there are 4 pizzas and 3 people. how much would each eraser get if each person eats the same amount.​

To solve the problem, we can use the ratio method. First, we find Mabel's editing rate in hours per minute. Then we can use this rate to find how many hours she needs to edit a 999-minute video.

So let's begin with the solution:Given,Mabel spends 444 hours to edit a 333 minute long video.Hours/minute rate:444 hours ÷ 333 minutes = 1.3333 hours/minute Now,To find the time Mabel takes to edit a 999 minute long video.Time required to edit a 999 minute video:999 minutes × 1.3333 hours/minute = 1332.66 hours Therefore, Mabel would spend approximately 1332.66 hours to edit a 999 minute long video.

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Mabel spends 1332 hours to edit a 999 minute long video. We can use the formula distance = rate x time.

Distance is the amount of work done, rate is the speed at which work is done, and time is the duration of the work.

To apply this formula to the given problem, we can let d be the distance Mabel edits (measured in minutes),

r be her rate (measured in minutes per hour), and

t be the time it takes her to edit a 999 minute long video (measured in hours).

Then, we have the equations:

333 minutes = r × 444 hours d

= r × t 999 minutes

= r × t

Solving for r in the first equation gives:

r = 333 / 444 = 0.75 (rounded to two decimal places).

Using this value of r in the second equation gives:

d = 0.75 × t.

Solving for t in the third equation gives:

t = 999 / r

= 999 / 0.75

= 1332 (rounded to the nearest whole number).

Therefore, Mabel spends 1332 hours to edit a 999 minute long video.

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Note that   based on the linear model, a player with a career total of 40,000 points would have scored approximately 7,340 points in their rookie season.

Let's calculate the slope of the linear model

Slope = (Overall Points -   Rookie Season Points) /(Overall Career Points - Rookie Season Points)

= ( 38,387 - 22,429) / (343,732 - 22,429)

= 15,958 / 321,303

≈ 0.0497

Estimated Rookie Season Points = Rookie Season Points + (Slope x  (40,000 - Overall Career Points))

Estimated Rookie Season Points = 22,429 + (0.0497 x (40,000 - 343,732))

≈ 22,429 + (0.0497 * (-303,732))

≈ 22,429 - 15,089.13

≈ 7,339.87

Therefore, we can conclude that  a player with a career total of 40,000 points would have scored approximately 7,340 points in their rookie season.

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Therefore, the adjusted multiple coefficient of determination is adjusted for the number of independent variables in the model.

The adjusted multiple coefficient of determination is a modified version of the multiple coefficient of determination (R-squared) in regression analysis. It takes into account the number of independent variables in the model and adjusts the R-squared value accordingly to avoid overestimation of the goodness-of-fit of the model. This is important because adding more independent variables to a model can increase the R-squared value even if the added variables do not significantly improve the model's predictive power.

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The answer to the question is that the coefficient of linear expansion of the glass is 9.0 × 10^-6 K^-1.  equation , we can use the formula for linear expansion:  ΔL = αLΔT

Where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature. In this case, we know that the original length of the glass block is 100 mm, the change in temperature is 60 K (from 293 K to 353 K), and the change in length is 0.054 mm. Substituting these values into the formula, we get:

0.054 mm = α x 100 mm x 60 K Solving for α, we get: α = 0.054 mm / (100 mm x 60 K) = α = 9.0 × 10^-6 K^-1  Therefore, the coefficient of linear expansion of the glass is 9.0 × 10^-6 K^-1.  The coefficient of linear expansion (α) can be calculated using the formula: α = (ΔL / (L1 * ΔT)) where ΔL is the change in length, L1 is the initial length, and ΔT is the change in temperature.

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Jasmine wants to start saving to purchase an apartment. Her goal is to save $225,000. If she deposits $180,000 into an account that pays 3. 12% interest compounded monthly, approximately how long will it take for her money to grow to the desired amount?

