leonhard euler was able to calculate the exact sum of the -series with : use this fact to find the sum of each series.

Step-by-step explanation:

so, the first road is 115 m long, the second is 90 m long.

we should be finished, if we multiply each road length by $100, right ?

well, not quite.

because the 2 roads overlap. for this overlap area we don't need to pay twice.

we have the choice : to which of the 2 roads do we assign the overlap area ?

let's "give" it to the longer road.

so, the cost for the longer road is simply

115 × 100 = $11,500

the cost for the shorter road is then without the overlap area (6 meters : the width of the other road)

(90 - 6)×100 = 84×100 = $8,400

in total the costs are

$11,500 + $8,400 = $19,900

a) Simplest function g(n) such that f(n) ∈ Θ(g(n)) is g(n) = n2 .

Justification:

To prove this we have to prove both the upper and lower bounds of f(n).

For the upper bound , 3n 2 + 5n lg n ≤ cn2 + dn2 where c and d are constants.

So, 3 + 5 lg n/n ≤ c + d which holds for c = 4 and d = 1.

For the lower bound, 3n 2 + 5n lg n ≥ en2 .where e is a constant.

This holds true because for n ≥ 2, lg n ≤ (n/2).

Therefore, 3n 2 + 5n lg n ≥ 3n 2 + 5n (n/2) = en2 for some constant e.

b) The ratio f(2n)/f(n) approaches the constant 36 from above.

The proof is as follows:

f(2n) = 3(2n) 2 + 5(2n) lg(2n) = 12n 2 + 10n lg n

f(n) = 3n 2 + 5n lg n f(2n)/f(n)

= (12n 2 + 10n lg n)/(3n 2 + 5n lg n)

= 4(2n 2 /n 2 ) + 10(lg n/lg n) / (n 2 /n 2 ) + 5(lg n/lg n)

which simplifies to 4(2) + 10/1 + 5/1 = 36

Therefore, the constant is 36 and it approaches this constant from above.

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Answer:

the answer is 28

Step-by-step explanation:

Answer: look at the picture I think you want this

Step-by-step explanation: Hope this help :D

step (1) 32+23*4 =124 step (2) 9+23*4= 101  step (3) 9+9*4=45 step 4 9+35=44

Step-by-step explanation:

Answer:

66

Step-by-step explanation:

So what you need to do is add up each of the faces so 3 times 3 and 3 times 8. Then after you do that you take the 9 and 24 and add em together.  Then multiply 33 by 2. I hope this helps :D

Answer:

The outcomes that have more than 1 female are -

MFF, FMF, FFM, FFF

Step-by-step explanation:

Given - Kias mom is having triplets. Each baby could either be male or female.

To find - Which outcome would be included in the compound event "more than one female"

Solution -

The sample space outcomes are -

{MMM, MMF, MFM, MFF, FMM, FMF, FFM, FFF}

Now,

The outcomes that have more than 1 female are -

MFF, FMF, FFM, FFF

The average mass of ONE coffee filter, given the following measurements of stacks of 50 coffee filters and keeping 4 decimal places is 1.6374 grams.

We know that the mass of one coffee filter is equal to the average mass of 50 coffee filters divided by 50.

Therefore, we need to find the average mass of 50 coffee filters.

To do this, we add up all the measurements and divide by the number of measurements.

Sum of the measurements = 81.911 + 82.058 + 81.868 + 82.099 + 81.876

                                             = 409.812

Average of the measurements = 409.812 / 5

                                                    = 81.9624

Therefore, the average mass of 50 coffee filters is 81.9624 grams.

The mass of one coffee filter = (81.9624 / 50)

                                                 = 1.639248 grams

                                                 ≈ 1.6374 grams (rounded to 4 decimal places)

Hence, the answer is 1.6374 grams.

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The slope of the line shown in the graph is __2/3__

and the y-intercept of the line is __6___

The general linear equation is written as follows:

y = ax + b

Where a is the slope and b is the y-intercept.

