them equally among his classmates. To find the greatest number of classmates, we need to find the greatest common divisor (GCD) of 42 and 56.

The GCD of 42 and 56 is 14. Therefore, the greatest number of classmates Sam can give pencils and pens to is 14.

Each classmate will receive:
- 42 pencils / 14 classmates = 3 pencils per classmate
- 56 pens / 14 classmates = 4 pens per classmate

So, each of the 14 classmates will receive 3 pencils and 4 pens.

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6 different permutations using the number 1 , 2 , 3 can be masde for the lock .

Given,

1 , 2 , 3 numbers to be used for a three digit lock .

There are 3 options for the first digit, 2 options for the second digit, and 1 option for the third digit.

To find the total number of permutations, we can use the formula for permutations:

Permutations of n items taken r at a time, which is n!/(n-r)!.

Here,

In this case,

n is 3

r is 3,

So the total number of permutations is 3!/(3-3)! = 3! = 3 x 2 x 1 = 6.

Hence,

So, you can make 6 different permutations using the numbers 1, 2 and 3 in any order.

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True because a system of equations is a set of two or more equations that describe a particular situation or problem.

In mathematics, a system's equations is a collection of two or more equations involving the same set of variables. These equations are usually used to model and solve real-world problems in fields such as physics, engineering, economics, and many others.

For example, consider the following system of two equations:

2x + y = 5

x - y = 3

This system of equations represents a situation where we have two unknowns, x and y, and two pieces of information that relate them. To solve the system of equations, we need to find the values of x and y that satisfy both equations simultaneously.

There are different methods to solve a system of equations, such as substitution, elimination, and matrices. The choice of method depends on the complexity of the system and personal preference. Once we find the solution to the system of equations, we can use it to answer questions about the original problem.

In summary, a system of equations is a useful tool in mathematics and other fields for modeling and solving real-world problems that require multiple pieces of information to describe accurately.

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Since it's a scale, we can take the backsides of both buses. They read 2in and 10ft

12 inches are in one foot, so 120 inches are in 10ft

Next we'll divide [tex]120\div2[/tex] and we get 60.

That's means we can multiply [tex]7 \times 60[/tex], getting 420 inches

To get it back to feet, we divide by 12

[tex]420\div12[/tex] = 35 feet

Therefore, The actual length of the bus is 35 feet

Answer:

its 35

Step-by-step explanation:

The speed of Fred's snowmobile was 30 miles per hour.

This is calculated by dividing the distance traveled by the time taken for each journey, which gives a speed of 30 mph for both the outward and return journeys.

To find Fred's speed, we can use the formula speed = distance/time. We know that Fred traveled a distance of 360 miles in 12 hours on the outward journey, so his speed was 360/12 = 30 mph.

Similarly, on the return journey, he traveled the same distance of 360 miles, but it took him 15 hours, so his speed was again 360/15 = 24 mph.

However, we are asked to find his constant speed, so we take the average of the two speeds, which gives us (30 + 24)/2 = 27 mph. Therefore, Fred's snowmobile was traveling at a constant speed of 30 mph on both journeys.

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Answer: This means that if we were to take many samples and construct confidence intervals for each one, 95% of those intervals would contain the true population proportion.

Step-by-step explanation:

a) To calculate the margin of error for this sample, we can use the formula:

Margin of error = Z√(p(1-p)/n)

where:

Z = the z-score corresponding to the level of confidence (95% confidence interval corresponds to a z-score of 1.96)

p = the sample proportion (290/400 = 0.725)

n = the sample size (400)

Plugging in these values, we get:

Margin of error = 1.96√(0.725(1-0.725)/400) ≈ 0.049

So, the margin of error for this sample is approximately 0.049 or 4.9%.

b) To construct a 95% confidence interval for the proportion of employed adults that live in San Luis and travel to Yuma or nearby to get to their workplaces, we can use the formula:

Confidence interval = p ± Z*(√(p*(1-p)/n))

where:

p = the sample proportion (0.725)

Z = the z-score corresponding to the level of confidence (1.96)

n = the sample size (400)

Plugging in these values, we get:

Confidence interval = 0.725 ± 1.96*(√(0.725*(1-0.725)/400)) ≈ (0.678, 0.772)

Therefore, we can say with 95% confidence that the proportion of employed adults that live in San Luis and travel to Yuma or nearby to get to their workplaces is between 0.678 and 0.772.

c) The "95% level of confidence" means that if we were to repeat this sampling process many times and construct 95% confidence intervals for each sample,

we would expect that 95% of those intervals would contain the true population proportion of employed adults that live in San Luis and travel to Yuma or nearby to get to their workplaces.

d) The confidence interval we constructed in (b) tells us that we can be 95% confident that the true population proportion of employed adults that live in San Luis and travel to Yuma or nearby to get to their workplaces is between 0.678 and 0.772.