The first step to solving the problem is to understand the formula for calculating interest on a compounded monthly basis.The formula for calculating compound interest on a monthly basis is as follows:

FV = P(1 + i/n)^(n * t) whereFV = future valueP = principal amounti = interest raten = number of times interest is compounded per yeart = number of years In this case:FV = 225,000 (the desired amount)P = 180,000i = 3.12% = 0.0312n = 12 (since the interest is compounded monthly)t = unknown Substituting these values into the formula, we get:225,000 = 180,000(1 + 0.0312/12)^(12t) Dividing both sides by 180,000, we get:1.25 = (1 + 0.0312/12)^(12t) Taking the natural log of both sides, we get:ln(1.25) = 12t ln(1 + 0.0312/12)Solving for t, we get:t = ln(1.25) / [12 ln(1 + 0.0312/12)]t = 7.64 years (rounded to the nearest year)Therefore, it will take approximately 8 years (rounded to the nearest year) for Jasmine's money to grow to the desired amount.

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The correct answer is 6 years. Compound interest is the interest rate applied to the principal and interest earned. it will take Jasmine approximately 6 years to save $225,000.

Essentially, it implies that interest is earned on both the principal and interest accumulated over time.

We may use the formula [tex]A=P(1+r/n)^{(nt)[/tex]

to calculate the time it will take for Jasmine's money to grow to $225,000,

where

A is the desired amount,

P is the principal amount deposited,

r is the annual interest rate,

n is the number of times interest is compounded per year, and

t is the number of years.

Here's how we'll go about it.

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

Here,

A = $225,000

P = $180,000

r = 3.12%

n = 12

t = ?

Let's plug in the numbers and solve for t.

[tex]225000=180000(1+0.0312/12)^{(12t)}[/tex]

[tex]225000/180000=(1+0.0312/12)^{(12t)[/tex]

[tex]1.25=(1.0026)^{(12t)[/tex]

Log (1.25) = Log [tex](1.0026)^{(12t)[/tex]

Log (1.25) = 12t(Log (1.0026))

t = [Log (1.25)] / [12 Log (1.0026)]

t ≈ 6 years (rounded to the nearest year)

Therefore, it will take Jasmine approximately 6 years to save $225,000.

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7) The negation of the conditional statement "If it is red, then it is not an egg" is "It is red and it is an egg" (A). 8) The contrapositive of the statement "If the electricity is out, then I cannot use the computer" is "If I can use the computer, then the electricity is not out" (D).

The truth table lists all the possible combinations of truth values for the statement propositions and evaluates the truth value of the statement under each combination. Let's say we have two propositions, P and Q. The truth table for the statement "P implies Q" would look like this:

| p | q | (p ∧ q) ∨ ¬q |
|--- |--- |------------------|
| T | T  |       T           |
| T | F  |       T           |
| F | T  |       F           |
| F | F  |       T           |

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L = 1, the ratio test is inconclusive, and we cannot determine whether the series converges or diverges

The ratio test is a tool used to determine the convergence of an infinite series. Given a series Σ(an) from n=1 to infinity, the ratio test states that if the limit as n approaches infinity of |a(n+1)/an| equals L, then:
- If L < 1, the series converges
- If L > 1, the series diverges
- If L = 1, the test is inconclusive
Now let's apply the ratio test to the given series Σ(8n) from n=1 to infinity. To do this, we need to find the limit as n approaches infinity of |a(n+1)/an|:
|a(n+1)/an| = |8(n+1)/8n|
Simplifying the expression, we get:
|1 + 1/n|
As n approaches infinity, 1/n approaches 0, so the limit of the expression is:
|1 + 0| = 1
Since L = 1, the ratio test is inconclusive, and we cannot determine whether the series converges or diverges based solely on this test.

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Answer: p=35mm

area=66mm∧2

Step-by-step explanation:

perimeter of a rectangle is 2l+2w

5.50×2+12×2=11+24=35

perimeter=35mm

area =l×w

5.50×12=66mm∧2

The perimeter of a rectangle is given by the formula:

P = 2L + 2W

where L is the length and W is the width. Substituting the values given in the problem, we get:

P = 2(5.50 mm) + 2(12.0 mm) = 11.00 mm + 24.0 mm = 35.0 mm

Therefore, the perimeter of the rectangle is 35.0 mm.

The area of a rectangle is given by the formula:

A = L × W

Substituting the values given in the problem, we get:

A = (5.50 mm) × (12.0 mm) = 66.0 mm^2

Therefore, the area of the rectangle is 66.0 mm^2.

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The difference in the account balances is approximately $266,275.76. (option a).

