Solve for the equation of the tangent line. Question is attached below. Please show full working.

Answer:

Step-by-step explanation: Option A is correct.

To find the distance from a point P to a plane defined by three points Q, R, and S, we can use the following formula:

distance = |(A * x + B * y + C * z + D) / √(A^2 + B^2 + C^2)|

where (A, B, C) are the coefficients of the normal vector of the plane, (x, y, z) are the coordinates of the point P, and D is the constant in the equation of the plane.

To find the coefficients of the normal vector, we can use the cross product of two vectors lying in the plane, say, QR and RS.

Let's first find the vectors QR and RS:

QR = (5 - 0, 0 - (-1), 3 - 7) = (5, 1, -4)

RS = (7 - 0, 5 - (-1), 0 - 7) = (7, 6, -7)

Next, we find the cross product of QR and RS:

cross product of QR and RS = (B, C, A) = (7, -30, 15)

So, the normal vector of the plane is (15, -30, 7).

Now we substitute the values for the coefficients and the coordinates of the point P into the formula:

distance = |(15 * 3 + (-30) * 1 + 7 * 7 + D) / √(15^2 + (-30)^2 + 7^2)|

To find the value of D, we can use any of the three points on the plane and substitute their coordinates into the equation of the plane in the form Ax + By + Cz + D = 0.

Let's use the point Q:

15 * 5 + (-30) * 0 + 7 * 3 + D = 0

Expanding and solving for D, we get:

75 + 21 + D = 0

D = -96

Finally, substituting the values back into the formula for distance, we get:

distance = |(15 * 3 + (-30) * 1 + 7 * 7 + -96) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-24 + -30 + 49 - 96) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(15^2 + (-30)^2 + 7^2)|

distance = |(-101) / √(225 + 900 + 49)|

distance = |(-101) / √1274|

distance = |(-101) / 35.6|

distance = 2.85

So, the distance between the point P and the plane is approximately 2.85.

Answer:

661,068.8j

Step-by-step explanation:

The car has 661068.8 J of potential energy at a height of 62 m having a mass of 1088 grams.

Potential energy is a form of stored energy that is dependent on the relationship between different system components. When a spring is compressed or stretched, its potential energy increases.

We know, The gravitational constant is 9.8 m/s².

From the given information, m = 1088 grams, h = 62 meters.

Therefore, The amount of potential energy the car has is,

PE = mgh.

PE = 1088×9.8×62 J.

PE = 661068.8 J.

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Answer:

Given that s = 1.5, the expressions that have a value of 13.5 are:

(1) 5s + 8 = 5 × 1.5 + 8 = 13.5

(2) 2s^2 + 9 = 2 × 1.5^2 + 9 = 13.5

Note: The value of any other expression will depend on the specific formula and the value of other variables used in the expression.

Answer:

slope

Step-by-step explanation:

Step-by-step explanation:

always find the quantity of 100%.

then

1% = 100%/100

then calculate the absolute quantity change.

then divide the quantity change by 1% to see how many % fit into it.

and that is your answer.

so, if the original quantity is q1, the new quantity is q2, then the % of quantity change is

|q1 - q2| / (q1/100)

or

100 × |q1 - q2| / q1

where || means the absolute value.

The lateral surface area of the mailer is 234 In²

It is important to note that the formula for the lateral area of a triangular prism is expressed as;

A = (a + b+ c) h

Given that the parameters are enumerated as;

Now, substitute the values into the equation, we have;

Lateral area, A = (4 + 4 + 5) 18

Add the values

Lateral area = (13)(18)

Multiply the values

Lateral area = 234 In²

Hence, the value is 234 In²

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The formula to calculate the nth term of the sequence is an = -3n+18.

The difference between any two consecutive integers in an arithmetic progression (AP) sequence of numbers is always the same amount. It also goes by the name Arithmetic Sequence.

Given that the arithmetic series is 15,12,9,6.....

The nth term of the arithmetic sequence will be calculated as:-

an = a₁ + (n-1)d

The first term is 15 and the common difference for the series is 12-15 = -3.

The nth term will be calculated as:-

an = a₁ + (n-1)d

an = 15 + (n-1)-3

an = 15 -3n +3

an = -3n + 18

Therefore, an = -3n+18 is the formula to find the nth term in the sequence.

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The ratio 8.64 : 37.76 as a fraction will be 27/118.

