find the root of -3x^{3\:}-21x^2+72x+540=0

Answer:

Step-by-step explanation:

A. To find the formula for f'(x) using the limit definition of the derivative, we need to evaluate the following limit:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Substituting f(x) = 3x^2 - 5, we get:

f'(x) = lim(h->0) [3(x + h)^2 - 5 - (3x^2 - 5)] / h

Expanding the square, simplifying, and canceling out the constant terms, we get:

f'(x) = lim(h->0) [6xh + 3h^2] / h

Canceling out the h's, we get:

f'(x) = lim(h->0) (6x + 3h)

Taking the limit as h approaches 0, we get:

f'(x) = 6x

Therefore, the formula for f'(x) is:

f'(x) = 6x

B. To evaluate f'(-1), we simply substitute x = -1 into the formula we found in part A:

f'(-1) = 6(-1) = -6

Therefore, f'(-1) = -6.

Answer: $400,000

Step-by-step explanation:

$500,000-20%=$400,000

The sum of the series is approximately -2.977069 x 10³⁷.

The sum of the series is calculated by applying the following methods as shown below.

We can write the sum as:

∑k=252(3)k = 3^252 + 3^253 + 3^254 + ...

Notice that this is an infinite geometric series with first term a = 3^252 and common ratio r = 3.

To find the sum of an infinite geometric series, we use the formula:

S = a/(1-r)

Applying this formula, we get:

S = 3^252/(1-3) = 3^252/(-2)

So the sum of the series is -0.5 times 3 raised to the power of 252, which is a very large negative number.

Specifically, it is approximately equal to: -2.977069 x 10³⁷.

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The meaning of the slope from the breeder's equation is that each month, the weight of the puppy increases by 0.25 pounds.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

For the context of this problem, we have that the slope is of 0.25, which is the monthly increase in the weight of the puppy, in pounds.

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Answer: 2x + 4 (I'm not sure because you didn't show the choices)

Answer:

Step-by-step explanation:

324-68=256

256-60=196

196-52=144

144-44=100

Yep...you were correct!

Answer:

Shorter section is (mm):
159

Longer section is (mm):
211

Step-by-step explanation:

Let x represent the longer section length and y the shorter section length

We are given x : y = 4:3

We can rewrite in fractional form as

[tex]\dfrac{x}{y} = \dfrac{4}{3}[/tex]

Multiply both sides by y to get
[tex]\dfrac{x}{y} \cdot y = \dfrac{4}{3}y\\\\x = \dfrac{4}{3}y[/tex]

We know that the combined length of both sections is 370 mm
x + y = 370

Substitute the expression for x in the above equation
[tex]\dfrac{4}{3}y + y = 370\\\\\dfrac{4}{3}y + y = y(\dfrac{4}{3} + 1})\\= y(\dfrac{4}{3} + \dfrac{3}{3})\\\\= y(\dfrac{7}{3})\\\\= \dfrac{7}{3}y[/tex]

Therefore we get
[tex]\dfrac{7}{3} y= 370\\\\[/tex]

Multiply both sides by [tex]\dfrac{3}{7}[/tex]

[tex]\dfrac{7}{3} y \times \dfrac{3}{7} = 370 \times \dfrac{3}{7} \\\\y = 158.57\; mm\\[/tex]

= 159 mm (to 3 significant figures)

Therefore the longer section = 370 - 159 = 211 mm

Check the ratio to verify

211/159 = 1.327 ≈ 1.33 which is 4/3

The sigma notations when evaluated are Σ(3X - 1) = 33 and Σ(2X)² = 248

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

3, 7, 2

The notation Σ(3X - 1) can be represented as

Σ(3X - 1) = (3(3) - 1) + (3(7) - 1) + (3(2) - 1)

Evaluate

Σ(3X - 1) = 33

Using the above as a guide, we have the following operation

Σ(2X)² = (2 *(3))² + (2 * (7))² + (2 * (2))²

Evaluate

Σ(2X)² = 248

Hence, the value of Σ(2X)² is 248

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Therefore , the solution of the given problem of domain comes out to be (a) Domain of fog: {1, 6} and (b) Range of fog: {1, 6} .

The range of potential values that a function can take is known as its domain. These numbers serve as a representation for the x-values of an equation like f. (x). The range of potential numbers on which a function can be used is known as its domain. The result that the function gives following the entry of the x value is this set.