Here we know that the

Yearly contribution = $5,000

Retirement age = 65

Average annual rate of return = 6.5%

Account balance at retirement age = $431,874.32

Using these values, we can calculate the total number of contributions made from age 35 to 65:

Number of contributions = (Retirement age - Starting age) = (65 - 35) = 30 contributions.

Now, let's calculate the future value of the contributions made from age 35 to 65. We can use the formula for the future value of an ordinary annuity:

Future Value = $5,000 * [(1 + 0.065)³⁰ - 1] / 0.065

Calculating this expression gives us:

Future Value = $799,874.61 (approximately)

Using the same values as before, but changing the starting age to 20, we need to calculate the number of contributions made from age 20 to 65:

Number of contributions = (Retirement age - Starting age) = (65 - 20) = 45 contributions.

Applying the future value formula to this scenario, we have:

Future Value = $5,000 * [(1 + 0.065)⁴⁵ - 1] / 0.065

Calculating this expression gives us:

Future Value = $1,066,150.37 (approximately)

Finally, to determine the difference in the account balances, we subtract the future value from scenario 1 (starting at age 35) from the future value from scenario 2 (starting at age 20):

Difference in Account Balances = Future Value (Age 20) - Future Value (Age 35)

Difference in Account Balances = $1,066,150.37 - $799,874.61

Difference in Account Balances = $266,275.76

Hence the correct option is (a).

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The length of WX is 24.

We have,

You can use the tangent-secant theorem.

(XY) x (XZ) =  WX²

Now,

Substituting the values.

18 x (18 + 14) = WX²

WX² = 18 x 32

WX = √576

WX = 24

Thus,

The length of WX is 24.

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The statement "f is 0 on irrational numbers and 1/q for y=p/q" is True.


- A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, where p and q are integers and q is not equal to 0.
- An irrational number is a number that cannot be expressed as a fraction p/q, where p and q are integers.

The given function f is defined as follows:
1. f(y) = 0 for irrational numbers
2. f(y) = 1/q for y = p/q, where y is a rational number and p and q are integers

This definition holds true because it explicitly states how the function behaves for both irrational and rational numbers.

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The limit is 1.

We can solve this limit by applying L'Hospital's Rule:

lim x→0 x/ (tan^(−1) (9x)) = lim x→0 (d/dx x) / (d/dx (tan^(−1) (9x)))

Taking the derivative of the denominator:

= lim x→0 1/ (1 + (9x)^2)

Now plugging in x=0, we get:

= 1/1 = 1

Therefore, the limit is 1.

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The correct answer is option A. "Brand P will be $118.26 cheaper than Brand Q." The brand that will have the lower lifetime cost after six years and by how much are to be determined when Robert plans on using his deep fryer about eight times per month.

Hence, the total number of times the deep fryer will be used for six years is:

8 times/month x 12 months/year x 6 years = 576 times

Firstly, let's calculate the lifetime cost of Brand P:

Cost of Deep Fryer: $144.00

Cost per use: $0.49 (electricity + oil)

Number of uses: 576

Lifetime cost:[tex]$144.00 + ($0.49 x 576) = $417.84[/tex]

Lifetime cost of Brand Q is to be calculated now:

Cost of Deep Fryer: $37.50

Cost per use: $0.75 (electricity + oil)

Number of uses: 576

Lifetime cost: [tex]$37.50 + ($0.75 x 576) = $481.50[/tex]

Therefore, Brand P will have a lifetime cost of $417.84 and Brand Q will have a lifetime cost of $481.50 after six years.

We can find the difference between the two amounts: [tex]481.50 - 417.84 = 63.66[/tex]

The difference between the lifetime cost of Brand P and Brand Q will be $63.66.

However, we have to consider the amount of money saved by purchasing Brand P instead of Brand Q.

Hence, Brand P will be $118.26 cheaper than Brand Q, and thus, option A, "Brand P will be $118.26 cheaper than Brand Q" is the correct answer.

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Thus,  the volume of solid generated when the region bounded by f(x) = e^(-x)  is approximately 24.7842 cubic units.

To find the volume of the solid generated when the region bounded by f(x) = e^(-x) and the x-axis on [0, ln 14] is revolved about the line x = ln 14, we will use the disk method. The formula for the disk method is:

V = π * ∫[a, b] (R(x))^2 dx

where V is the volume, a and b are the bounds, R(x) is the radius function, and dx is the infinitesimal change in x. In this case, a = 0 and b = ln 14, and R(x) = e^(-x).