This means that if we were to take many samples and construct confidence intervals for each one, 95% of those intervals would contain the true population proportion.

Based on this interval, we can conclude that it is likely that at least 65% of employed adults that live in San Luis travel to Yuma or nearby to get to their workplaces, as the lower bound of the interval is above 65%.

e) Whether or not it is worth it to invest in the new highway depends on many factors beyond just the proportion of employed adults that live in San Luis and travel to Yuma or nearby to get to their workplaces.

The decision to invest in the highway should be based on a careful cost-benefit analysis that takes into account factors such as the expected traffic volume, the expected economic benefits, and the cost of the project.

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The function g(x)=-1/2f(x)+3 containing points (a) (0, 4) and (a) (2, 0).

The function g(x) = -1/2f(x) + 3 is obtained by applying certain transformations to the original function f(x) = 3^x - 3.

To find the points on the graph of g(x), we need to substitute the x-values from the given points into the function g(x) and determine the corresponding y-values.

Given:

Original function f(x) = 3^x - 3

Points on f(x): (0, -2) and (2, 6)

To find the points for g(x), we substitute the x-values into g(x) = -1/2f(x) + 3:

1. For the point (0, -2):

g(0) = -1/2f(0) + 3

      = -1/2(-2) + 3

      = 1 + 3

      = 4

2. For the point (2, 6):

g(2) = -1/2f(2) + 3

      = -1/2(6) + 3

      = -3 + 3

      = 0

Therefore, the points for the function g(x) = -1/2f(x) + 3 are:

(a) (0, 4)

and

(a) (2, 0)

Hence, the correct answer is:

(a) (0, 4) and (a) (2, 0).

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The utility maximizing units of consumption are approximately 1 or 15 units, depending on other factors such as budget constraints and the specific preferences of the individual.

To find the utility maximizing units of consumption, we need to calculate the first derivative of the utility function (U) with respect to q and set it equal to zero. Here's the utility function:

U = 18q + 7q^2 - (1/3)q^3

Now, we'll find the first derivative (dU/dq):

dU/dq = 18 + 14q - q^2

To find the utility maximizing units, set dU/dq to zero and solve for q:

0 = 18 + 14q - q^2

Rearrange the equation:

q^2 - 14q + 18 = 0

Now, we'll solve for q using the quadratic formula:

q = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -14, and c = 18. Plug these values into the formula:

q = (14 ± √((-14)^2 - 4 * 18)) / 2

q = (14 ± √(196 - 72)) / 2

q = (14 ± √124) / 2

The two possible solutions for q are:

q1 ≈ 1.27
q2 ≈ 14.73

Since the individual consumes discrete units, the utility maximizing consumption will be the whole number closest to these values.

Therefore, the utility maximizing units of consumption are approximately 1 or 15 units, depending on other factors such as budget constraints and the specific preferences of the individual.

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The closest answer choice is  [tex]27.52 units^2.[/tex]

The area of the circle, we need to find the area of one sector and then multiply it by 16 since there are 16 congruent sectors.

To find the area of one sector, we use the formula:

[tex]Area of sector = (angle/360) * \pi*r^2[/tex]

Since we know the base and height of the highlighted sector, we can use the Pythagorean theorem to find the radius of the circle:

[tex]r^2 = (1.56/2)^2 + (3.92)^2[/tex]

r ≈ 3.969 units

Now we can find the angle of one sector using the formula:

angle = (base/radius) x 180/π

angle ≈ 22.5 degrees

Plugging in the values for angle and radius in the area of sector formula, we get:

[tex]Area of sector =(22.5/360) *\pi (3.969)^2[/tex]

Area of sector ≈ 0.491π

Multiplying this by 16, we get the approximate area of the circle:

Approximate area of circle ≈ 16 x 0.491π

Approximate area of circle ≈ 7.8π

Using a calculator to approximate π as 3.14, we get:

Approximate area of circle ≈ [tex]24.46 units^2[/tex]

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Sneha's current age is 16 years old. Sneha's mother is 44 years old.

Let's assume Sneha's current age is x.

Sneha's mother's current age = 2x + 12

After 8 years, Sneha's age = x + 8

After 8 years, Sneha's mother's age = 2x + 12 + 8 = 2x + 20

After 8 years, Sneha's mother's age will be 20 less than three times Sneha's age: 2x + 20 = 3(x + 8) - 20

Now we can solve for x:

2x + 20 = 3(x + 8) - 20

2x + 20 = 3x + 24 - 20

2x + 20 = 3x + 4

x = 16

Therefore, Sneha's current age is 16 years old.