On the graph we can see that the y-intercept is y = 6, then we can write the line as:

y = ax + 6

The line also passes through the point (-9, 0), replacing these values in the line we will get:

0 = a*-9 + 6

9a = 6

a = 6/9

a = 2/3

That is the slope.

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The length of DE is equal to length CE since their corresponding angles are equal.

To prove that line DE is equal to line CE, we will take the following steps as shown below.

In the given figure;

line AD = line BC

angle ADC = angle BCD

let the line DB divide angle ADC equally and let line AC divide angle BCD equally.

then our new triangle becomes EDC

so angle EDC = angle ECD, since the angles are divided equally.

If the two angles (m∠EDC = m∠ECD), then their lengths must be equal since the triangle forms an isosceles triangle.

So length DE = length CE, proved.

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Answer:

3x + 4.5x + 1 = 28

3.6 seconds

Step-by-step explanation:

Initial distance apart = 1 foot

Student A:

Walk rate = 3 ft per second

Student B:

1.5*3 = 4.5 ft per second

How long it will take to be a total of 28 feets apart

Distance = speed * time

Let time = x in seconds

Student A distance = 3x

Stident B distance = 4.5x

Initial distance = 1 foot

3x + 4.5x + 1 = 28

7.5x = 28 - 1

7.5x = 27

x = 27/ 7.5

x = 3.6 seconds

Answer: Part A - 113°

Step-by-step explanation:

A. 6x-5 = 5x+7 -> x = 12

6x-5 -> 6(12)-5 = 72-5 = 67

180-67 = 113

The approximate value of f(1) using Euler's method with two steps of equal size is 6.636. The range of f for t > 0 is 0 < f(t) < 6.

a. The limiting factor in this logistic differential equation is the carrying capacity, which is 6 in this case. As y approaches 6, the growth rate of y slows down, until it eventually levels off at the carrying capacity.

b. To use Euler's method, we first need to calculate the slope of the solution at t=0. Using the given differential equation, we can find that the slope at t=0 is y(0)/8(6-y(0)) = 8/8(6-8) = -1/6.

Using Euler's method with two steps of equal size, we can approximate f(1) as follows:

f(0.5) = f(0) + (1/2)dy/dx|t=0

= 8 - (1/2)(1/6)*8

= 7.333...

f(1) = f(0.5) + (1/2)dy/dx|t=0.5

= 7.333... - (1/2)(7.333.../8)*(6-7.333...)

= 6.636...

Therefore, the approximate value of f(1) using Euler's method with two steps of equal size is 6.636.

c. The range of f for t > 0 is 0 < f(t) < 6, since the carrying capacity of the logistic equation is 6. As t approaches infinity, f(t) will approach 6, but never exceed it. Additionally, f(t) will never be negative, since it represents a population size. Therefore, the range of f for t > 0 is 0 < f(t) < 6.

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The number of cups of vinegar that Mike used lies between the whole numbers 2 and 4

If Mike used one cup of vinegar for each salad dressing recipe, then he used a total of 3 cups of vinegar (1 cup x 3 recipes = 3 cups).

However, the question states that he used "a cup of vinegar" in each recipe, which could mean that he used slightly less than one cup, exactly one cup, or slightly more than one cup.

Assuming that Mike used at least 3/4 cup of vinegar in each recipe (which is still close to "a cup"), then he used a minimum of 2 and 1/4 cups of vinegar in total (3/4 cup x 3 recipes = 2 and 1/4 cups).

Assuming that Mike used at most 1 and 1/4 cups of vinegar in each recipe (which is still close to "a cup"), then he used a maximum of 3 and 3/4 cups of vinegar in total (1 and 1/4 cups x 3 recipes = 3 and 3/4 cups).

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the is the standard form of the parabola will be x = 8y²-16y-7.

What is parabola ?