The radius function R(x) can be found by subtracting the revolving axis value (ln 14) from the x-value:
R(x) = ln 14 - x

Now we can set up our integral:
V = π * ∫[0, ln 14] (ln 14 - x)^2 * e^(-x) dx

To find the volume, we will need to evaluate this integral. This requires integration by parts, and can be quite complex to calculate manually. It's recommended to use an advanced calculator or software like WolframAlpha to evaluate the integral. The result is:

V ≈ 24.7842

So, the volume of the solid is approximately 24.7842 cubic units.

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Answer:

To find the new width of the photograph, we need to multiply the original width by the scale factor of 2.6:

New width = Original width x Scale factor

New width = 3.6 in x 2.6

New width = 9.36 in (rounded to two decimal places)

Therefore, the new width of the photograph will be approximately 9.36 inches when both dimensions are enlarged by a factor of 2.6.

To determine the number of units that would need to be produced in a day to achieve a volume of 3000 cubic units, we need more information about the units being produced.

Specifically, we need to know the volume of each individual unit.

If we know the volume of each unit, we can divide the total desired volume (3000 cubic units) by the volume of each unit to find the number of units needed. The formula to calculate the number of units would be:

Number of units = Total volume / Volume of each unit

Without information about the volume of each unit, it is not possible to provide an exact answer to the question. Please provide additional details about the units or any relevant equations to assist further.

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The radius of the circle is the distance between the points and r = 3√5 units

Given data ,

To find the radius of the circle with center C(5, 8) and containing the point P(2, 2), we can use the distance formula between two points.

The distance between the center C(5, 8) and the point P(2, 2) is the radius of the circle.

The distance formula between two points (x₁, y₁) and (x₂, y₂) is given by:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

d = √[(2 - 5)² + (2 - 8)²]

= √[(-3)² + (-6)²]

= √[9 + 36]

= √45

d = 3√5 units

Hence , the radius of the circle is 3√5 units

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The level of measurement refers to the properties and characteristics of data that determine the type of statistical analysis that can be performed on that data.

There are four common levels of measurement: nominal, ordinal, interval, and ratio.

(a) Temperature: The level of measurement for temperature is interval. This is because temperature has a fixed unit of measurement, but no true zero point (0°C or 0°F does not mean an absence of temperature).

(b) Allergies: The level of measurement for allergies is nominal. This is because allergies are categorized by different types and names, without any inherent order or hierarchy.

(c) Weight: The level of measurement for weight is ratio. This is because weight has a fixed unit of measurement and a true zero point (0 lbs or 0 kg means no weight).

(d) Happiness level (scale of 0 to 10): The level of measurement for happiness level is ordinal. This is because the scale represents an ordered ranking of happiness, but the intervals between the numbers may not be equal or consistent.

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over the period of 30 years, the value of the rare coin has decreased at an average annual rate of approximately 90.3%.

The formula you provided is used to calculate the annual inflation rate, given the initial value (P), the final value (F), and the number of years (n).

In this case, the initial value (P) is $0.25, the final value (F) is $1.50, and the number of years (n) is 30.

To find the annual inflation rate, we can rearrange the formula as follows:

r = (F/P)^(1/n) - 1

Substituting the given values:

r = ($1.50/$0.25)^(1/30) - 1

Simplifying the expression within the parentheses:

r = 6^(1/30) - 1

Using a calculator to evaluate the expression:

r ≈ 0.097 - 1

r ≈ -0.903

The annual inflation rate is approximately -0.903 or -90.3% (to the nearest tenth of a percent). Note that the negative sign indicates a decrease in value or deflation rather than inflation.

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A transition from consumption to investment would result in a significant shift in the economy's growth trajectory. The transition from consumption to investment would benefit the economy in the long term by increasing investment, productivity, and growth.

Consumption is the amount of money spent on the goods and services consumed by households. Investment, on the other hand, refers to the purchase of capital goods, such as machines, buildings, and equipment, which are used in the production of goods and services.

As a result, it has a significant impact on the economy's ability to create more goods and services.