Here,

To find the domain and range of the composition fog, we need to evaluate fog(x) for all x in the domain of g that maps to the domain of f.

Since g maps 1 to 5 and 6 to 4, we have:

fog(1) = f(g(1)) = f(5) = 6

fog(6) = f(g(6)) = f(4) = 1

Therefore, the domain of fog is {1, 6} and the range of fog is {1, 6}, written in set notation as:

(a) Domain of fog: {1, 6}

(b) Range of fog: {1, 6}

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Answer:

[tex]\huge\boxed{\sf x = -2}[/tex]

Step-by-step explanation:

2(3x - 3) = -18

6x - 6 = -18

6x = -18 + 6

6x = -12

x = -12/6

[tex]\rule[225]{225}{2}[/tex]

Answer: x=-2

Step-by-step explanation:

Step 1. Multiply the 2×3x and 2×3 which would make the problem 6x-6=-18

Step 2. Add the -6 to the -18 and multiply that answer of -12 by 6 to get your final answer of x=-2

Answer: I took the test and got it right

Tthe matrix representing this system of equations is:

[6 11 | -4]

[5 -9 | 1]

The equations in the image can be expressed as a matrix as follows:

[6 11] [x] [-4]

[5 -9] * [y] = [1]

The constants on the right-hand side of each equation are represented by the matrix on the right, while the coefficients of the variables x and y are represented by the matrix on the left. We use square brackets to surround the coefficients matrix and vertical bars to demarcate the variables and constants in order to represent this in matrix form. As a result, the matrix used to describe this set of equations is:

[6 11 | -4]

[5 -9 | 1]

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The percentage decrease in total principal and interest paid between a 30-year term mortgage and a 15-year mortgage with an APR of 6.5% and a principal balance of $242,300.00 is about 31.1%

An APR is an acronym for the Annual Percentage Rate, which is the interest on al loan amount expressed as a percentage of the amount on loan annually.

The loan amount, the annual percentage rate, and the term of the loan are presented as follows;

Loan amount, P = $242,300.00

Term of the loan, t = 30 years and 15 years

Annual Percentage Rate (APR), r = 6.5%

The amount of the loan after the period can be obtained by using the formula;

A = P·(1 + r)^t

The amount of each loan can therefore be calculated as follows;

The monthly payment is; M = P·(r·(1 + r)^t)/((1 + r)^t - 1)

M = 242,300×((0.065/12) × (1 + (0.065/12))^(30×12))/((1 + ((0.065/12)))^(30×12) - 1) ≈ 1531.5

The total payment on the 30-year loan ≈ 360 × 1531.5 = 551,340

The monthly payment for the 15-year loan is therefore;

M = 242,300×((0.065/12) × (1 + (0.065/12))^(15×12))/((1 + ((0.065/12)))^(15×12) - 1) ≈ 2110.7

The total payment on the 15-year loan ≈ 360 × 2110.7 = 379962

551,340

The percentage change is therefore;

((551,340 - 379962)/551,340) × 100 ≈ 31.1%

The percentage decrease in the total principal and interest paid between the 30-year term mortgage and a 15-year mortgage is about 31.1%

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Answer:

C⁰ = 28⁰

Step-by-step explanation:

^^^^ The answer of the question is given above in the image

Answer:

the inverse of f(x) = 3^(sqrt(x-4)) is f^-1(x) = ((ln x)/ln 3)^2 + 4.

Step-by-step explanation:

To find the inverse of f(x) = 3^(sqrt(x-4)), we can follow these steps:

Step 1: Replace f(x) with y:

y = 3^(sqrt(x-4))

Step 2: Swap x and y:

x = 3^(sqrt(y-4))

Step 3: Solve for y:

Take the natural logarithm (ln) of both sides to bring down the exponent:

ln x = ln(3^(sqrt(y-4)))

Using the rule that ln(a^b) = b ln(a), we can simplify the right side:

ln x = (sqrt(y-4)) ln 3

Divide both sides by ln 3:

(sqrt(y-4)) = (1/ln 3) ln x

Square both sides:

y - 4 = ((ln x)/ln 3)^2

Add 4 to both sides:

y = ((ln x)/ln 3)^2 + 4

Step 4: Replace y with f^-1(x):

f^-1(x) = ((ln x)/ln 3)^2 + 4

Therefore, the inverse of f(x) = 3^(sqrt(x-4)) is f^-1(x) = ((ln x)/ln 3)^2 + 4.

Answer:

Blaise Pascal is known for inventing the mechanical calculator, also called Pascaline, in 1642. The Pascaline was a type of mechanical calculator that could perform addition and subtraction by using a series of gears and wheels. It was one of the first machines capable of performing arithmetic operations automatically and quickly, without the need for manual calculations. The invention of the Pascaline was a significant contribution to the field of mathematics and engineering, and it paved the way for the development of more advanced calculating machines in the future.

Step-by-step explanation:

The probability that a randomly chosen young adult who are male do not use a grocery delivery app is 16.39%.

The probability refers to the chance or likelihood that an expected outcome, event, or success occurs given that there are many possible outcomes, events, or successes.

Probability is a fractional value that lies between zero and one depending on the degree of certainty or otherwise.

The total number of young adults surveyed = 140

The total number of young adults who do not use the app = 78

The number of male young adults who do not use the app = 5

The total number of male young adults surveyed = 17

The probability of a young adult not using the app = 78/140

The probability of a male young adults not using the app = 5/17

Thus, the probability that a randomly chosen young adult who are male do not use a grocery delivery app = 5/17 x 78/140 = 16.39%.

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Total of 3/5 can of black beans and  pinto beans used in preparing the chili by Ronald.

An element of a whole is a fraction. The number is represented mathematically as a quotient, where the numerator and denominator are split. Both are integers in a simple fraction. A fraction appears in the numerator either denominator of a complex fraction.

Fraction of beans used by Ronald to make chili.

black beans = 1/2  can

pinto beans = 7/10 can

Thus,

total beans = 1/2  can + 7/10 can

total beans = (1/2 + 7/10) can

Simplifying:

total beans = (5*1 + 7)/ 2*10  can

total beans = 12/20 can

total beans = 3/5 can

Thus, total of 3/5 can of black beans and  pinto beans used in preparing the chili by Ronald.

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The new balance is the sum of the unpaid balance, the finance charge, and any new purchases or fees:

New balance = Unpaid balance + Finance charge + Purchase

= $35.51 + $0.84 + $19.27

= $55.62.

We must first identify the balance remaining after the payment and purchase in order to calculate the outstanding balance:

Balance after payment = $50.51 - $15.00 = $35.51

Amount remaining after payment: $35.51 + $19.27 = $54.78

The balance remaining after the payment is made, which comes to $35.51, is the unpaid balance.

We must first determine the average daily balance before we can calculate the finance charge using the unpaid balance technique. Finding the daily balance and dividing the total by the number of days in the billing cycle will allow us to achieve this:

$50.51 times 14 days in April equals $707.

14 April 15–24: $35.51 multiplied by 10 days equals $355.

10 April 25–30: $54.78 multiplied by six days is $328.68

Balance total: $707.14 plus $355.10 plus $328.68 ($1,390.92)

Average daily balance: $1,390.92 divided by 30 equals $46.36.

Hence, the finance charge can be determined as follows:

Finance fee equals $46.36 * (0.18 / 365) * 30 = $0.84. Finance charge formula: Average daily balance * Daily periodic rate * Number of days in billing cycle

The total of the unpaid debt, the finance charge, and any additional purchases or fees is the new balance:

Purchase + Finance fee + New balance = $35.51 + $0.84 + $19.27 = $55.62.

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Answer:

The account balance on April 1st is $50.51. On April 15th a payment of $15.00 is made. On April 25th a purchase of $19.27 is made. The annual rate is 18%. What is the unpaid balance? What is the finance charge using the unpaid balance method? What is the new balance? Unpaid balance = $ Finance charge = $ New balance = $

Step-by-step explanation:

To calculate the unpaid balance, we first need to calculate the balance after the payment and the balance after the purchase:

Balance after payment on April 15th: $50.51 - $15.00 = $35.51

Balance after purchase on April 25th: $35.51 + $19.27 = $54.78

Next, we need to calculate the average daily balance for the billing period. The billing period runs from April 1st to April 30th, which is 30 days. We can split this into two parts:

April 1st to April 15th (15 days)

April 16th to April 30th (15 days)

For the first part of the billing period, the balance is $50.51 for 15 days. For the second part of the billing period, the balance is $54.78 for 15 days. The average daily balance is therefore:

($50.51 x 15 + $54.78 x 15) / 30 = $52.64

The unpaid balance is the average daily balance minus the payment, so:

$52.64 - $15.00 = $37.64

The finance charge using the unpaid balance method is the unpaid balance multiplied by the daily periodic rate and the number of days in the billing cycle. The daily periodic rate is the annual rate divided by 365 days, so:

Daily periodic rate = 0.18 / 365 = 0.000493

Finance charge = $37.64 x 0.000493 x 30 = $0.56

The new balance is the previous balance (after the purchase) plus the finance charge, so:

New balance = $54.78 + $0.56 = $55.34

Therefore, the unpaid balance is $37.64, the finance charge is $0.56, and the new balance is $55.34.

The angle ∠5 and ∠6 are 135° and 45° respectively.

The study of points, lines, and shapes, as well as their characteristics and connections in two and three dimensions, is the subject of the mathematic branch known as geometry. It contains ideas like polygons, circles, triangles, angles, and solids.

For, Carla's adjustment,

angle ∠5 = 2 • angle ∠6

Then, ∠5 + ∠6 = 180° ( by linear pair)

2 × ∠6 + ∠6 = 180°

3 × ∠6 = 180°

∠6 = 60°

Then, ∠5 = 120°

For, Carla's assistant Joe adjustment

∠5 = 3 × ∠6

Again, ∠5 + ∠6 = 180° ( by linear pair)

3 ×∠6 + ∠6 = 180°

4 ×∠6 = 180°

∠6 = 45°   Then, ∠5 = 135°

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Answer:

Step-by-step explanation:

You want to know the intervals on which f(x) = sin(x)/(2+cos(x)²) is increasing and decreasing, and the relative extremes.

The quotient rule can be used to find the derivative of f(x). Where the derivative is positive, the function is increasing.

  f'(x) = ((2+cos(x)²)cos(x) +sin(x)(2cos(x)sin(x)))/(2+cos(x)²)²

  f'(x) = (cos(x)(2 +cos(x)² +2sin(x)²)/(2+cos(x)²)²

  f'(x) = cos(x)(3+sin(x)²)/(2+cos(x)²)²

We observe that the factors (3+sin(x)²) and (2+cos(x)²) are both positive for all x. This means the sign of the derivative will match the sign of cos(x).

The function is increasing where cos(x) > 0, on the intervals ...

  (0, π/2) ∪ (3π/2, 2π)

The function is decreasing where cos(x) < 0, on the interval ...

  (π/2, 3π/2)

The first derivative test tells us the function will have a relative maximum where the function goes from increasing to decreasing, at x = π/2. The function value at that point is ...

  f(π/2) = sin(π/2)/(2 +cos(π/2)²) = 1/2

The relative maximum is at (π/2, 1/2).

The first derivative test tells us the function will have a relative minimum where the function goes from decreasing to increasing, at x = 3π/2. The function value at that point is ...

  f(3π/2) = sin(3π/2)/(2 +cos(3π/2)²) = -1/2

The relative minimum is at (3π/2, -1/2).

If Mrs lam worked a total of 180 hours in 9 months she worked 5 days each month then, Mrs. Lam worked 4 hours each day.

Bar graphs or bar charts are visual depictions of groups of data that are made up of vertical or horizontal rectangular bars with lengths that are equal to the data's measure.

The variable quantity is shown on one of the axes, and the drawn bars are all of the same width. Moreover, the other axes show the variable's measure. These graphs are also used to compare different numbers since the heights or lengths of the bars represent the value of the variable. Bar charts make it simple to grasp the data and show frequency distribution tables, which facilitates computations.

Let us suppose the number of hours worked each day = x.

Given that, Mrs lam worked a total of 180 hours in 9 months she worked 5 days.

Thus,

Total hours = 180 = 5 * 9 * x

180 = 45x

x = 4

Hence, Mrs. Lam worked 4 hours each day.

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Answer: 792,000

Step-by-step explanation: 330,000 x 2.4%by 20 years = 792,000

Answer:

Step-by-step explanation:

To prove the statement "A Set S with two or more vectors is linearly independent if and only if no vector in S is expressible as a linear combination of the other vectors in S," we must show that both directions of the statement are true. That is, we must show that if a set S is linearly independent, then no vector in S is expressible as a linear combination of the other vectors in S, and conversely, if no vector in S is expressible as a linear combination of the other vectors in S, then the set S is linearly independent.