Sneha's mother's current age = 2x + 12

= 2(16) + 12 = 44

So, Sneha's mother is 44 years old.

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Answer:

let amount of games won and lost be x and y respectively

y+18=x

x+y=78

y+y+18=78

2y=78-18

2y=60

y=30

x=30+18

x=48

thus, games won is 48

Answer:

BC= 47.424

I believe that’s correct, but if you really need an answer just get a triangle calculator

The volume of the oblique cylinder whose base and height is given would be = 9,646.08 m³. That is option B.

To calculate the volume of a cylinder, the formula that should be used is given as follows:

Volume of a cylinder = πr²h

π = 3.14

R = diameter/2 = 16/2 = 8m

Height = 8²+48² (using the Pythagorean formula)

= 64+2304

=√ 2368

= 48.66cm³

The volume of the cylinder = 3.14 × 8×8×48.66

= 9,646.08 m³

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The number of times she should expect to pull a yellow marble is 13

From the question, we have the following parameters that can be used in our computation:

Color of Marble Frequency

Yellow 26

Green 34

Purple 18

Red 22

This means that

Times she should expect to pull a yellow marble is

Yellow = P(Yellow) * 50

So, we have

Yellow = 26/(26 + 34 + 18 + 22) * 50

Evaluate

Yellow = 13

Hence, the number of times she should expect to pull a yellow marble is 13

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part a.

the percentage of eggs between 42 and 45mm is 48.48%

part b.

The median width is approximately (42+45)/2 = 43.5mm.

The median length is approximately (56+59)/2 = 57.5mm.

part c.

The width of grade A chicken eggs has a range of about 24mm.

part d.

I think its impossible to determine because we don't have the value for the standard deviation.

The second option should be correct.

A histogram is described as an approximate representation of the distribution of numerical data.

part a.

From the histogram, we see that the frequency for the bin that ranges from 42 to 45mm is 4 and we have  a total of 33 eggwe use this values and calculate the percentage of eggs between 42 and 45mm is 48.48%.

part b.

we have an  estimation  that the median of the width is 48mm and the median of the length is around 60mm.

part c.

Also from the histogram, we notice  that the smallest value is around 36mm and the largest value is around 66mm, hence the width of grade A chicken eggs has a range of about 24mm.

In a histogram, the range is the width that the bars cover along the x-axis and these are approximate values because histograms display bin values rather than raw data values.

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The missing number for box a transaction amount account balance are a = 10, b = 210, c = 210.

Using the information provided in the table, we can fill in the missing numbers as follows:

For transaction 3: The account balance after transaction 2 was $100, and transaction 3 had an amount of $90. Therefore, the account balance after transaction 3 is $190. Hence, the missing number in box a is 190.

For transaction 4: The account balance after transaction 3 was $190, and transaction 4 had an amount of -$200. Therefore, the account balance after transaction 4 is -$10. Hence, the missing number in box b is -10.

For transaction 5: The account balance after transaction 4 was -$10, and transaction 5 had an amount of $c. Therefore, the account balance after transaction 5 is 0. Hence, the missing number in box c is 10.

Therefore, the completed table is:

transaction amount account balance

1 150 150

2 50 100

3 90 190

4     -200-10

5 10 0

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The maps scale is 1 inch = 6 miles. This means that for every inch on the map, it represents 6 miles in real-life distance.

To determine the scale, we need to know the relationship between the distance on the map and the actual distance. In this case, the distance on the route map between Hugo's house and his work is 3 inches, and the actual distance he travels by bus is 18 miles.

To find the scale, we can set up a proportion using the given information:

(distance on the map in inches) / (actual distance in miles) = (1 inch) / (x miles)

Now, we can plug in the known values:

(3 inches) / (18 miles) = (1 inch) / (x miles)

To solve for x, we can cross-multiply:

3 inches * x miles = 18 miles * 1 inch

3x = 18

Now, divide both sides by 3:

x = 6

So, the scale of the map is 1 inch = 6 miles.

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Answer:3,15,18.

Step-by-step explanation:

6:1:5 total ratio =6+1+5=12 so you’ll take all the numbers at different times so 6 will be divided by 12 and multiplied by36 (6/12)36= 18so the first number is nine do the same thing for the next ratio (1/12)36=3 thirdly(5/12)36=15 now add the three numbers to check whether they sum up to36(18+3+15=36)