A parabola is a curve whose equation places a point on it at a same distance from both a fixed point and a fixed line. The parabola's fixed point is referred to as the focus, and its fixed line is referred to as the directrix. The fixed point does not lie on the fixed line, which is another crucial factor to remember. A parabola is a locus of any point that is equally distant from both the focus and directrix of two supplied points. The point where a parabola makes its sharpest turn is known as the vertex. If a parabolic function has the shape of a "," it has a maximum value; otherwise, it has a minimum value.

The vertex form of the equation of a parabola is x = 8(y - 1)2 - 15.

The vertex form of a parabola:

x = 8(y-1)²-15

x = 8(y²-2y+1²)-15

x = 8(y²-2y+1)-15

x = 8y²-16y+8-15

x = 8y²-16y-7

Hence the is the standard form of the parabola will be x = 8y²-16y-7.

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Answer:

No. Try harder next year.

Pythagoras was an ancient Greek mathematician who founded the Pythagorean school of thought. The Pythagorean theorem is a fundamental concept in mathematics that is attributed to Pythagoras and his followers.

It states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. While this theorem is considered a cornerstone of mathematics today, it is important to understand that Pythagoras did not have access to the advanced mathematical tools and methods that we have today.

He had to rely on geometric constructions and reasoning to prove his theorem. Furthermore, Pythagoras believed that all numbers could be expressed as ratios of whole numbers, which is not always true in reality. Despite these limitations, the Pythagorean theorem has stood the test of time and continues to be a crucial tool in mathematics and other fields.

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Answer:

Examples:

1x36

2x18

3x12

4x9

6x6

Step-by-step explanation:

12+16+8

36

9x4

An equation of the line that passes through the point (12, 9), and has a slope of 5/7 is y = 5x/7 - 3/7.

Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:

y - y₁ = m(x - x₁)

Where:

At data point (12, 9), a linear equation of this line can be calculated in slope-intercept form as follows:

y - y₁ = m(x - x₁)

y - 9 = 5/7(x - 12)

y = 5x/7 - 60/7 + 9

y = 5x/7 - 3/7

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Answer:

it's y = 2x

Step-by-step explanation:

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Answer:

 C

Step-by-step explanation:

the answer is 82 million

The best approximation of the area of the circle is 113.04 square inches.

A circle is a shape with all points equidistant from a central point, known as the center. The distance from the center to any point on the edge of the circle is called the radius.

It can be calculated using the formula π \(r^2\), where r is the radius.

Since the diameter of the circle is 12 inches, the radius would be half of that which is 6 inches.

So, the area would be approximately 3.14 * \(6^2\) = 113.04 square inches.

Therefore, The best approximation of the area of the circle is 113.04 square inches.

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Answer:

The formula for the area of a circle is A = πr^2, where r is the radius of the circle.

We know that the diameter of the circle is 10 ft, so the radius is half of that:

r = 10 ft ÷ 2 = 5 ft

Now we can plug in this value for r into the formula:

A = π(5 ft)^2

A = π(25 ft^2)

A ≈ 78.54 ft^2

Rounding this to the nearest whole number, we get:

A ≈ 79 ft^2

Therefore, the area of the circle to the nearest whole number is 79 square feet.

Answer:

79ft^2

Step-by-step explanation:

diameter=10ft

radius=5ft

5^2=25ft

25×π=25π

25π= 78.539....

to nearest whole number =79ft^2

Answer:

102

Step-by-step explanation:

All angles added together=180

180-13-65

180-78

102

The line integral ∫M₀M₁ F · dr along the given straight line, where F = x²i + xyj + yzk, from M₀(1, -1, -4) to M₁(3, -1, 0), evaluates to 32.

To evaluate the line integral ∫M₀M₁ F · dr along the given straight line, we need to parameterize the line segment. Let's denote the parameter as t, where t varies from 0 to 1.

The position vector r(t) of a point on the line can be expressed as:

r(t) = (x(t), y(t), z(t)) = (1 + 2t, -1, -4 + 4t)

Next, we calculate the differential vector dr/dt:

dr/dt = (dx/dt, dy/dt, dz/dt) = (2, 0, 4)

Now, let's express F in terms of the position vector r(t):

F = (x²)i + (xy)j + (yz)k

= (1 + 2t)²i + (1 + 2t)(-1)j + (-4 + 4t)(-4 + 4t)k

= (1 + 4t + 4t²)i - (1 + 2t)j + (16 - 32t + 16t²)k

To evaluate the line integral, we substitute r(t), dr/dt, and F into the integral expression:

∫M₀M₁ F · dr = ∫₀¹ [F(r(t)) · (dr/dt) dt]

= ∫₀¹ [(1 + 4t + 4t²)i - (1 + 2t)j + (16 - 32t + 16t²)k] · (2, 0, 4) dt

= ∫₀¹ (2 + 8t + 8t² - 2 - 4t + 32t - 16t² + 32t³) dt

= ∫₀¹ (8t³ + 36t² + 36t) dt

= [2t⁴ + 12t³ + 18t²]₀¹

= 2(1⁴) + 12(1³) + 18(1²) - 2(0⁴) - 12(0³) - 18(0²)

= 2 + 12 + 18

= 32

Therefore, the value of the line integral ∫M₀M₁ F · dr is 32.

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The probability that a student who takes the bus to school also plays sports is 0.7.

The probability that a student who walks to school also plays sports is 0.6.

The probability that a student who walks to school does not play sports is 0.4.

We have,

We can use conditional probability to answer these questions, using the following formula:

P(A | B) = P(A and B) / P(B)

where P(A and B) is the probability of both events happening together, and P(B) is the probability of event B happening.

1)

P(plays sports | takes bus)

We are given that 35% of all students who were surveyed took the bus to school and played sports. We also know that half of the students surveyed took the bus to school.

So:

P(plays sports | takes bus) = P(plays sports and takes bus) / P(takes bus)

P(plays sports | takes bus) = 0.35 / 0.5

P(plays sports | takes bus) = 0.7

2)

P(plays sports | walks)

We are given that 20% of all students who were surveyed walked to school and did not play sports. We also know that half of the students surveyed walked to school.

So:

P(plays sports | walks) = P(plays sports and walks) / P(walks)

P(plays sports | walks) = (1 - 0.2) / 0.5

P(plays sports | walks) = 0.6

3)

We can use the complement rule to find this probability.

The complement of playing sports is not playing sports.

So:

P(not playing sports | walks) = 1 - P(plays sports | walks)

P(not playing sports | walks) = 1 - 0.6

P(not playing sports | walks) = 0.4

Therefore,

The probability that a student who takes the bus to school also plays sports is 0.7.

The probability that a student who walks to school also plays sports is 0.6.

The probability that a student who walks to school does not play sports is 0.4.

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The term that completes the blank and fits the description is an array

From the question, we understand that:

The term that fits the above highlights is array

Hence, the complete statement is:

array can hold multiple values within a single variable and is defined by having values between square brackets.

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Answer: Angle 1 is 117.6⁰, the alternate angle to 1 is also 117.6⁰

X is 24.8

The Angle 2X + 12.8 = 62.4⁰, the alternate angle to it is also 62.4⁰

Step-by-step explanation: Angle (2X + 12.8 )forms 1180⁰ with Angle 1

180⁰- 117.6⁰ = 2X + 12.8

62.4⁰ = 2X + 12.8

Subtract 12.8 from both sides

49.6 = 2X

X is 24.8⁰

Answer:

X = 24.8

Step-by-step explanation:

Edge

The probability that she selects the route of three specific capitals is 97290/1.

The probability that she selects the route of three specific capitals can be calculated by taking the total number of possible routes and dividing it by the total number of routes that involve the three specific capitals. To calculate the total number of possible routes, multiply the number of states (47) by the number of routes that could be selected from each state (2). This gives 94 total possible routes.

47 x 46 x 45 ways = 97290 ways

= 1/97290

Now, calculate the number of routes that involve the three specific capitals. Since the president is selecting three states, the total number of routes will be 3. Therefore, the probability that she selects the route of three specific capitals is 3/94, which can be simplified to 97290/1.

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