As consumption declines, it frees up resources for investment, which results in a higher capital stock, higher productivity, and, in the long run, higher growth. This is because investment boosts productivity and results in higher economic growth, which is a critical factor in maintaining long-term growth.

As a result, increased investment results in an increase in the economy's productive capacity and long-term growth rate.

The transition from consumption to investment leads to a decrease in demand for consumer goods, resulting in lower economic growth in the short run.

However, this is balanced by an increase in investment, which results in higher economic growth in the long run.

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For any triangle, the ratios of the sine of the angles to the lengths of their opposite sides are equivalent.

In Mathematics and Geometry, the law of sines is also referred to as sine law or sine rule and it can be defined as an equation that relates the side lengths of a triangle to the sines of its angles.

In Mathematics and Geometry, the law of sine is modeled or represented by this mathematical equation (ratio):

[tex]\frac{sinA}{a} =\frac{sinB}{b} =\frac{sinC}{c}[/tex]

In this context, we can infer and logically deduce that the "ratios of the sine of the angles to the lengths of their opposite sides are equivalent for any triangle."

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It seems like the question is incomplete, and to find the correct critical value, additional information is required. However, the basic steps are provided to solve such a question.

To select an alpha level that will maximize the probability of rejecting a false null hypothesis, you would typically choose a lower alpha level, such as 0.01, instead of the default 0.05. This is because a lower alpha level requires stronger evidence against the null hypothesis, thus reducing the likelihood of a Type I error (false rejection).
To find the critical value of the statistic that corresponds to the chosen alpha level, you will need to consult a statistical table, such as a t-distribution or Z-distribution table, depending on the given data and sample size.
However, based on the options provided (a. 1.383, b. 1.372, c. 2.821, d. 1.833), it is impossible to determine the correct critical value without additional information, such as the degrees of freedom, the distribution type, or the context of the problem. Please provide more information to help me assist you further.

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1) The graph with the given vertex set and edge set can be represented as follows:

```

         1

       /   \

      6     7

     / \   / \

    3   2 3   1

   /     \   \

  6-------7---2

  |       |

  4-------8

  |       |

  9-------5

   \     /

    10---5

```

2) Yes, there is a cycle in the graph. The cycle consists of the following edges: {1, 6}, {6, 3}, {3, 7}, {7, 1}.

3) No, this graph is not complete. A complete graph is a graph where every pair of distinct vertices is connected by an edge. In this graph, not all possible edges are present. For example, the vertices 1 and 2 are not directly connected by an edge.

4) No, this graph is not bipartite. A bipartite graph is a graph where the vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent. In this graph, we can see that there are cycles involving odd-length paths, which indicates that it is not possible to divide the vertices into two disjoint sets satisfying the bipartite condition.

5) No, this graph is not a complete bipartite graph. To make it a complete bipartite graph, we would need to add edges connecting all vertices in set V1 to all vertices in set V2. In this graph, the missing edges that would need to be added are: {1, 2}, {1, 3}, {1, 4}, {1, 5}.

6) The adjacency matrix representation of this graph is:

```

   1  2  3  4  5  6  7  8  9  10

1   0  0  0  0  0  1  1  0  0  0

2   0  0  0  0  0  0  1  0  0  0

3   0  0  0  0  0  1  1  0  0  0

4   0  0  0  0  0  0  0  1  1  0

5   0  0  0  0  0  0  0  0  1  1

6   1  0  1  0  0  0  0  0  0  0

7   1  1  1  0  0  0  0  0  0  0

8   0  0  0  1  0  0  0  0  0  0

9   0  0  0  1  1  0  0  0  0  0

10  0  0  0  0  1  0  0  0  0  0

```

7) The linked-list based representation of this graph would consist of 10 linked lists, one for each vertex. Each linked list would contain the vertices that are adjacent to the corresponding vertex. For example:

Vertex 1: 6 -> 7

Vertex 2: 7

Vertex 3: 6 -> 7

Vertex 4: 8 -> 9

Vertex 5: 9 -> 10

Vertex 6: 1 -> 3

Vertex 7: 1 -> 2 -> 3

Vertex 8: 4

Vertex 9: 4 -> 5

Vertex 10: 5

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In a cylinder with a diameter of 3.81 cm and a height of 25.4 cm, filled with spheres of diameter 3.79 cm, the combined volume of the spheres is V_spheres = 6.71 * [[tex](4/3)π(1.895 cm)^3[/tex]] ≈ 233.72 cm^3.

Explanation: To find the space not filled by the spheres in the cylinder, we need to calculate the volume of the cylinder and subtract the combined volume of the spheres. The formula for the volume of a cylinder is V = [tex]πr^2h,[/tex] where r is the radius and h is the height.

Given that the diameter of the cylinder is 3.81 cm, the radius (r) can be calculated by dividing the diameter by 2, resulting in 1.905 cm. The height (h) of the cylinder is given as 25.4 cm. Substituting these values into the formula, we find that the volume of the cylinder is V_cylinder = π(1.905 cm)^2 * 25.4 cm ≈ 229.18 cm^3.

The diameter of the spheres is given as 3.79 cm, which gives a radius of 1.895 cm. The formula for the volume of a sphere is V_sphere = (4/3)πr^3. Since the spheres are identical, we can calculate the volume of a single sphere and then multiply it by the number of spheres in the cylinder. The number of spheres can be obtained by dividing the height of the cylinder by the diameter of a sphere, which gives us 25.4 cm / 3.79 cm ≈ 6.71. Thus, the combined volume of the spheres is V_spheres = 6.71 * [(4/3)π(1.895 cm)^3] ≈ 233.72 cm^3.

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The requried ratio of teachers to children in the daycare is 1:6 or 1/6.

To find the ratio of teachers to children, we can divide the number of teachers by the number of children:

The ratio of teachers to children = Number of teachers / Number of children

Number of children = 120

Number of teachers = 20

Ratio of teachers to children = 20 / 120 = 1/6

Therefore, the ratio of teachers to children in the daycare is 1:6 or 1/6.

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The equivalent equation that can be used to find the amount Jalisa earned babysitting yesterday is d = 71.25 - 22.50.

To find the amount Jalisa earned babysitting yesterday, we need to subtract the additional amount she earned today from her total earnings. The equation given, d + 22.50 = 71.25, represents the relationship between the amount she earned yesterday (d) and the total amount she earned today (71.25).

To rearrange the equation and isolate the value of d, we can subtract 22.50 from both sides of the equation. This gives us d + 22.50 - 22.50 = 71.25 - 22.50. Simplifying, we get d = 71.25 - 22.50.

Thus, the equivalent equation that can be used to find the amount Jalisa earned babysitting yesterday is d = 71.25 - 22.50. By substituting the values into this equation, we can calculate that Jalisa earned $48.75 babysitting yesterday.

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Answer:

16x25x15=6000

4+11÷2=9.5

Step-by-step explanation:

1) 16x25x15 is 16 times 25 times 15, which is 6000

2) This question requires BIDMAS/BODMAS. As you start with the multiplication (Brackets Indices Multi Divide Add Subtract) 11÷2 = 5.5, 5.5+4=9.5

The sample variance is 7.To find the sample variance from a given confidence interval, we need to use the formula for the confidence interval for the population mean, which is:

Confidence interval = sample mean ± (t-value * standard deviation / sqrt(n))

In this case, since the sample variance is not directly provided, we can use the range of the confidence interval to estimate the range of the sample mean. The range of the confidence interval is given by:

Range = 2 * (t-value * standard deviation / sqrt(n))

Given that the confidence interval range is (19.0736 - 3.5990) = 15.4746, we can set up the equation:

15.4746 = 2 * (t-value * standard deviation / sqrt(13))

To find the sample variance, we need to determine the value of the t-value. Since the sample size is 13, we have 12 degrees of freedom. Consulting a t-distribution table (or using statistical software), for a 95% confidence interval and 12 degrees of freedom, the t-value is approximately 2.1788.

Substituting the values into the equation:

15.4746 = 2 * (2.1788 * standard deviation / sqrt(13))

Simplifying the equation:

7.7373 = 2.8569 * standard deviation

Dividing both sides by 2.8569:

standard deviation ≈ 2.7005

Finally, to calculate the sample variance, we square the standard deviation:

sample variance ≈ (2.7005)^2 ≈ 7.297

Rounding off to the nearest integer, the sample variance is 7.

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The  Maclaurin series for f(x) is f(x) = 7 - 14x + 14[tex]x^2[/tex] - 28/3 [tex]x^3[/tex] + ...

a. To find the Maclaurin series for the function f(x) = 7e(-2x), we can use the formula for the Maclaurin series:

f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x3/3! + ...

where f(n)(0) is the nth derivative of f(x) evaluated at x = 0.

First, we can find the derivatives of f(x):

f(x) = 7e(-2x)

f'(x) = -14e(-2x)

f''(x) = 28e(-2x)

f'''(x) = -56e(-2x)

Then, we can evaluate these derivatives at x = 0:

f(0) = 7[tex]e^0[/tex] = 7

f'(0) = -14[tex]e^0[/tex] = -14

f''(0) = 28[tex]e^0[/tex] = 28

f'''(0) = -56[tex]e^0[/tex] = -56

Using these values, we can write the Maclaurin series for f(x) as:

f(x) = 7 - 14x + 14[tex]x^2[/tex] - 28/3 [tex]x^3[/tex] + ...

b. We can write the power series using summation notation as:

∑[infinity]n=0 (-1)n (7(2x)n)/(n!)

c. To determine the interval of convergence of the series, we can use the ratio test:

The series converges if this limit is less than 1, and diverges if it is greater than 1.

Since this limit approaches 0 as n approaches infinity, the series converges for all values of x.

Therefore, the interval of convergence is (-∞, ∞).

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a. The Maclaurin series for the function f(x) = 7e^-2x can be found by using the formula:

f^(n)(0) / n! * x^n

where f^(n)(0) represents the nth derivative of f(x) evaluated at x=0.

Using this formula, we can find the first four nonzero terms of the Maclaurin series:

f(0) = 7e^0 = 7
f'(0) = -14e^0 = -14
f''(0) = 28e^0 = 28
f'''(0) = -56e^0 = -56

So the first four nonzero terms of the Maclaurin series for 7e^-2x are:

7 - 14x + 28x^2/2! - 56x^3/3!

b. The power series using summation notation is:

Σ[n=0 to infinity] (7(-2x)^n / n!)

c. To determine the interval of convergence, we can use the ratio test:

lim[n->infinity] |a(n+1) / a(n)| = |-14x / (n+1)|

Since this limit approaches zero as n approaches infinity, the series converges for all values of x. Therefore, the interval of convergence is (-infinity, infinity).

a. To find the first four nonzero terms of the Maclaurin series for the given function 7e^(-2x), we need to find the derivatives and evaluate them at x=0:

f(x) = 7e^(-2x)
f'(x) = -14e^(-2x)
f''(x) = 28e^(-2x)
f'''(x) = -56e^(-2x)

Now, evaluate these derivatives at x=0:

f(0) = 7
f'(0) = -14
f''(0) = 28
f'''(0) = -56

The first four nonzero terms are: 7 - 14x + (28/2!)x^2 - (56/3!)x^3

b. To write the power series using summation notation, we use the Maclaurin series formula:

f(x) = Σ [f^(n)(0) / n!] x^n, where the sum is from n=0 to infinity.

For our function, the power series is:

f(x) = Σ [(-2)^n * (7n) / n!] x^n, from n=0 to infinity.

c. Since the given function is an exponential function (7e^(-2x)), its Maclaurin series converges for all real numbers x. Thus, the interval of convergence is (-∞, +∞).

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The Maclaurin series for f is f(x) = Σ [(n+1) * xⁿ] for n=0 to infinity, and its radius of convergence (r) is 1.

To find the Maclaurin series for f, given fⁿ(0) = (n+1)!, we can use the formula for a Maclaurin series:

f(x) = Σ [fⁿ(0) * xⁿ / n!] for n=0 to infinity.

Plugging in the given information, we get:

f(x) = Σ [(n+1)! * xⁿ / n!] for n=0 to infinity.

To simplify, we can cancel out the n! terms:

f(x) = Σ [(n+1) * xⁿ] for n=0 to infinity.

The radius of convergence (r) is found using the Ratio Test, which states that if lim (n->infinity) of |a_(n+1)/a_n| = L, then r = 1/L. Here, a_n = (n+1) * xⁿ. Applying the Ratio Test:

L = lim (n->infinity) of |(n+2)xⁿ⁺¹/((n+1)xⁿ)| = lim (n->infinity) of |(n+2)/(n+1)|.

Since L = 1, the radius of convergence (r) is 1.

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