What precisely is ratio?

Ratios and proportions are typically described using fractions. When a fraction is presented as a:b, a ratio is shown, whereas a percentage suggests that two ratios are equal. In this case, a and b can be any two numbers. The ratio and proportion are two essential notions that serve as a foundation for understanding other concepts in mathematics and science.

In our daily lives, we apply the notion of ratio and proportion, such as in business when dealing with money or in cooking.

As a consequence, when b is greater than zero, the ratio specifies the connection between two variables, such as a:b. The 2:4 ratio, for example, is written as 2:4 = 1:2. 

Now,

Given ratio is  8.64 : 37.76

we can write this as 8.64/37.76

=864/3776

=432/1888

=216/944

=54/236

=27/118

Hence,

             The ratio 8.64 : 37.76, as a fraction will be 27/118.

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The Equation x² − 15x + 12 = 0 have two Real solution and 3x² − 6x + 4 = 0 have Imaginary solution.

The discriminant, denoted by the equation b² - 4ac, can be used to ascertain if the solutions are real, repetitive, or complex:

1) If the discriminant is smaller than 0, there are two imaginary roots to the problem (s).

2) The equation has one repeating real solution if the discriminant equals zero (s).

Given:

x² − 15x + 12 = 0

Solving the Quadratic Equation

x² − 15x + 12 = 0

D = b² - 4ac

D = (-15)² - 4(1)(12)

D =  225 - 48

D = 177 >0

So, the Equation will have two Real solution.

b) 3x² − 6x + 4 = 0

Solving the Quadratic Equation

3x² − 6x + 4 = 0

D = b² - 4ac

D = (-6)² - 4(3)(4)

D =  36 - 48

D =  -12 >0

So, the Equation will have Imaginary solution.

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Answer:32768/7

Step-by-step explanation:

double every term

The median of the following salary list is 29763.

The median is the average in the centre of a data collection, which means that 50% of the data points have a value less than or equal to the median, and 50% of the data points have a value greater than or equal to the median. For a small data collection, calculate the number of data points (n) and organize them in ascending order.

The middle score in the group is called the median. Begin by sorting all of the data sets from lowest to highest before calculating the median. The median in an odd-numbered set will be the value that falls exactly in the center of the list. You must determine the average of the two middle integers in a set of even numbers.

Arrange in ascending order

21955, 25963, 27507, 32019, 38850, 43715

Take the two mid terms

27507 and 32019

Median = [tex]\dfrac{27507 + 32019}{2}[/tex]

Median = 29763

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The point that lies in the solution to the system of inequalities is (5,-2). The correct answer would be an option (B).

An Inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.

The system of inequalities is given in the question, as follows:

y ≤ x -5 and y ≥ -x-4

Since the required point is to satisfy the system of inequalities.

Then the graph attached below represents a solution to the system of inequalities.

Point A (-5,2) does not lie in the solution to the system of inequalities.

Point B (5,-2) lies in the solution to the system of inequalities.

Point C (-5,-2) does not lie in the solution to the system of inequalities.

Point D (5,2) does not lie in the solution to the system of inequalities.

Therefore, the point that lies in the solution to the system of inequalities is (5,-2).

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Answer:

12 and 5

Step-by-step explanation:

12+5=17

12x5=60

Answer:

Step-by-step explanation:

5 12 step by step

Answer:x^3+x^2-8x-32

Step-by-step explanation:

The solution is, 12 (long division) 924 = 77.

Division is the process of splitting a number or an amount into equal parts.

Division is one of the four basic operations of arithmetic, the ways that numbers are combined to make new numbers. The other operations are addition, subtraction, and multiplication.

here, we have,

given that,

12 (long division) 924

so, we get,

12 | 924  | 77

     84

----------------

       84

       84

-----------------

       0

Hence, The solution is, 12 (long division) 924 = 77.

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Answer:

It will turn on again, just charge it. All phones come with a charger.

Answer:

$3.25 per hotdog

Step-by-step explanation:

This question can be written as a system of equations:

x = price of hotdogs

y = price of drinks

First family:

4x + 4y = 18

Second family

5x + 3y = 20

Multiply each equation so that one of the variables can cancel out:

-3(4x + 4y = 18)

4(5x + 3y = 20)

First equation:

-12x - 12y = -54

Second equation

20x + 12y = 80

Add the equations together

-12x - 12y = -54

20x + 12y = 80

8x = 26

Divide

x = $3.25

Insert the x into one of the original equations to find y:

4(3.25) + 4y = 18

13 + 4y = 18

4y = 5

y = $1.25

Answer:

5(3-2)

Step-by-step explanation:

15-10

5×3-5×2

The equation of the inner boundary of the circular walkway around the monument is x² + y² = 900.

