I suck at math lol. Could someone give me the answers in detail? Picture attached! Thanks

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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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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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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Answer:

point A that is the first point represents the smallest value

Step-by-step explanation:

find point A represent in this box plot

first point is the smallest value of all the data

last point is the largest value of all the data

second point is the first quartile

Middle point is the second quartile

fourth point is the third quartile

So point A that is the first point represents the smallest value

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.

A regular polygon with 32 sides has 16 lines of symmetry.

A polygon is a plane figure made up of line segments connected to form a closed polygonal chain.

We need to find the number of  lines of symmetry does a regular polygon with 32 sides have.

The imaginary line or axis along which you can fold a figure to obtain the symmetrical halves is called the line of symmetry

A regular polygon with 32 sides has 16 lines of symmetry.

This is because each side is equal in length and angles, creating a mirrored effect when each side is divided in half.

Hence, a regular polygon with 32 sides has 16 lines of symmetry.

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To find the percentage of high-rise buildings with rooftop gardens, divide the number of buildings with gardens (29) by the total number of buildings (810), and then multiply by 100. Round the final result to the nearest tenth, which is approximately 3.6%.

To calculate the percentage of high-rise buildings with rooftop gardens, we use the formula:

Percentage = (Part / Whole)  imes 100

Where the Part is the number of buildings with rooftop gardens, and the Whole is the total number of high-rise buildings.

Substituting the given values:

Percentage = (29 / 810) times 100

Carrying out the division first gives us:

Percentage ≈ 0.035802469 times 100

Finally, multiplying by 100 to find the percentage, we get:

Percentage ≈ 3.58

After rounding to the nearest tenth of a percent, we obtain:

Percentage ≈ 3.6%

3.6% of the high-rise buildings in the city have rooftop gardens.

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

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.

Answer:

f(x) = (x – 8)(x – 4)(x – 4)(x + 3)

Step-by-step explanation:

A polynomial function with roots [tex]x_{1}, x_{2}, ..., x_{n}[/tex] has the following format:

[tex]f(x) = a(x - x_{1})(x - x_{2})...(x - x_{n})[/tex]

In which a is the leading coefficient.

In this problem, we have that:

Leading coefficient 1, so [tex]a = 1[/tex]

roots -3 and 8 with multiplicity 1, so [tex](x + 3)(x - 8)[/tex].

root 4 with multiplicity 2, so [tex](x - 4)^{2} = (x - 4)(x - 4)[/tex]

So the correct answer is:

f(x) = (x – 8)(x – 4)(x – 4)(x + 3)

Answer:

The correct option is A. The new mean age will be less than 34.8  

Step-by-step explanation:

The mean of a curriculum committee is 34.8 years

Now, A 15-year-old student representative is added to the committee

Number of students in the curriculum committee is not known so if a student of 15 year age is added into the society we cannot find the exact new mean of the curriculum committee.

But, As the 15 is added to the sum of all observations but only 1 is added to the number of observations

So, The mean of the curriculum committee will obviously decrease

Since, Th mean of the curriculum committee is 34.8

Therefore, the new mean of the curriculum committee will be less than 34.8

Hence, The correct option is A. The new mean age will be less than 34.8

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.

Calculate the probability that the change in heart rate is less than 10 beats per minute using z-scores and the standard normal distribution table.

The probability (P) that the change in heart rate (y) is less than 10 beats per minute is calculated by finding the z-score for 10, then using the z-table or a calculator to find the corresponding probability.

First, calculate the z-score: z = (10 - 7.3) / 11.1 = 0.2432. Next, find the probability by looking up this z-score in the standard normal distribution table, which corresponds to approximately 59.93%.

Therefore, the probability that the change in heart rate is less than 10 beats per minute is approximately 59.93%.

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.

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.

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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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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