The formula for centripetal acceleration, a, is given below, where v is the velocity of the object and r is the object's distance from the center of the circular path. Solve the formula for the velocity. the formula is a= v^2/ r

Answer:

8/13 - 1/13i

Step-by-step explanation:

Final answer:

To convert the expression (1-2i)/(2-3i) to standard form, multiply by the complex conjugate of the denominator (2+3i). After simplification, the standard form of the complex number is 0.8 - 0.1i.

Explanation:

To write the expression (1-2i)/(2-3i) as a complex number in standard form, we need to eliminate the imaginary unit i from the denominator. This can be achieved by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of a complex number 2-3i is 2+3i.

Multiplying the expression by the complex conjugate, we get:

(1-2i)(2+3i) / (2-3i)(2+3i).

Now, we can expand both the numerator and denominator:

Numerator: (1 × 2) + (1 × 3i) - (2i × 2) - (2i × 3i) = 2 + 3i - 4i - 6(i²)

Denominator: (2 × 2) + (2 × 3i) - (3i × 2) - (3i × 3i) = 4 - 6(i²)

Recall that i² = -1, so we can simplify the expression further.

Numerator: 2 + 3i - 4i + 6 = 8 - i

Denominator: 4 + 6 = 10

Therefore, our complex number in the standard form is:

(8 - i) / 10 = 0.8 - 0.1i.

Using part 1 of the fundamental theorem of calculus to find the derivative of the function. The derivative of the given function is:

[tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]

Consider the given function:

[tex]\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}[/tex]

The objective is to find  [tex]\mathbf{\dfrac{dy}{dx}}[/tex]  by using the fundamental theorem of calculus.

Using chain rule:

[tex]\mathbf{\dfrac{dy}{dx} = \dfrac{dy}{dv}\times \dfrac{dv}{dx}}[/tex]

[tex]\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{4-3x} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx}}}[/tex]

[tex]\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{v} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} \ \ \ \ \ since \ v \ = 4 - 3x} }[/tex]

[tex]\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} }[/tex]

[tex]\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ (-3)}[/tex]

[tex]\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }[/tex]

From the fundamental theorem of calculus;

 [tex]\mathbf{\dfrac{d}{dx} \Big( \int^x_1 \ g(t) dt \Big) = g(x)}[/tex]

∴

[tex]\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }[/tex] will be:

[tex]\mathbf{ =3\times \Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]

∴

[tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]

Therefore, we can conclude that the derivative of [tex]\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}[/tex]

using the fundamental theorem of calculus is   [tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]

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Acetic acid, with the molecular formula C2H4O2, has a carbon to hydrogen atom ratio of 2:4, which simplifies to a 1:2 ratio.

The student's question pertains to the ratio of carbon atoms to hydrogen atoms in an acetic acid molecule. Acetic acid, which is also known as vinegar, has the molecular formula C2H4O2. The ratio of carbon to hydrogen atoms can be found by considering the number of carbon atoms and the number of hydrogen atoms in the formula. There are 2 carbon atoms and 4 hydrogen atoms, resulting in a 2:4 ratio. However, this ratio can be simplified by dividing both numbers by the greatest common divisor, which is 2. After simplifying, the ratio of carbon to hydrogen atoms becomes 1:2.

Answer:

$154,960,863.44

Step-by-step explanation:

Add the 5 earnings numbers using a suitable calculator.

_____

In this case, a "suitable calculator" is one that will display numbers of 11 digits or more. Apparently the one at the Google search box is up to the task.

The total earnings of all five movies in the given week were approximately $155,950,863.44.

To find the total earnings of all five movies in the given week, you can simply add up their individual earnings:

Total Earnings = Earnings of Movie A + Earnings of Movie B + Earnings of Movie C + Earnings of Movie D + Earnings of Movie E

Total Earnings = $26,088,808.74 + $60,394,938.12 + $23,659,617.52 + $34,311,887.98 + $10,505,611.08

Now, calculate the sum:

Total Earnings = $155,950,863.44

So, the total earnings of all five movies in the given week were approximately $155,950,863.44.

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Answer:

You would shift this to the right 8 places

Step-by-step explanation:

The area of a sector bounded by a 45° arc in a circle with a radius of 2 kilometers is π/² square kilometers.

To calculate the area of a sector bounded by a 45° arc in a circle with a radius of 2 kilometers, we will use the formula for the area of a sector, which is (θ/360) × π × r², where θ is the central angle in degrees and r is the radius of the circle. The central angle for our sector is 45° and the radius r is given as 2 km.

Plugging these values into the formula, we have:

Area of sector = (45/360) × π × (2²) = (1/8) × π × 4 = (1/2) × π = π/2 km².

Therefore, the area of the sector bounded by a 45° arc in a circle with a radius of 2 kilometers is π/²square kilometers.

Answer:

Option B is correct

-2.3

Step-by-step explanation:

Average rate of change(A(x)) for the function f(x) over the interval [a, b] is given by:

[tex]A(x) = \frac{f(b)-f(a)}{b-a}[/tex]           ....[1]

We have to find the  average rate of change from x = 1 to x = 4.