The equation of the outer boundary of the circular walkway around the monument is x² + y² = 1369.

A circle is a conic section whenever the plane cuts the cone parallel to its base. Every point of the circumference lies at the same distance from its center. The equation of a circle is (x-h)² + (y-k)² = r², where (h, k) is the center and r is the radius.

a.

According to the diagram, the radius of the inner boundary is 30 feet and the center is (0, 0).

Substitute h = 0, k = 0, and r = 30 into the equation of circle (x-h)² + (y-k)² = r².

(x-0)² + (y-0)² = 30²

x² + y² = 900

Therefore, the obtained answer is x² + y² = 900.

b.

According to the diagram, the radius of the outer boundary is 37 feet (30 feet + 7 feet) and the center is (0, 0).

Substitute h = 0, k = 0, and r = 37 into the equation of circle (x-h)² + (y-k)² = r².

(x-0)² + (y-0)² = 37²

x² + y² = 1369

Therefore, the obtained answer is x² + y² = 1369.

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The complete question is mentioned in the image below.

Answer:

[tex]\textsf{C.} \quad \sin x[/tex]

Step-by-step explanation:

Given rational expression:

[tex]\dfrac{\sec^2x \csc x}{\sec^2x + \csc^2 x}[/tex]

Rewrite the numerator and denominator of the given rational expression using the following trigonometric identities:

[tex]\boxed{\boxed{\begin{array}{c}\underline{\sf Trigonometric\;Identities}\\\\\boxed{\sec^2 x=\dfrac{1}{\cos^2x}} \qquad \boxed{\csc^2 x=\dfrac{1}{\sin^2 x}}\qquad \boxed{\csc x=\dfrac{1}{\sin x}}\\\\\end{array}}}[/tex]

Therefore:

[tex]=\dfrac{\dfrac{1}{\cos^2x} \cdot \dfrac{1}{\sin x}}{\dfrac{1}{\cos^2x}+\dfrac{1}{\sin^2x}}[/tex]

Multiply the fractions in the numerator, and make the denominators of the fractions in the denominator the same:

[tex]=\dfrac{\dfrac{1}{\sin x\cos^2x}}{\dfrac{\sin^2x}{\sin^2x\cos^2x}+\dfrac{\cos^2x}{\sin^2x\cos^2x}}[/tex]

[tex]=\dfrac{\left(\dfrac{1}{\sin x\cos^2x}\right)}{\left(\dfrac{\sin^2x+\cos^2x}{\sin^2x\cos^2x}\right)}[/tex]

[tex]\textsf{Apply the trigonometric identity}\;\;\boxed{\sin^2x + \cos^2 x = 1}:[/tex]

[tex]=\dfrac{\left(\dfrac{1}{\sin x\cos^2x}\right)}{\left(\dfrac{1}{\sin^2x\cos^2x}\right)}[/tex]

[tex]\textsf{Apply\:the\:fraction\:rule}\;\;\boxed{\dfrac{\frac{a}{b}}{\frac{c}{d}}=\dfrac{ad}{bc}}:[/tex]

[tex]=\dfrac{\sin^2x\cos^2x}{\sin x\cos^2x}[/tex]

Cancel the common factor cos²x:

[tex]=\dfrac{\sin^2x}{\sin x}[/tex]

Simplify:

[tex]= \sin x[/tex]

Therefore:

[tex]\large\textsf{$\dfrac{\sec^2x \csc x}{\sec^2x + \csc^2 x}=$}\;\boxed{\boxed{\sin x}}[/tex]

The vertex of the given function is (0,4)  

What is a quadratic equation?

A quadratic equation is a second-order polynomial equation with a single variable such as x, where ax²+bx+c=0. with a ≠ 0 . Given that it is a second-order polynomial equation, the algebraic fundamental theorem ensures that it has at least one solution. Real or complicated solutions are both possible.

identify the coefficients a and b of the quadratic function

y =ax² +bx+c-------(1)

a=2, b=0

x-coordinate of the vertex is x= -b/2a = -(0)/(2(2)) = 0

Substitute  x=0 into given equation  to get y-coordinate of the vertex

y= 2(0)² + 4 = 0+4 = 4

Vertex= (x,y) = (0,4)

The vertex of the given function is (0,4)

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The mixture should contain 12 pounds of cashews and 18 pounds of Brazil nuts.

A physical combination of two or more unrelated substances is referred to as a mixture. For instance, water and salt are two different substances that, when combined, form a combination.

Let's call the amount of cashews used in pounds "x". The amount of Brazil nuts used in pounds can be calculated as 30 - x.

We know that the total cost of the mixture is $6.08 per pound * 30 pounds = $182.40.

The cost of the cashews in the mixture is $7.70 per pound * x pounds =$7.70x.

The cost of the Brazil nuts in the mixture is $5.00 per pound * (30 - x) pounds = $150 - $5.00x.

The total cost of the cashews and Brazil nuts in the mixture must equal $182.40, so we can set up the equation:

$7.70x + ($150 - $5.00x) = $182.40

Expanding and simplifying the equation:

$2.70x = $32.40

Dividing both sides by 2.70:

$x = 12

So, the mixture should contain 12 pounds of cashews and 18 pounds of Brazil nuts.

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The values of x and y would be   [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].

What are trigonometric ratios?

Trigonometric ratios are mathematical functions that describe the relationship between the angles and sides of a right triangle. The three main trigonometric ratios are the sine, cosine, and tangent.

In the given figure, by using trigonometric ratios, we can find x

tan60° = 20/x

√3 = 20/x

x = 20/√3

=> [tex]x=\frac{20 \sqrt{3}}{3}[/tex]

Now by using the ratio sin60°

sin60° = 20/y

√3/2  = 20/y

y = 40/√3

[tex]y=\frac{40\sqrt{3}}{3}[/tex]

Hence, the values of x and y would be   [tex]x=\frac{20 \sqrt{3}}{3}[/tex] and [tex]y=\frac{40\sqrt{3}}{3}[/tex].

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The number of ways to arrange all the letters if the arrangement must end in a vowel is; 25,200.

The number of arrangement using only five letters is; 6720.

Since there are 5 options for the last letter, and there a 7 letters left to make an arrangement of 7.

Hence, The number of ways can you arrange all of the letters if the arrangement must end in a vowel is; 5 × 7!

= 5 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 25,200.

b). The number of arrangements if one can only use five letters is; ⁸P₅

= 6720.

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The GCF of the polynomials are

1. for 15x²+18 , the GCF is 3

2. for -18y⁴+6y³+24y² ,the GCF is 6y²

The GCF stands for the “greatest common factor”. The GCF is defined as the largest number that is a factor of two or more numbers. For example, the GCF of 24 and 36 is 12, because the largest factor that is shared by 24 and 36 is 12. 24 and 36 have other factors in common, but 12 is the largest.

The GCF is the highest factor that can be used to divide through 2 or more terms

1. For 15x²+18, 3 is a factor of 15 and 18 and it's the greatest factor, therefore the GCF is 3

2. For -18y⁴+6y³+24y², between 18,6 and 24, 6 is the greatest factor Ang between y⁴ ,y³, y², y² is the greatest factor. Therefore the GCF of the expression is 6y².

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The amount after 13 years with a compound interest of 6.9 % is given by the equation A = $ 5,149.69

Compound interest is interest based on the initial principle plus all prior periods' accumulated interest. The power of compound interest is the ability to generate "interest on interest." Interest can be added at any time, from continuously to daily to annually.

The formula for calculating Compound Interest is

A = P ( 1 + r/n )ⁿᵇ

where A = Final Amount

P = Principal

r = rate of interest

n = number of times interest is applied

b = number of time periods elapsed

Given data ,

Let the amount after compound interest be represented as A

Now , the equation will be

The principal amount P = $ 2,100

The rate of interest r = 6.9 %

The number of years b = 13 years

when compounded continuously ,

The compound interest amount A = Pe^rt

Substituting the values in the equation , we get

A = 2,100.00 ( 2.71828 ) ^ ( 0.069 ) ( 13 )

A = 2,100 ( 2.71828 )^( 0.897 )

On further simplification , we get

A = $ 5,149.69

Hence , the amount after 13 years is $ 5,149.69

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