evaluate the integral. (use c for the constant of integration.) ∫ tan^3(x)sec^6(x)dxI am stuck on the first step.According to my book, it says "if the power of the secant is even, save a factor of sec^2(x) and use sec^2(x) = 1 + tan^2(x) to express the remaining factors in terms of tan(x)If the power of tangent is odd, save a factor of sec(x)tan(x) and use tan^2(x) = sec^2(x) - 1 to express the remaining factors in terms of sec(x).The power of tangent is odd and the power of secant is even, what should I do next? Can I apply both rules to the equation? Or will it be wrong? Please show steps and help guide me.

Janet should multiply both sides of the equation by the denominator of the fraction to eliminate the fraction on the right side,

To solve this equation, Janet needs to find the value of "y" that makes the equation true. In order to do this, she needs to get all of the terms with "y" on one side of the equation and all of the constants on the other side.

One way to do this would be to multiply both sides of the equation by the denominator of the fraction on the right side. This will eliminate the fraction on the right side and give Janet an equation with only one term on each side. The denominator of the fraction on the right side is (y+1), so Janet should multiply both sides of the equation by (y+1).

This will give her the equation:

(y+1)(y+startfraction y squared minus 5 over y squared minus 1 end fraction) = (y+1)(start fraction y squared + y + 2 over y + 1 end fraction)

Multiplying out the terms on each side will give Janet a polynomial equation in "y" that she can solve using standard techniques.

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Answer:

y=37

Step-by-step explanation:

-8 + (y+3) =32

-8+y+3=32

-5+5+y=32+5

y=37

The midpoint of the line segment is (4, -3).

The slope of the line segment is -2/5.

The equation of the perpendicular bisector is y = 5/2x - 13.

To find the equation of the perpendicular bisector of a line segment, we need to determine two things:

The midpoint of the line segment and the slope of the perpendicular bisector.

Midpoint:

The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:

[(x₁ + x₂) / 2, (y₁ + y₂) / 2]

Using the endpoints (-1, -1) and (9, -5), we can find the midpoint:

x = (-1 + 9) / 2 = 4

y = (-1 - 5) / 2 = -3

Slope:

The slope of the line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Using the endpoints (-1, -1) and (9, -5), we can find the slope:

m = (-5 - (-1)) / (9 - (-1))

= (-5 + 1) / (9 + 1)

= -4 / 10

= -2 / 5

Perpendicular Bisector:

The slope of the perpendicular bisector will be the negative reciprocal of the slope of the line segment.

The slope of the perpendicular bisector will be 5/2.

Using the midpoint (4, -3) and the slope 5/2, we can find the equation of the perpendicular bisector using the point-slope form:

y - y₁ = m(x - x₁)

Plugging in the values, we have:

y - (-3) = 5/2(x - 4)

y + 3 = 5/2(x - 4)

y + 3 = 5/2x - 10

y = 5/2x - 10 - 3

y = 5/2x - 13

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Answer:

y = -4x + 9

Step-by-step explanation:

y = mx + b

First, we will find 'm', aka slope,

([tex]y_{2}[/tex] - [tex]y_{1}[/tex] ) / ([tex]x_{2}[/tex] - [tex]x_{1}[/tex]) = m

(-7 -7)/ (4 - (-3)) = m

-14/ 7 = m

∴ m = -2

Then we can substitute m = -2 and point 1 in to find 'b', aka y-intercept,

7 = -2 (-3) + b

7 = 6 + b

7 - 6 = b

∴ b = 1

Hence the equation is,

y = -2x + 1

The average velocity would be,

[tex]V_{avg}= \frac{-10h^2 - 80h + 8t -32}{h}[/tex]

We know that the average velocity is nothing but the total displacement/ total time taken.

As we know the average velocity over the time interval [t1, t2] is given by, 

[tex]v_{avg}= \frac{s(t_2) - s(t_1)}{ t_2 - t_1}[/tex]

Here, we have been given the disaplcement function s(t) = -10t² + 8t + 4

And time interval is [4, 4 + h]

For t = 4,

s(4) =  -10(4)² + 8(4) + 4

s(4) = -10(16) + 32 + 4

s(4) = -160 + 32 + 4

s(4) = -124

For t = 4 + h

s(4 + h) =  -10(4 + h)² + 8t + 4

s(4 + h) = -10(16 + 8h + h²) + 8t + 4

s(4 + h) = -160 - 80h - 10h² + 8t + 4

s(4 + h) = -10h² - 80h + 8t - 156

Using abova formula the average velocity V_(ave) would be,

[tex]V_{avg}= \frac{s(4+h) - s(4)}{ 4+h - 4}\\\\V_{avg}=\frac{-10h^2 - 80h + 8t - 156 - (-124)}{h} \\\\V_{avg}=\frac{-10h^2 - 80h + 8t - 156 + 124}{h} \\\\V_{avg}= \frac{-10h^2 - 80h + 8t -32}{h}[/tex]

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Answer:

Step-by-step explanation:

1.4

Answer:

1 2/5

Step-by-step explanation:

simplified: 1.4

The value of a and b are 9 and 14 when the two triangle ΔJHK congruent to ΔTRS.

Given that,

We have two triangle. ΔJHK congruent to ΔTRS

We must evaluate what a and b are worth.

We know that,

In the picture we have two triangle

ΔJHK has ∠H = 51° and ∠K = 7b-15

ΔTRS has ∠S= 83° and ∠T = 6a-3

When all three corresponding sides are the same length and all three corresponding angles are the same size, two triangles are said to be congruent.

We have ∠ H ≅ ∠ T,

∠ J ≅ ∠ R and

∠ K ≅ ∠ S.

So,

∠ H ≅ ∠ T,

51°=6a-3

6a=51+3

6a =54

a=9

∠ K ≅ ∠ S.

7b-15=83

7b=83+15

7b=98

b=14

Therefore, The value of a and b are 9 and 14 when the two triangle ΔJHK congruent to ΔTRS.

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The linear function rule to model how many inches of water w are in the bucket after any number of minutes will be w = (1/3)t + 14

An equation simply has to do with the statement that illustrates the variables given. In this case, it is vital to note that two or more components are considered in order to be able to describe the scenario. An equation expresses the relationship between the variables that are given

In this situation, At time t=0, water begins to drip out of a pipe into an empty bucket. After 42 minutes, there are 14 inches of water in the bucket

The linear function rule to model how many inches of water w are in the bucket after any number of minutes will be:

w = (14/42)t + 14

w = (1/3)t + 14

Therefore, based on the calculation, the equation that can be used to represent the information will be w = (1/3)t + 14

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The probability that Marta makes at least 2 of the 3 free throws is 0.73.

1. Marta makes all three free throws :

Probability of making one free throw = 0.90 (90%)

Chance of converting all three free throws =  [tex]\(0.90 \times 0.90 \times 0.90\)[/tex]

2. Marta makes two free throws and misses one

Probability of making one free throw = 0.90 (90%)

Missing one free throw has a probability = 1 - 0.90

                                                              = 0.10 (10%)

Probability of making two free throws and missing one

[tex]=\(0.90 \times 0.90 \times 0.10 + 0.90 \times 0.10 \times 0.90 + 0.10 \times 0.90 \times 0.90\)[/tex]

3. Marta makes one free throw and misses two (OXO, OOX, XOO):

Probability of making one free throw = 0.90 (90%)

Missing one free throw has a probability = 1 - 0.90

                                                              = 0.10 (10%)

Probability of making one free throw and missing two

= [tex]\(0.90 \times 0.10 \times 0.10 + 0.10 \times 0.90 \times 0.10 + 0.10 \times 0.10 \times 0.90\)[/tex]

Now, add the probabilities of all these scenario

[tex]\[\text{Probability of making at least 2 free throws} \\= \text{Probability of making all three} \\+ \text{Probability of making two and missing one} \\+ \text{Probability of making one and missing two}\][/tex]

= 0.729

Thus, the probability is 0.73.

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Answer:

hard question

Step-by-step explanation:

Answer:

Add 3 to both sides

Divide -9 from each side

Step-by-step explanation:

-9d - 3 = -12

      +3 = +3

-9d = -9

then, divide each side by -9 and you have your answer

Hope that helps!

Answer:

Add 3 to both sides

Divide both sides by -9

Step-by-step explanation:

-9d -3 = -12

-9d - 3 + 3 = -1 + 3  Add 3 to both sides

-9d = -9  Divide both sides by -9

[tex]\frac{-9d}{-9}[/tex] = [tex]\frac{-9}{-9}[/tex]

d = 1

The minimum sample size required for the estimate is 7.

It is the average value of the set given.

It is calculated as:

Mean = Sum of all the values of the set given / Number of values in the set

We have,

Mean = 10.3

Standard deviation = S.D = 2.2

Sample size = n = 0.18

90% confidence interval Z value = 1.65

Now,

( {Mean - Z value (S.D/√n)}, {Mean + Z value (S.D/√n} )

( {10.3 - 1.65 (2.2/√0.18), 10.3 + 1.65 (2.2/√0.18} )

( {10.3 - 8.64}, 10.3 + 8.64} )

(1.66, 18.94)  

Minimum value:

= 1.66

= 7

Thus,

The minimum sample size is 7.

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Answer:

The scale factor is 4

Step-by-step explanation:

Assuming that the center of the dilation is the origin (0,0).  I hope the picture is clear enough to read.

The required length of LM and LK are given as 4.709 cm and 3.151 cm respectively.

These are the equation that contains trigonometric operators such as sin, cos.. etc. In algebraic operations.

Here,
Let LM = x and LK = y

Applying trig operators,

(1)

tan48 = 3.5 / y
y = 3.5 / tan48
y =  3.151 cm

(2)

sin48 = 3.5 / x
x = 3.5/sin 48
x = 4.709 cm

Thus, the required length of LM and LK are given as 4.709 cm and 3.151 cm respectively.

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the values or root of x and y. are

• x = 9

• y = 6√2

The right triangles are all similar, so the ratios of hypotenuse to short side are the same:

27/x = x/3

x^2 = 81 . . . . . multiply by 3x

x = 9 . . . . . . . . take the square root

Then y can be found from the Pythagorean theorem:

x^2 = y^2 + 3^2

81 - 9 = y^2 = 72 . . . . . subtract 9

y = √72 = 6√2 . . . . . . .take the square root

The values of x and y are 9 and 6√2, respectively.

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Answer:

Simplify the following:

(10 x)^(-3)

Multiply each exponent in 10 x by -3:

1/(10^3 x^3)

10^3 = 1000:

Answer: 1/(1000 x^3)

Step-by-step explanation:

Answer:

x = 17.6

Step-by-step explanation:

This is a right triangle.

To find the value of x, the length of the hypotenuse, we write the following equation using Pythagoras theorem:

[tex] {a}^{2} + {b}^{2} = {c}^{2} [/tex]

[tex] {12}^{2} + {13}^{2} = {x}^{2} [/tex]

[tex]144 + 169 = {x}^{2} [/tex]

[tex]313 = {x}^{2} [/tex]

x = 17.6 approximately

An equivalent fraction with a rational denominator can multiply the original fraction by a fraction with a rational denominator.

To produce an equivalent fraction with a rational denominator, you can multiply the original fraction by a fraction with a rational denominator. For example, if the original fraction is [tex]$\frac{3}{5}$[/tex] , you can multiply it by [tex]$\frac{2}{2}$[/tex]  to get the equivalent fraction [tex]$\frac{3 \cdot 2}{5 \cdot 2} = \frac{6}{10}$[/tex]  . The denominator of this equivalent fraction, 10, is a rational number.

In general, to produce an equivalent fraction with a rational denominator, you can multiply the original fraction by a fraction with a rational denominator. The specific fraction you choose will depend on the desired denominator.

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Multiplying [tex]\frac{3}{\sqrt{17}-\sqrt{2} }[/tex] by  [tex]\frac{\sqrt{17}+\sqrt{2}}{5}[/tex] will produce an equivalent fraction with a rational denominator.

Consider given  expression [tex]\frac{3}{\sqrt{17}-\sqrt{2} }[/tex]

First we rationalize a denominator of  fraction [tex]\frac{3}{\sqrt{17}-\sqrt{2} }[/tex]

To rationalize the denominator, multiply and divide the expression by [tex]{\sqrt{17}+\sqrt{2} }[/tex]

So given fraction would be,  

[tex]\frac{3}{\sqrt{17}-\sqrt{2} }\\\\= \frac{3\times (\sqrt{17}+\sqrt{2})}{(\sqrt{17}-\sqrt{2})\times (\sqrt{17}+\sqrt{2}) }\\\\= \frac{3\times (\sqrt{17}+\sqrt{2})}{(\sqrt{17}^2-\sqrt{2}^2) }\\\\ = \frac{3\times (\sqrt{17}+\sqrt{2})}{(17-2) }\\\\= \frac{3\times (\sqrt{17}+\sqrt{2})}{15 }\\\\= \frac{3\times (\sqrt{17}+\sqrt{2})}{3\times 5 }\\\\= \frac{\sqrt{17}+\sqrt{2}}{5 }[/tex]

Therefore, the required fraction is:  [tex]\frac{\sqrt{17}+\sqrt{2}}{5}[/tex]

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A complete question is:

Multiplying 3/sqrt 17 - sqrt 2 by which fraction will produce an equivalent fraction with a rational denominator

The area of the regular hexagon will be 216√3 square units. Then the correct option is C.

Let 'a' be the apothem and 'p' be the perimeter of the regular polygon.

Then the area of the regular polygon will be

A = (1/2) × a × p

A regular hexagon has a side length of 12. Then the perimeter of the regular hexagon is given as,

P = 6 x 12

P = 72

Then the length of the apothem will be given as,

a = 6 x tan 60

a = 6√3

Then the area of the regular hexagon will be given as,

A = (1/2) x (6√3) x (72)

A = 216√3

A regular hexagon has a side length of 12. Then the area of the regular hexagon will be 216√3 square units. Then the correct option is C.

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The angles of the parallelogram are

The measure of ∠A = 4y° = 108°

The measure of ∠B = ( 3x - 18 )° = 72°

The measure of ∠C = 3z° = 108°

The measure of ∠D = ( 2x + 12 )° = 72°

A parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure

The four types are parallelograms, squares, rectangles, and rhombuses

Properties of Parallelogram

Opposite sides are parallel

Opposite sides are congruent

Opposite angles are congruent.

Same-Side interior angles (consecutive angles) are supplementary

Each diagonal of a parallelogram separates it into two congruent triangles

The diagonals of a parallelogram bisect each other

Given data ,

Let the parallelogram be represented as ABCD

where

The measure of ∠A = 4y°

The measure of ∠B = ( 3x - 18 )°

The measure of ∠C = 3z°

The measure of ∠D = ( 2x + 12 )°

From the parallelogram theorem , we get

The measure of ∠B = The measure of ∠D   be equation (1)

And ,

The measure of ∠A = The measure of ∠C   be equation (2)

Substituting the values in the equation , we get

( 3x - 18 )° = ( 2x + 12 )°  

3x - 18 = 2x + 12

Adding 18 on both sides of the equation , we get

3x = 2x + 30

Subtracting 2x on both sides of the equation , we get

x = 30

Substituting the value of x in equation ( 1) , we get

3x - 18 = 3 ( 30 ) - 18

The measure of ∠A = 72°

So , the measure of ∠B = the measure of ∠D = 72°

And ,

The measure of ∠A = The measure of ∠C   be equation (2)

So , 4y° = 3z°

The total internal angle = 360°

So , 4y° + 3z° = 360° - 72° - 72°

4y° + 3z° = 216°   be equation (3)

From equation (2) , we get

2 ( 4y° ) = 216°

Divide by 2 on both sides of the equation , we get

( 4y ) ° = 108°

So , y = 27

And , ( 3z )° = 108°

So , z = 36

And , The measure of ∠A = The measure of ∠C = 108°

Hence , the measures of the angles of the parallelogram are 108° , 72° ,  108° and 72°

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The equation of the sphere with center (6, 4, 9) that just touches the xy-plane is (x - 6)^2 + (y - 4)^2 + (z - 9)^2 = 0.

To find the equation of the sphere with center (6, 4, 9) that just touches the xy-plane, we can first find the equation of the xy-plane and then use the point-normal form of the equation of a sphere to find the desired equation.

The equation of the xy-plane is z = 0. Now, we can use the point-normal form of the equation of a sphere, which is

(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2,

where (h, k, l) is the center of the sphere and r is the radius of the sphere. Since the sphere is just touching the xy-plane, the z-coordinate of the center is equal to 0, so the equation of the sphere can be written as (x - 6)^2 + (y - 4)^2 + (z - 0)^2 = r^2.

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The graph of y=3x^4-16x^3+24x^2+48 is concave down for x values greater than or equal to 0.

The graph of y=3x^4-16x^3+24x^2+48 is an example of a polynomial function. To determine the concavity of a polynomial function, we must first identify the intervals where the function is increasing and decreasing. In this case, the function is increasing for all x values greater than or equal to 0.

Next, we must find the second derivative and determine the intervals where the second derivative is negative. If the second derivative is negative, then the graph is concave down. For this polynomial, the second derivative is y'' = -48x + 48, which is negative for all x values greater than or equal to 0. This means that the graph of y=3x^4-16x^3+24x^2+48 is concave down for all x values greater than or equal to 0.

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This is because the Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

Pythagorean Theorem is an essential mathematical principle that has been used for centuries. It states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is named after the ancient Greek mathematician Pythagoras. Using this theorem, one can calculate the length of any side of a right triangle if the lengths of the other two sides are known.

This relationship between the side lengths and the angle measures can be used to calculate the exact angle measures when all three side lengths are known.

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Answer:

I am not sure I hope this helps!

Selling price- $500.00

Rate of discount- 20%

Discount- $100.00 (20% of $500.00)

Sale price- $400.00 ($500.00 - $100.00)

100 meters was he under the sea at the end of the 10 minutes. This can be solved using the concept of multiplication.

Along with addition, subtraction, and division, multiplication is one of the four fundamental mathematical operations. Multiply in arithmetic refers to the continuous addition of equal-sized groups.

Multiplication in mathematics is the addition of equal groups. The number of items there in group expands while we multiply. An example of a multiplication issue would be the product and the two components. The components in the multiplication problem 8 x 9 = 72 are the numbers 8 and 9, while the result is the number 72.

At the end of 10 minutes, the deep-sea diver would have been at a depth of 100 meters under the sea.

This is because 10 meters per minute multiplied by 10 minutes

=  100 meters.

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The equations for the perpendicular and parallel lines are obtained as y = 2/3x + 12 and y = -3/2x - 15/2 respectively.

A straight line can be written in the slope-intercept form as, y = mx + c.

In order to obtain the slope, the ratio of the difference of the coordinates are taken and c is the y-intercept which can be found by substituting x = 0 in the equation.

The equation of the line is given as y = -3/2x + 6.

(a) Compare the given equation with y = mx + c to obtain,

m = -3/2

Then, the slope of the perpendicular line is given as,

⇒ -1 ÷ -3/2 = 2/3

Plug x = -9 and y = 6 in y = mx + c to get,

6 = 2/3 × -9 + c

⇒ c = 12

Thus, the required equation can be given as y = 2/3x + 12.

(b) The slope of the line parallel to y = -3/2x + 6 is -3/2.

Plug x = -9 and y = 6 in y = mx + c to get,

6 = -3/2 × -9 + c

⇒ c = -15/2

Thus, the required equation can be given as y = -3/2x - 15/2.

Hence, the equations of the lines for both part are given as  y = 2/3x + 12 and y = -3/2x - 15/2 respectively.

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Answer:

14

Step-by-step explanation:

because ab means 2(5), then add 4 you get your answer as 14

A linear function which has the greatest y-intercept include the following: D. y = 3x + 4

In Mathematics, the y-intercept of any graph such as a linear function, generally occur at the point where the value of "x" is equal to zero (x = 0).

Mathematically, the slope-intercept form of a line can be calculated by using this equation:

y = mx + c

Where:

Based on the information provided regarding the equations and graphs, the y-intercept are as follows:

Therefore, y = 3x + 4 is a linear function with the greatest y-intercept.

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Complete Question:

Which linear function has the greatest y-intercept?

y = 6x + 1

On a coordinate plane, a line goes through points (0, 2) and (5, 0).

On a coordinate plane, a line goes through points (1, 2) and (0, negative 3).

y = 3x + 4

Step-by-step explanation:

as it is a right angled triangle ,you can use the formula above .

Answer:

y + 4= -3(x + 1)

y = -3x - 7

Step-by-step explanation: