4/3 pi (2)^3 + 1/3 (2)(4)

The Solution of the equation y = C₁e⁴ˣ + C₂e⁻ˣ

The derivative of a function is a function that indicates how quickly another function is changing. Differentiation is a term used to describe the process of finding a derivative. In an equation, it is shown as the dependent variable with respect to the independent variable. A value's derivative is a change in value relative to its input. The symbol for a derivative is f'(x).

Here we have

d²y/dx²-3(dy/dx) - 4y = 0

As we know (d/dx) = D

=> D²-3D - 4 = 0

=> D²- 4D + D - 4 = 0

=> D(D - 4) + (D - 4) = 0

=> (D - 4) (D + 1) = 0

=> (D - 4) = 0 and (D + 1) = 0  

=>  D = 4 and  D = -1  

Here, roots are real and distinct

Therefore,

Solution of the equation y = C₁e⁴ˣ + C₂e⁻ˣ

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Answer: 23 years old

Step-by-step explanation:  31-8=23

The volume of the cube with the given side length is 1/125b⁹

From the question, we are to determine the volume of the given cube.

The volume of a cube is given by

V = s³

Where V is the volume of the cube

and s is the side length of a side of the cube

From the given information,

s = 1/5b³

Thus,

The volume of the cube will be

V = 1/5b³ × 1/5b³ × 1/5b³

V = 1/5 × 1/5 × 1/5 × b³ × b³ × b³

V = 1/125 × b⁹

V = 1/125b⁹

Hence,

The volume is 1/125b⁹

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The depth of the bottom of the hole after the second day is 425 feet using addition operation.

What is addition?

In math, addition is the process of adding two or more integers together. The terms addends and sum describe the input and output of the addition operation, respectively.

Given the depth on the first day is 232 feet.

Depth on the second day = 193 feet more than on the first day i.e. 193 feet + depth on the first day

This implies, depth on the second day = 193+232

=  425 feet

The hole's bottom will therefore be 425 feet deep after the second day.

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The formula that can be used to estimate the population of California in 2020 is P = Poe^rt.

Population growth has to do with how the number of people in a place is increasing in a given time. We know that the population growth is a positive exponential function.

Initial population (Po) = 37 million

Population growth rate (r) = 1.0%

Time interval (t) = 10 years

Now using the formula;

P = Poe^rt

P = 37 * 10^6 e^0.01 * 10

P = 41 million

There would be about 41 million people within the next ten years.

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Answer:

11.25 litres

Step-by-step explanation:

75 ×15/100

=11.25 so the answer should be 11.25 litres

The values of a = 6.3 and b = -5

Presenting extremely big or extremely small numbers in a simpler format is done through the use of scientific notation. The full numbers can, as we all know, go on forever, but we are unable to write such enormous quantities on paper. In addition, there was a need for a simpler way to express numbers that appear in the million's place.    

Here we have

(a × 10ᵇ) x (3.1 × 10⁹) = 1.953 x 10⁵

=> [tex](a\times 10^{b} ) = \frac{1.953 \times10^{5} }{ (3.1\times 10^{9} )}[/tex]  

=> (a × 10ᵇ) = 0.63 × 10⁵ × 10⁻⁹  

=> (a × 10ᵇ) = 0.63 × 10⁻⁴      

=> 6.3 × 10⁻⁵    

Here 6.3 × 10⁻⁵  is in the form of a × 10ᵇ

=> a = 6.3 and b = -5

Therefore,

The values of a = 6.3 and b = -5  

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The perpendicular of the right triangle is 2.62441 inches and the base of the triangle is 3.0164 inches.

According to the question,

We have the following information:

A right triangle has a hypotenuse of length 4 inches. And one angle is 41°.

Now, we will use trigonometric functions of Sin and Cos to find the length of each leg of this triangle.

Sin 41 = Perpendicular/hypotenuse

Perpendicular = 0.6561*4

Perpendicular = 2.62441 inches

Cos 41 = base/hypotenuse

Base = 0.7541*4

Base = 3.0164 inches

Hence, the perpendicular of the right triangle is 2.62441 inches and the base of the triangle is 3.0164 inches.

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The number of fashion magazines subscription that were sold is approximately 58.

In mathematics, a percentage is a number or ratio that can be expressed as a fraction of 100.

In other words, percentage is a fraction or a ratio in which the value of whole is always 100.

Therefore, Carolyn’s homeroom sold 207 magazine subscriptions. Of the magazine subscriptions sold, 28 percent were fashion magazines.

The number of fashion magazine subscriptions that were sold can be calculated as follows:

number of fashion magazine sold = 28% of 207

number of fashion magazine sold = 28 / 100 × 207

number of fashion magazine sold = 5796 / 100

number of fashion magazine sold = 57.96

Therefore, about 58 fashion magazines were sold.

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The value of the 10th term for the sequence 11,12,13,14 is 20.

It's important to note that this is an arithmetic sequence. The formula for an arithmetic sequence is:

= a + (n - 1)d

where a = first term = 11

d = common difference = 1

n = number of terms = 10

The 10th term will be:

= a + (n - 1)d

= 11 + (10 - 1)1

= 11 + (9 × 1)

= 11 + 9

= 20

The value is 20.

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Complete question

The first four terms of an arithmetic sequence are 11,12,13,14. Find the 10th term.

The point-slope forms of the equation from the information given is 1. y + 2 = ï½(x-1), 2.y - 3 = 2(x+1) , 3.y - 5 = 1(x-8), 4. y + 7 = -5/7(x+1) and 5.y - 3/2 = -1(x-1/2).

1.

slope m = ï½ and  (h, k) = (1, -2)

The point slope is y + 2 = ï½(x-1)

2.

slope = 2 , point (-1, 3)

y - 3 = 2(x+1)

3.

The line goes through the points (8,5) and (9, 6).

slope m = 6-5 / 9-8

= 1/1

= 1

y - 5 = 1(x-8)

4.

The line goes through the points (-1, -7) and (-8, -2)

slope m = -2+7 / -8+1

= 5/-7

= -5/7

y + 7 = -5/7(x+1)

5.

The line goes through the points (1/2, 3/2) and (-1/4, 5/4)

slope m = 5/4 - 3/2 / -1/4 - 1/2

= 5 - 6 / 4 / -1 + 2 / 4

= -1/1

= -1

y - 3/2 = -1(x-1/2).

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a) She has three quadratic equations as one is the standard form, the other gives the roots, and the other gives the vertex.

b) If the distributive property is applied and the like terms combined, all the equations will have the same result.

c) The most helpful equations will be given for each point.

d) Maximum height of 64 feet after 1 second from h(t)=-16(t-1)^2+64.

e) The shell splashes into the river after a time of three seconds, from h(t)=-16(t+1)(t-3).

f) The shell was at a height of 48 feet when tossed into the river, from h(t)=-16t^2+32t+48.

In standard format, the quadratic function is given as follows:

h(t) = -16t² + 32t + 48.

From this, the relevant information is the initial height from which the shell was tossed off the cliff, which is the c-coefficient of 48 feet.

The roots of the equation are given from the rule presented as follows:

h(t)=-16(t+1)(t-3).

Hence the shell splashes into the river at the positive root of h(t), that is:

t - 3 = 0 -> t = 3.

Expanding the equation, we have that it is equivalent to the first one, as:

-16(t + 1)(t - 3) = -16(t² - 2t - 3) = -16t² + 32t + 48.

The vertex-form quadratic equation is given as follows:

h(t) = -16(t - 1)² + 64.

Meaning that a maximum height of 64 feet was reached after one second.

Expanding the equation, we have that:

h(t) = -16(t - 1)² + 64 = -16(t² - 2t + 1) + 64 = -16t² + 32t - 16 + 64 = -16t² + 32t + 48.

Which is also equivalent to the first one.

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The value of x in the given trigonometric function is 7.37

Trigonometric ratios are the ratios of the length of sides of a triangle. These ratios in trigonometry relate the ratio of sides of a right triangle to the respective angle. The basic trigonometric ratios are sin, cos, and tan, namely sine, cosine, and tangent ratios. The other important trig ratios, cosec, sec, and cot, can be derived using the sin, cos, and tan respectively.

