Find the equation of the line tangent to the curve defined by f(x) = x^2+1 / square root x at the point (1,2). Show all your work.

The equation of line parallel to 2x +3y = 9 and passing through the point (6,4) is 3y + 2x = 24

Gradient or slope is the ratio of the change in y to the change in x between two points in a line.

Gradient tells how steep a line is. A perfectly vertical or horizontal line has no slope.

2x + 3y = 9

making y the subject,

3y = 9 - 2x

y = 9/3 -2x/3

Comparing the above equation, with equation of a straight line, y = mx + c

m, which is slope = -2/3

The new line will also have a slope of -2/3 because they are parallel.

Equation for new line,

y-[tex]y_{1}[/tex] / x - [tex]x_{1}[/tex] = slope

The new point is (6,4)

y - 4 / x -6 = -2/3

By cross multiplying

3(y-4) = -2(x - 6)

3y - 12 = -2x + 12

3y + 2x = 24

In conclusion, the equation o the new line is 3y + 2x = 24

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Equation of the parabola that passes through the points (-4, 7), (-3, 9), and (4,-5) is y = -1/2[tex]x^{2}[/tex] - 3/2 x + 9.

Given points:

(-4, 7), (-3, 9), and (4,-5).

Let parabola equation be y = a[tex]x^{2} +bx+c[/tex].

substitute (-4,7)

7 = a(16)+b(-4)+c

16a-4b+c-7 = 0.

substitute (-3,9)

9 = a(9)+b(-3)+c

9a-3b+c-9 = 0

substitute (4,-5)

-5 = 16a+4b+c

16a+4b+c+5=0.

solving three equations we get,

a = -1/2 , b = -3/2 and c = 9

substitute a,b and c values we get,

y = (-1/2)[tex]x^{2}[/tex] + (-3/2)x + 9

y = -1/2[tex]x^{2}[/tex] - 3/2 x + 9.

Therefore the equation of the parabola that passes through the points (-4, 7), (-3, 9), and (4,-5) is y = -1/2[tex]x^{2}[/tex] - 3/2 x + 9.

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Answer:

The answer is C: 20 ÷ f

Step-by-step explanation:

Answer:

Step-by-step explanation:

The measure of h from the triangle is h = 55

A triangle is a plane figure or polygon with three sides and three angles.

A Triangle has three vertices and the sum of the interior angles add up to 180°

Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

The area of the triangle = ( 1/2 ) x Length x Base

For a right angle triangle

From the Pythagoras Theorem , The hypotenuse² = base² + height²

if a² + b² = c² , it is a right triangle

if a² + b² < c² , it is an obtuse triangle

if a² + b² > c² , it is an acute triangle

Given data ,

Let the triangle be represented as ΔABC

And , the measure of ∠ABC = 40°

Now , Let the Triangle be ΔABC , such that

∠A + ∠B + ∠C = 180°

So , the measure of ∠ACB = ∠BAC = y° = 70°

And , 2h + y° = 180° ( angles on a straight line )

On simplifying , we get

2h + 70° = 180°

Subtracting 70 on both sides , we get

2h = 110

Divide by 2 on both sides , we get

h = 55

Hence , the measure of h = 55

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The complete question is attached below :

(2h)

40°

(2h)

What is the value of h of the triangle?

O h=20

O h=35

O h=55

O h=70

Answer:

x is 70°

hopefully this will help u

The plumber will work for 4hrs and 30min. Which we can call in many ways as “four and half hours”.

The right side of the equation is represented by the expression to the right of the "=" sign, and the left side by the expression to the left of the "=" sign.

For instance, in, the left-hand side (LHS) is represented by x + 5 and the right-hand side (RHS) by y + 8.

      Plumber charges for coming home : $35

                Charge for work per hour: $60

      Total  charges of the plumber          : $305

Total  charges =   (charges for coming home) + ( Charge for work per hour) x (hours he worked)

The equation is 35 + 60X =305

305 = 35 + 270

270 = ( Charge for work per hour) × (X)      [Let hours he worked = x]

270 = 60 × (X)

270 ÷ 60 = (X)

X = 4.5

The plumber has worked for 4.5 hrs .

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Answer:

14-3x=25

Step-by-step explanation:

-3x=25+14

-3x=39

-3 -3

x = -13

The measure of the angle 'x' will be 41°. Then the correct option is c.

The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360 °.

Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.

The measure of the angles will be 76° and 63°.

By the property Supplementary angle, the equation is given as,

76° + x + 63° = 180°

Simplify the equation, then we have

x + 139° = 180°

x = 180° - 139°

x = 41°

The measure of the angle 'x' will be 41°. Then the correct option is c.

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Answer: The answer is A

Step-by-step explanation:

-5/8 is literally equal to -0.625 meaning this asnwer is equal not less

-0.5 is Greater than -0.625 because -0.6 is much larger

-8/5 is -1.6 which is Less than -0.625

Answer:

Scalene

Step-by-step explanation:

a scalene triangle has three sides different angles none of its sides are equal in length

Answer:

A scalene triangle or quadrilateral has no equal sides.

The value of the sets are A ∩ ∅  = ∅ and A' ∪ U  = {f, g, h, p, q, r}.

Given: U = {f, g, h, p, q, r} and A = {g, p, r}

We are to find: (a)A ∩ ∅  and (b) A' ∪ U

(a) A ∩ ∅  

The symbol ∩ means intersection. This tells us what sets have in common. Also, the symbol ∅ means an empty set or null. That is a set that does not have any element

The intersection of a set with the empty set is the null set. Thus:

A ∩ ∅  = ∅

(b) A' ∪ U

The set U is called the universal set

A' means elements not in set A but in set U

The symbol ∪ means union, A' ∪ U will give a list of elements in A' and U combined without repetition. Thus:

A' ∪ U = {f, h, q} ∪  {f, g, h, p, q, r}  = {f, g, h, p, q, r}

Therefore, A ∩ ∅  = ∅ and A' ∪ U  = {f, g, h, p, q, r}. They should be entered into the boxes accordingly

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12)  (2, 3) over the x - axis becomes, (2, -3).

13)  (7, -2) over the y - axis becomes, (-7, -2).

14) (4, 5) 90° about the origin becomes, (-5, 4).

15) (-3, -2) 270° about the origin becomes, (-2, 3).

16) (17, -23) over the line y = x becomes, (-23, 17).

What is transformation of points?

The reflection is the type of alteration that takes place when each point in the shape is reflected over a line. The picture is on the opposite side of the line from the pre-image when the points are reflected across a line, but it is still at the same distance from the line as the pre-image. The reflection of each point (p, q) onto an image point (q, p).

12) Reflect (2, 3) over the x axis:

Since, the x-coordinate of a point that is reflected across the x-axis stays constant, but the y-coordinate is assumed to be the additive inverse.

The x-axis' reflection of point (x, y) is (x, -y).

So, (2, 3) becomes, (2, -3).

13) Reflect (7, -2) over the y axis:

Since, the y-coordinate of a point that is reflected across the y-axis stays constant, but the x-coordinate is assumed to be the additive inverse.

So, (7, -2) becomes, (-7, -2).

14) Rotate (4, 5) 90° about the origin:

(x, y) → (-y, x) is the formula for a 90° rotation about the origin.

So, (4, 5) becomes, (-5, 4).

15) Rotate (-3, -2) 270° about the origin:

(x, y) → (y, -x) is the formula for a 270° rotation of the origin.

So, (-3, -2) becomes, (-2, 3).

16) Reflect (17, -23) over the line y=x.

The angle at which the point (x, y) is reflected across the line y = x is (y, x).

So, (17, -23) becomes, (-23, 17).

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Answer:  The correct answer is 17 bags of fertilizer to cover the lawn

Step-by-step explanation:

1 bag of fertilizer covers 3000 SF of lawn

How many bags are needed for an area of 290 feet x 170 feet

Calculate the Square Feet:

290 x 170 = 49,300 SF

Divide space (SF) by bag coverage (SF):

49,300 / 3,000 = 16.43 bags of fertilizer

Round up answer to 17 bags of fertilizer

Applying the definition of exterior angle of a triangle, the value of m is: m = 32.

An exterior angle of a triangle can be defined as the angle formed outside the triangle when one of its side is extended.

The exterior angle to an interior angle, is supplementary to the interior angle. That is, they have a sum of 180 degrees when we add their angle measures together.

Given the following:

Measure of angle Z = 4m - 15

Measure of the exterior angle to angle Z = 2m + 3

Therefore:

4m - 15 + 2m + 3 = 180

Combine like terms

6m - 12 = 180

6m = 180 + 12

6m = 192

6m/6 = 192/6

m = 32

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Answer:

5 is added each time. the next numbers are, 29, 34 and 39

Step-by-step explanation:

9+5 = 14 14+5=19 19+5=24

The total amount of interest and the final debt owed under the US Rule is $191.09.

Given that,

Comet Lee took out a $16,000 loan with a 90-day, 6% note. Comet settled the note for $2,000 after 20 days. Comet settled the note for $4,000 on day 50.

We have to find what is the total amount of interest and the final debt owed under the US Rule.

First,

To calculate $2000 per 20 days.

We know that,

i=p×r×t

Here,

i is interest.

p is principal amount.

r is rate of interest

t is time.

So, p is $1600

r is 6%

t is 20 days is 20/360

So,

i= 1600×0.06×20/360

i= $53.33

Apply partial payment to interest due,

$2000-$53.33= $1946.67

Next,

Subtract remainder of payment from principal is adjusted balance

Adjusted balance =$16000-$1946.67

Adjusted balance =$14053.33

Now, Second,

To calculate $4000 per 50 days.

We know that,

i=p×r×t

So, p is $14053.33

r is 6%

t is 50 days is 30/360

So,

i= 14053.33×0.06×30/360

i= $70.27

Apply partial payment to interest due,

$4000-$70.27= $3929.73

Next,

Subtract remainder of payment from principal is adjusted balance

Adjusted balance =$14053.33-$3929.73

Adjusted balance =$10123.6

Now, for 40 days

i= 10123.6×0.06×40/360

i= $67.49

So, balance is 10123.6+67.49= $10191.09

When max make two makes partial payments,

The total interest is $53.33+$70.27+$67.49= $191.09

Therefore, the total amount of interest and the final debt owed under the US Rule is $191.09.

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For the given two sets A and B, the set B is equal to A.

Set:

A set is the collection of well-defined objects or elements. It is represented by a capital letter symbol and the number of elements in the finite set is represented as the cardinal number of a set in a curly bracket {…}.

Given.

Here we have the two sets

A =  {piano, guitar, violin) and

B = (guitar, piano, violin)

Now, we need to find the relation between  B and A.

While we looking into the given sets,

Set A has 3 elements in it and

Set B has 3 elements in it.

So, the size of both sets are equal.

And now we have to looking at the elements in it.

While we checking the elements, instead of the order both have the same elements.

Therefore, the set B is equal to set A.

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Answer: [tex]6/(x-7) +( -5)/(x-5)[/tex]

Step-by-step explanation:

Let

   [tex](x+5)/(x-7)(x-5) = A/(x-7) +B/(x-5)[/tex]

⇒[tex](x+5)/(x-7)(x-5)=( A(x-5) +B(x-7))/(x-7)(x-5)[/tex]

Comparing both the sides

[tex](x+5)= A(x-5)+B(x-7)\\x+5=Ax-5A+Bx-7B\\x+5= (A+B)x + (-5A-7B)\\[/tex]

Comparing both sides of the equation we get,

[tex]A+B= 1[/tex]   ....eq(1)

[tex]-5A-5B=5[/tex]   ..... eq(2)

solving eq(1) and eq(2),

we get B=-5 and A=6.

Hence the solution is [tex]6/(x-7) +( -5)/(x-5)[/tex]

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The number of square inches of the wrapping paper left over is 380 square inches

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

Box:

Dimension = 8 inches by 9 inches by 10 inches

Rectangular-shaped wrapping paper

Dimension = 2 feet by 3 feet.

The surface area of the box is

Box = 2(lw + lh + wh)

So, we have

Box = 2 * (8 * 9 + 8 * 10 + 9 * 10)

Evaluate

Box = 484 square inches

The surface area of the wrapping paper is

Wrapper = lw

So, we have

Wrapper = 2 * 3

Evaluate

Wrapper = 6 square feet

Convert to square inches

Wrapper = 864 square inches

The amount of square inches left is

Amount left = Wrapper - Box

Amount left = 864 square inches - 484 square inches

This  gives

Amount left = 380 square inches

Hence, the amount left is 380 square inches

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Answer:

==============

Since both points have same x-coordinate, the distance between them is the difference of y-coordinates:

Correct choice is D

Answer:

32 units

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Distance between two points}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two points.\\\end{minipage}}[/tex]

Define the two given points:

Substitute the two points into the distance formula and solve for d:

[tex]\implies d=\sqrt{(-7-(-7))^2+(8-(-24))^2}[/tex]

[tex]\implies d=\sqrt{(-7+7)^2+(8+24)^2}[/tex]

[tex]\implies d=\sqrt{(0)^2+(32)^2}[/tex]

[tex]\implies d=\sqrt{32^2}[/tex]

[tex]\implies d=32[/tex]

Note: As the two given points have the same x-coordinate, the line segment (between the points) is a vertical line. Therefore, to find the distance between the two given points, simply calculate the difference between the y-coordinates:  

[tex]\implies 8 - (-24) = 8+23=32[/tex]

The solution is x > -2 and x > 3

x is greater than -2 and x is greater than 3.

This can be represented by the number line Option D.

It shows a relationship between two numbers or two expressions.

There are commonly used four inequalities:

Less than = <

Greater than = >

Less than and equal = ≤

Greater than and equal = ≥

We have,

(x + 2) (x - 3)² > 0

This can be written as,

x + 2 > 0 and (x - 3)² > 0

x + 2 > 0

x > -2 _____(1)

(x - 3)² > 0

Squaring both sides.

x - 3 > 0

x > 3 _____(2)

From (1) and (2) we get,

x > -2 and x > 3

Thus,

The solution is x > -2 and x > 3

x is greater than -2 and x is greater than 3.

This can be represented by the number line Option D.

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The height of the stack of paper is 3145.73 in.

Since the Notebook paper is about 0.003 in. thick, and you cut a sheet of paper in half, stack one piece on top of the other, cut those in half, stack all the pieces together, repeating this process 20 times.

Since each piece of paper is halved, that means that the number of paper sheets doubles every time.

So, after halving n times, we would have 2 sheets of paper. Since the thickness of the Notebookpaper is 0.003 in, then the height after cuttin half n times is h = 2ⁿ × 0.003 in.

Since the paper is folded 20 times, n = 20.

So, the height h = 2ⁿ × 0.003 in.

h =  2²⁰ × 0.003 in.

h = 1048576  × 0.003 in.

h = 3145.728 in

h ≅ 3145.73 in

So, the height of the stack of paper is 3145.73 in.

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Answer:

LCM = 42

Step-by-step explanation:

There are many ways to do this.

Listing Multiples:

Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.

Multiples of 14:

14, 28, 42, 56, 70

Multiples of 21:

21, 42, 63, 84

Therefore,

LCM(14, 21) = 42

Prime Factorization:

List all prime factors for each number.

Prime Factorization of 14 is:

2 x 7  =>  21 x 71

Prime Factorization of 21 is:

3 x 7  =>  31 x 71

For each prime factor, find where it occurs most often as a factor and write it that many times in a new list.

The new superset list is

2, 3, 7

Multiply these factors together to find the LCM.

LCM = 2 x 3 x 7 = 42

In exponential form:

LCM = 21 x 31 x 71 = 42

LCM = 42

Therefore,

LCM(14, 21) = 42

Cake / Ladder:

Divide your numbers by prime numbers as long as both numbers are evenly divisible by that prime.

Cake / Ladder

7 14 21

2       3

When there are no more primes that evenly divide into both numbers you are done.

The LCM is the product of the numbers in the L shape, left column and bottom row.

LCM = 7 x 2 x 3

LCM = 42

Therefore,

LCM(14, 21) = 42

Division Method:

Write your numbers in the top row.

Divide your numbers by prime numbers as long as at least one of your numbers is evenly divisible by a prime number.

Division Table

2 14 21

3 7 21

7 7 7

1       1

Bring down any numbers that are not evenly divisible by the prime number.

When the last row of results is all 1's you are done.

Find the product of the prime numbers in the first column to get the LCM:

LCM = 2 x 3 x 7

LCM = 42

Therefore,

LCM(14, 21) = 42

GCF Method:

LCM(14,21) = (14 × 21) / GCF(14,21)

= (14 × 21) / 7

= 294 / 7

= 42

Therefore,

LCM(14, 21) = 42

Venn Diagram:

Inside circle 14: 2

In between circles: 7

Inside circle 21: 3

Then, use prime factorization.

Find the set of prime factors for each number.

Prime Factorization Set for 14 is:

2, 7

Prime Factorization Set for 21 is:

3, 7

Identify the intersections of prime numbers and put them into the Venn Diagram where their circles intersect. Remove the numbers from the original sets.

14 ∩ 21

7

With the remaining prime numbers in each set, if any, put them into their respective circles, not in intersections.

14

2

21

3

Find the union of the sets in the Venn Diagram. This is the superset of all of the prime numbers in the diagram.

14 ∪ 21

The new superset list is

2, 3, 7

Multiply these factors together to find the LCM.

LCM = 2 x 3 x 7 = 42

LCM = 42

Therefore,

LCM(14, 21) = 42

A few points (co-ordinates) for the given equations are :

(1,-1/2),(-1,1/4),(0,1),,(2,-1/4)

These points are called Coordinates, where the horizontal line denotes the x-axis and the verticle line is the y-axis.

Each point is identified by both x and y coordinates. A point's x-coordinate is a value that indicates how far the point is from the horizontal or x-axis origin.

y denotes the distance of the point on the vertical line from the origin

x denotes the distance of the point on the horizontal line from the origin

In the given equation y=-1/3x-1

We can substitute x with any integer (i.e any number among the positive or negative numbers)

Putting, x=1 in the given equation we get

y=-1/(3x1)-1

⇒ y=-1/3-1

⇒y=-1/2

Thus the coordinate when x=1 is (1,-1/2)

Similarly by putting x=0, 1, 2, -1 ;we get the following co-ordinates respectively

(0,1), (1,-1/2), (2,-1/4),(-1,1/4)

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Answer:

Step-by-step explanation:

x²-4x+2=0

x²-4x=-2

x²-4x+4=-2+4

(x-2)²=2

x-2=±√2

x=2±√2

x=2+√2≈2+1.41≈3.41

x=2-√2≈2-1.41≈0.59

so roots lie between 0 and 4.

When the students get reduced to 15 then each student will get 3 pens.

The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.

For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.

12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.

As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.

This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.

Given:

Total students = 25

Each Student get 2 pens.

So, 25 students get = 25 x 2= 50 pens

Now, number of children reduced = 25 -10= 15

So, each student get = 50/ 15 = 3.333 = 3 pens

Hence, each student will get 3 pens.

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