From the graph as shown below

For x = 1

f(1) = 3

and

For x = 4

f(4) = -3.9

Using [1] we have;

[tex]A(x) = \frac{f(4)-f(1)}{4-1}[/tex]

Substitute the given values we have;

[tex]A(x) = \frac{-3.9-3}{3}[/tex]

⇒[tex]A(x) = \frac{-6.9}{3}[/tex]

Simplify:

[tex]A(x) = -2.3[/tex]

Therefore, -2.3  value is closest to the average rate of change from x = 1 to x = 4

To find the ratio of corresponding sides for similar triangles, compare the lengths of the corresponding sides in both triangles. If the triangles are similar, these ratios will be equal. To simplify the ratio, divide each term by the greatest common factor.

In mathematics, to write the ratio of corresponding sides for similar triangles, you need to compare the lengths of the sides that have the same relative position. These corresponding sides are proportional in similar triangles. Let's consider a simple scenario where we have two similar triangles

A'B'C' and ABC.

If these two triangles are similar, the ratio of their corresponding sides will be equal. Hence our ratios will look like this:

a'/a = b'/b = c'/c.

This is the ratio of corresponding sides for the similar triangles. To reduce the ratio to the lowest term, you simply divide each term by the greatest common factor of all the terms.

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To find the ratio of corresponding sides of similar triangles, identify the corresponding sides and write a ratio comparing the lengths of these sides. For example, if side AB in triangle ABC corresponds to side DE in triangle DEF with lengths 3 and 6 respectively, the ratio is 3/6 or 1/2 when reduced.

To write the ratio of corresponding sides for similar triangles, we must first identify which sides correspond in the two triangles. Usually, triangles are labeled so that corresponding sides have the same label (like "a", "b", etc.). Once you've identified which sides correspond, simply write a ratio comparing the length of one side in the first triangle to the length of the corresponding side in the second triangle. This could look something like a1/a2.

For example, if triangle ABC is similar to triangle DEF and side AB corresponds to DE with lengths 3 and 6, respectively, then the ratio of the corresponding sides is 3/6 or 1/2 in reduced form.

Understanding the ratio of corresponding sides is key to understanding the properties of similar triangles and can be extremely helpful in solving problems involving these figures not just in geometry, but in other branches of mathematics as well.

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The number of distinct license plates can be formed if there are no restrictions on the digits or letters is 175,760,000.

Given that, in Virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits.

Permutations are different ways of arranging objects in a definite order. It can also be expressed as the rearrangement of items in a linear order of an already ordered set.

There are 26 letters A to Z in the alphabet. There are 10 digits 0 to 9

Assuming repetition allowed.

Now, 10×26×26×26×10×10×10

= 175,760,000

Hence, the number of distinct license plates can be formed if there are no restrictions on the digits or letters is 175,760,000.

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The answer to this question is A: 31

The length of AB is approximately 14.68 which is option B. 15.

To find the length of side AB in the congruent figures where AC is 6, angle C is 119 degrees, AB is 15, angle B is 22 degrees, and BC is 12, we can use the Law of Cosines.

The Law of Cosines states:

[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]

where:

- c is the side opposite the angle C,

- a and b are the other two sides,

- c is the angle opposite side c.

In this case, we have AC as side a, BC as side b, and AB as side c. We also know that angle C is 119 degrees.

[tex]\[ AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(C) \][/tex]

Substitute the given values:

[tex]\[ AB^2 = 6^2 + 12^2 - 2 \cdot 6 \cdot 12 \cdot \cos(119^\circ) \][/tex]

Now, calculate the expression to find the length of AB.

[tex]\[ AB^2 = 36 + 144 - 2 \cdot 6 \cdot 12 \cdot \left(\cos(119^\circ)\right) \]\\AB^2 = 36 + 144 - 2 \cdot 6 \cdot 12 \cdot \left(-0.492403\right) \]\\AB^2 = 36 + 144 + 35.4234 \]\\AB^2 = 215.4234 \][/tex]

Now, take the square root of both sides to find AB:

[tex]\[ AB = \sqrt{215.4234} \approx 14.68 \][/tex]

So, the length of AB is approximately 14.68.

Answer:

m∠DRM = 45°

Step-by-step explanation:

∵ PSTR is a parallelogram

∴ TS // RP ⇒ opposite sides

∴ m∠T + m∠R = 180° ⇒ (1) (interior supplementary angles)

∵ m∠T : m∠R = 1 : 3

∴ m∠R = 3 m∠T ⇒ (2)

- Substitute (2) in (1)

∴ m∠T + 3 m∠T = 180

∴ 4 m∠T = 180

∴ m∠T = 180 ÷ 4 = 45°

∴ m∠R = 3 × 45 = 135°

∵ m∠R = m∠S ⇒ opposite angles in a parallelogram

∴ m∠S = 135°

∵ RD ⊥ PS

∴ m∠RDS = 90°

∵ RM ⊥ ST

∴ m∠RMS = 90°

- In quadrilateral RMSD

∵ m∠S = 135°

∵ m∠RDS = 90°

∵ m∠RMS = 90°

∵ The sum of measure of the angles of RMSD = 360°

∴ m∠DRM = 360 - ( 135 + 90 + 90) = 45°

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Answer:

XY is the longest side in the given ΔXYZ.

Step-by-step explanation:

We are given the following information in the question:

We have to find the longest side in the given triangle ΔXYZ.

The three angles of the triangle are:

[tex]\angle X = 62^\circ\\\angle Y = 55^\circ\\\angle Z = 63^\circ\\[/tex]

We know that in a triangle the side opposite to smallest triangle is smallest and the side opposite to largest angle is longest in length.

Angle Z in the given triangle is the largest angle and therefore, the side opposite to this angle is the longest side of the triangle.

Hence, XY is the longest side in the given ΔXYZ.

The correct figure which has all sides of equal measure but not necessarily all angles of equal measure are ''Rhombus'' and ''parallelogram''.

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

Given that;

To find figure which has all sides of equal measure but not necessarily all angles of equal measure.

Now, We know that;

In a Parallelogram, A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length.

And, In a Rhombus, A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides the same length,

Thus, The correct figure which has all sides of equal measure but not necessarily all angles of equal measure are Rhombus and parallelogram.

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The marginal cost when 30 radios are produced is $8.

To find the marginal cost when 30 radios are produced, we need to differentiate the cost function with respect to x, which represents the number of radios produced. The cost function is given as c(x) = 400 + 20x - 0.2x^2. Differentiating c(x) with respect to x, we get c'(x) = 20 - 0.4x. Now, substitute x = 30 into c'(x) to find the marginal cost when 30 radios are produced. c'(30) = 20 - 0.4(30) = 20 - 12 = 8. Therefore, the marginal cost when 30 radios are produced is $8.

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Abra made 9/10 of her free throws which are equal to 90%, so Abra is the best shooter.

A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100.

Hannah made 0.7 of her free throws in a basketball game.

[tex]\rm = 0.7 \times 100\\\\=70 \ percent[/tex]

Hannah made 0.7 of her free throws in a basketball game which is equal to 70%.

Abra made 9/10 of her free throws.

[tex]\rm =\dfrac{9}{10}\times 100\\\\= 9\times 10\\\\=90 \ percent[/tex]

Abra made 9/10 of her free throws which are equal to 90%.

Dena made 3/4 of her free throws.

[tex]\rm =\dfrac{3}{4} \times 100\\\\=3\times 25\\\\=75 \ percent[/tex]

Dena made 3/4 of her free throws which are equal to 75%.

Hence, Abra made 9/10 of her free throws which are equal to 90%, so Abra is the best shooter.

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The probability of getting heads then a number less than 4 is, 0.15

"Probability is a branch of mathematics which deals with finding out the likelihood of the occurrence of an event."

The formula of the probability of an event A is:

[tex]P(A)=\frac{n(A)}{n(S)}[/tex] , where [tex]n(A)[/tex] is the number of favorable outcomes and [tex]n(S)[/tex] is the total number of events in the sample space.

For given situation,

Let, event A: getting heads on a coin toss

and event B: choosing a number less than four from 1-10

For event A, [tex]n(A)=1[/tex] and [tex]n(S)=2[/tex] (either head or tail)

So, [tex]P(A)=\frac{1}{2}[/tex]

Similarly, for event B:

choosing a number less than four = {1, 2, 3}

So, [tex]n(B)=3[/tex] and [tex]n(S)=10[/tex]

⇒ [tex]P(B)=\frac{3}{10}[/tex]

The probability of getting heads and choosing a number less than 4 is:

⇒ [tex]\frac{1}{2}[/tex] × [tex]\frac{3}{10}[/tex]

[tex]=\frac{3}{20}[/tex]

[tex]=0.15[/tex]

Hence, the probability of getting heads then a number less than 4 is 0.15

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Final answer:

Division by zero is undefined because a/0 would imply an infinite value not included in the real numbers, and 0/0 is an indeterminate form lacking unique value, requiring advanced techniques in calculus for evaluation.

Explanation:

The concept of division by zero, specifically when discussing expressions like a/0 and 0/0, is a fundamental aspect of mathematics that leads to the undefined nature of these operations. In the case of 1/0, this division is not defined within the real number system, as a non-zero number divided by zero would imply an infinite value, which is outside the bounds of real numbers.

Conversely, the expression 0/0 is an example of an indeterminate form because it doesn't present enough information to deduce a unique value for the division, as zero divided by zero could represent any number.

Indeterminate forms such as these necessitate a more nuanced approach, particularly in calculus where the evaluation of limits often brings these expressions into play. In some contexts, sophisticated mathematical techniques must be employed to determine the behavior of functions as they approach these forms.