5cos(5x) = 4

divide both sides by 5, it becomes,

cos(5x) = 4/5

cos(5x) = 0.8

taking Arc cos on both sides

5x = Arc cos 0.8

5x = 36.87

x = 36.87/5

x = 7.37 to 2 decimal places

In conclusion the value of x is 7.37

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Assume that Chris is not a housekeeper for anyone else. Assume that Angel paid $2,250 in wages for the fourth quarter of 2021. The social security tax is $1,116, Medicare tax is $261 and FUTA tax is $42.

Social security  =12.4%

Medicare = 2.9%

FUTA = 0.6%

a. Social security tax

Yearly salary = $750 × 12 months

Yearly salary  = $9,000

Social security tax  = $9,000 ×12.4%

Social security tax = $1,116

b.  Medicare tax

Yearly salary =$750 × 12 months

Yearly salary = $9,000.00

Medicare tax = $9,000×2.9%

Medicare tax = $261

c. FUTA tax

Yearly salary = $750 × 12 months

Yearly salary = $9,000

The first $7,000 will be  subject to FUTA tax.

FUTA tax = $7,000 ×0.6%

FUTA tax = $42

Therefore the social security tax is $1,116.

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Answer:

x<5

Step-by-step explanation:

1.  add 14 to both sides  

4x - 14  + 14 < 6 + 14

    4x<20

2.  divide both sides by 4

[tex]\frac{4x}{4} < \frac{20}{4}[/tex]

     x<5

                                               Answer = x<5

Liu will earn $472 on an investment of $1298.

Given, Liu earned $316 on an investment of $869.

We have to find how much he would earned if he invest $1298 in the same investment.

Now, $869 investment = $316 earned

$1 investment = 316/869 earned

$1298 investment = 316/869 × 1298 earned

$1298 investment = $472 earned

Hence, Liu will earn $472 on an investment of $1298.

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Answer:

leg lengths are 5 units and 3 units.

Step-by-step explanation:

ヾ(•ω•`)o

The results of the combinations are given as follows:

A. Number of possible outcomes with red exactly 3 times and Purple exactly 2: 10.

B. Number of possible outcomes with red at least ounce in 4 spins: 175.

C. Number of possible outcomes with red at least ounce in 5 spins: 781.

The outcomes are obtained in two ways, as follows:

The combination formula calculates the number of different combinations of x objects from a set of n elements, according to the equation presented as follows:

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

From item a, there are 5 outcomes with 3 and 2 repetitions, hence the number of results is obtained as follows:

[tex]C_{5,3} = \frac{5!}{3!2!} = 10[/tex]

It states that the number of outcomes of n independent trials is given by the multiplication of the number of outcomes for each trial.

For item b, we have that:

Hence the total number of outcomes, without considering the restriction is:

Total = 4^4 = 256.

With no red results, the number of outcomes is:

Total no red = 3^4 = 81.

Hence the number of outcomes with at least one red result, considering complementary events, is:

256 - 81 = 175.

Item c follows the same logic, only with 5 trials instead of four, hence:

4^5 - 3^5 = 781.

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a) The break-even quantity (in units) is 59 units.

b) If the company is selling no more than 34 units, A. The product should not be produced because it will lead to a loss of $625.

The break-even point is the production and sales point at which the total costs equal the total revenue.

At the break-even point, there is no profit or loss.

Total units produced = x

Let the cost of production = C

Let the revenue = R

C = 84x+1,475

R = 109x

At the break-even point, 109x = 84x + 1,475

109x - 84x = 1,475

25x = 1,475

x = 59 units

If x = 34 units:

Total cost = 84x+1,475

= 84(34) + 1,475

= 2,856 + 1,475

= $4,331

Total revenue = 109x

= 109(34)

= $3,706

Loss = $625 ($3,706 - $4,331)

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(a) Find the break-even quantity units (Simplify your answer.)

(b) Decide whether the product should be produced on the basis of whether it will earn a profit. (Profit = Revenue - Cost.)

Choose the correct answer below and, if necessary, in the answer box to complete your choice.

A. The product should not be produced because it will lead to a loss of  B. The product should be produced because it will lead to a profit of

The probability of any day being rainy last year is 21/73 .

In the question ,

it is given that ,

In past year ,

the number of days it rained in last past year = 105 days

total number of days in past year = 365 days .

So , the probability of any day being rainy last year is = (number of days it rained )/(total number of days ) .

Substituting the values , we get

probability of any day being rainy last year = 105/365

Simplifying further , we get

= 21/73

Therefore , The probability of any day being rainy last year is 21/73 .

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Answer:  Refer to the drawing below

Explanation:

Complementary angles always add to 90 degrees.

Draw a right angle, aka 90 degree angle. Plot points A, B and D on it as shown below. Then add point C somewhere inside the rectangular region formed by this right angle. Lastly, connect a segment from B to C. This helps form angles ABC and CBD. Note how these smaller angles combine to a 90 degree angle.

(angleABC) + (angleCBD) = angle ABD = 90 degrees

The angles are adjacent because they share a common wall, which in this case is segment BC.

Answer: 493/16

Step-by-step explanation:

Find the GCD (or HCF) of numerator and denominator

GCD of 493 and 16 is 1

Divide both the numerator and denominator by the GCD

493 ÷ 1

16 ÷ 1

Reduced fraction:

493

16

Therefore, 493/16 simplified to lowest terms is 493/16. or 30.8125

The box plot that could represent a data set that has an interquartile range of 5 is the box plot first option shown in the image.

What is Interquartile Range?

The interquartile range of a data set is the distance between the lower quartile and the upper quartile on a box plot.

Thus, a box plot that has a lower quartile of 3 and an interquartile range of 5 will have a distance of 5 from the lower quartile to the upper quartile.

Therefore, the box plot that could represent a data set that has an interquartile range of 5 is the box plot shown in the image attached below (see attachment).

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Answer:

option A - 2

Step-by-step explanation:

event A { 2 4 6 8 10 12 }

event B { 3 6 9 12 }

The set AnB means A intercept B, which simply means all the elements that are common in event A and event B

So we just have to look at all the elements in A and B, we can see that the common elements are 6 and 12 so therefore:

set AnB = { 6 12 }

and the number of elements in set AnB is 2

$841,500 is the maturity value.

Given in question,

MV = P(1+RT)

P = $750,000

R = 13.35%

  = 13.35/100

  = 0.1335

T= 11 months

 = 11/12 years

 = 0.917 years

Putting the value in equation,

MV = 750000(1+0.1335*0.917)

      = 750000(1+0.122)

      = 750000*1.122

      = 814500

Hence, Maturity Value is $814,500.

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The unit rate of discovery of the new species of marine animals are; 2000 per year.

There is independent quantity, and a quantity which depends on it (dependent quantity). When independent quantity moves by a unit measurement (single unit increment), the increment in dependent quantity is called rate of increment of dependent quantity per unit increment in independent quantity.

We have been given about 4000 new species of marine animals were discovered in the last 2 years.

We need to find the unit rate of discovery.

Total = 4000 new species

Time = 2 year.

Unit rate = new species of marine animals/ time period

Unit rate = 4000/ 2

Unit rate = 2000/1

Therefore, the unit rate of discovery of the new species of marine animals are; 2000 per year.

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The surface area of prism is 224 square centimeter

The surface area of the prim = 2 × Area of the hexagonal face + 6 × Area of the rectangular face  

The area of the hexagonal face = 42 square centimeter

The area of the two hexagonal face = 2×42

= 84 square centimeter

The area of the one rectangular face = length × width

= 4 × 8

= 32 square centimeter

The area of the 6 rectangular face = 6 × 32

= 192 square centimeter

The surface area of the prim = 32 + 192

= 224 square centimeter

Hence, the surface area of prism is 224 square centimeter

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The base of a cylinder is circular and it's area is given as B= πr^2

A cylinder is a three-dimensional shape consisting of two parallel circular bases, joined by a curved surface. The center of the circular bases overlaps each other to form a right cylinder.

The base of a cylinder is circular . If the base area is B, then the area of the base is πr^2.

where r is the radius of the circle and π is 22/7

The volume of a cylinder is V = B × h

where B is the base area and h is the height or length of the cylinder.

Therefore the volume of a cylinder is V= πr^2h

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