First, let's assume that the set S is linearly independent. This means that for any vectors v1, v2, ..., vn in S, the equation a1v1 + a2v2 + ... + anvn = 0 has only the trivial solution a1 = a2 = ... = an = 0. We will prove that no vector in S is expressible as a linear combination of the other vectors in S.

Suppose, for the sake of contradiction, that there exists a vector v in S that can be expressed as a linear combination of the other vectors in S. That is, there exist vectors v1, v2, ..., vn-1 in S such that v = b1v1 + b2v2 + ... + bn-1vn-1, where not all of the bi's are zero. Without loss of generality, assume that b1 is nonzero. Then we can write v1 as a linear combination of the other vectors in S:

v1 = (1/b1)v - (b2/b1)v2 - ... - (bn-1/b1)vn-1

Substituting this expression for v1 in the equation a1v1 + a2v2 + ... + anvn = 0, we get:

a1[(1/b1)v - (b2/b1)v2 - ... - (bn-1/b1)vn-1] + a2v2 + ... + anvn = 0

Multiplying both sides by b1 and rearranging, we get:

(a1/b1)v + (-a1b2/b1)v2 + ... + (-a1bn-1/b1)vn-1 + a2v2 + ... + anvn = 0

This is a linear combination of vectors in S that equals the zero vector, and not all of the coefficients are zero, since a1 is nonzero. But this contradicts the assumption that S is linearly independent, so our assumption that there exists a vector in S that can be expressed as a linear combination of the other vectors in S must be false. Therefore, no vector in S is expressible as a linear combination of the other vectors in S.

Now let's assume that no vector in S is expressible as a linear combination of the other vectors in S. We will prove that the set S is linearly independent. Suppose, for the sake of contradiction, that there exist vectors v1, v2, ..., vn in S such that the equation a1v1 + a2v2 + ... + anvn = 0 has a nontrivial solution, where not all of the ai's are zero. Without loss of generality, assume that a1 is nonzero. Then we can write v1 as a linear combination of the other vectors in S:

v1 = (-a2/a1)v2 - ... - (an/a1)vn

Substituting this expression for v1 in the equation a1v1 + a2v2 + ... + anvn = 0, we get:

0 = a1(-a2/a1)v2 + ... + a

This is because congruent triangles' corresponding angles are identical, and since triangle ABC and triangle DEF are congruent, their corresponding angles are equal.

A triangle is a two-dimensional closed geometric object that consists of three line segments called sides that intersect at three places called vertices. Triangles can be distinguished by their angled sides. Triangles, depending on their sides, can be mutually perpendicular (all sides equal), ellipse, or scalene. Triangles are classed as acutely (all angles less than 90 degrees), equal (one angle equal to 90 degrees), or ambiguous (all angles more than 90 degrees) (all angles greater than 90 degrees). The area of a triangle may be determined by applying the formula A = (1/2)bh, wherein There is the area, b is the base of the triangle, and h is the height of the triangle.

In triangle ABC, we have:

In triangle DEF, we have:

This is because congruent triangles' corresponding angles are identical, and since triangle ABC and triangle DEF are congruent, their corresponding angles are equal.

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The set representing the domain of the relation is given as follows:

{-4,-2,0,3,5}.

For the relation in this problem, the x-coordinates of each point are given as follows:

Hence the set representing the domain of the relation is given as follows:

{-4,-2,0,3,5}.

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Answer: If the store sold 25% of its books during the holiday, that means it has 75% of its books left in stock after the holiday. Let's represent the original number of books with the variable "x".

So, if 75% of x is equal to 600 books, we can write an equation:

0.75x = 600

To solve for x, we can divide both sides by 0.75:

x = 600 / 0.75

x = 800

Therefore, the store had 800 books before the holiday.

Step-by-step explanation:

Answer:

(3,-2)

Step-by-step explanation:

Answer:dear

Step-by-step explanation:

257,540.765

Answer:

The right triangle has sides of approximately 7.86 cm, 24 cm, and 24.88 cm, and angles of 90°, 72°, and 18°.

Answer:

A=18 degrees
a=7.80cm

C=24.54cm

Step-by-step explanation: