In triangle ABC, AP is an angle bisector of angle BAC. What is the length of PC? Round you answer to the nearest whole number.
A) 6
B) 7
C) 8
D) 9

The ABEF is a parallelogram because DC is extended to point F; AB ≅ FE and FA ≅ BE.

In two-dimensional geometry, it is a plane shape having four sides, in which two pairs of sides are parallel to each other and equal in length. The sum of all angles in a parallelogram is 360°.

We have ABCD is a parallelogram.

BE is perpendicular to FC

FA is perpendicular to FC

As we know in the parallelogram ABCD:

AB ≅ DC

AD ≅ CB

As the DC is extended to F

So, AB ≅ FE

And BE is perpendicular to FC

       FA  is perpendicular to FC  (given)

∴ FA ≅ BE

∴ ABEF is a parallelogram.

Thus, the ABEF is a parallelogram because DC is extended to point F; AB ≅ FE and FA ≅ BE.

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Answer: even

Step-by-step explanation:

A function is even if f(x)=f(-x), and odd if f(x)=-f(-x). If a function satisfies neither of these, it is neither even nor odd.

[tex]f(x)_=-6x^{2}-5x^{2}-2=-11x^{2}-2\\\\f(-x)=-11(-x)^{2}-2=-11x^{2}-2\\\\\therefore f(x)=f(-x)[/tex]

Therefore, the function is even.

The amount of ingredients necessary to serve 5 people is 3.75 cups spaghetti sauce, 2.5 cups ricotta cheese, and 2.5 cups mozzarella cheese.

It is described as the comparison of two quantities to determine how many times one obtains the other. The proportion can be expressed as a fraction or as a sign: between two integers.

We have:

A recipe for 8 servings lists: 6 cups spaghetti sauce 4 cups ricotta cheese and 4 cups mozzarella cheese.

First we take the ratio of ingredients for 8 servings:

spaghetti sauce : ricotta cheese : mozzarella cheese = 6:4:4

The ratio of ingredients for 1 servings:

spaghetti sauce : ricotta cheese : mozzarella cheese = 6/8 : 4/8 : 4/8

The ratio of ingredients for 5 servings:

spaghetti sauce : ricotta cheese : mozzarella cheese

= 5×0.75 : 5×0.5 : 5×0.5

= 3.75 : 2.5 : 2.5

The amount of ingredients necessary to serve 5 people:

Thus, the amount of ingredients necessary to serve 5 people is 3.75 cups spaghetti sauce, 2.5 cups ricotta cheese, and 2.5 cups mozzarella cheese.

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Answer:

The angle shows 90° so that means

90° - 42° = 48°

Therefore the angle of the depression is 48°

The angle of depression from C to point A 48°. The correct option is c. 48°

From the question, we are to determine the angle of depression from C to point A

The angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks downwards at an object

Thus, in the given diagram

The angle of depression is 90° - 42°

Angle of depression = 48°

Hence, the angle of depression from C to point A 48°. The correct option is c. 48°

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The percentage decrease will be equal to 99.30 %.

The Percentage is defined as representing any number with respect to the 100. It is denoted by the sign %.

Given that:-

The percentage decrease will be calculated as:-

Percentage decrease = [tex]\dfrac{2.88 - 0.02}{2.88}[/tex]

Percentage decrease = 0.9930 = 99.30%

Therefore the percentage decrease will be equal to 99.30 %.

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Answer:

4 and 7/12

Step-by-step explanation:

This is addition with fractions.

We must convert to the same denominator.

1/3 and 1/4 becomes 4/12 and 3/12

This makes 7/12

1 add 3 add 7/12 is 4 and 7/12

Answer:

Step-by-step explanation:

-5x+4y=-3

-6(-5x+4y)=-6×(-3)=18

Answer:

Solution: Here, the given equations are,

[tex]: \implies8x+5y=2...(i)\:and\:y-2=0...(ii)[/tex]

From equation,(ii); y-2=0

[tex] \therefore \: y = 2.........(iii)[/tex]

Substituting the value of y from equation (iii) to equation (i),we get

[tex]: \implies{8x + 5y = 2}[/tex]

[tex]: \implies{8x + 5(2) = 2}[/tex]

[tex]: \implies{8x = 2 - 10}[/tex]

[tex]: \implies{8x = - 8}[/tex]

[tex]: \implies{x = - 1}[/tex]

[tex] \therefore \: x = - 1[/tex]

Thus, x= -1 and y= 2 is the solution.

8x+5y=2...(i) y-2=0...(ii)

8(-1)+5(2)=2 2-2=0

2=2(True) 0=0(True)

The scale factor have been categorized into Expansion and Contraction.

It is the measure by which two figures are same in appearance but different in size.

It can be positive or negative as the figure is getting bigger or smaller.

Different scale factors are given and it has been asked they describe which type of dilation.

1.  -2/3  This will contract the figure

2. 3.25 This shows expansion

3. 2/3 This will contract the figure

4. -3 This will expand but the image will be form on the other side of Center of Enlargement.

Therefore the scale factor have been categorized into Expansion and Contraction.

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The values of the functions are:

1. y = f(-7) = 128

2. y = f(5) = 0.0313

We can evaluate a function by substituting the given value into the equation of the function, then simplify.

Given the function, y = f(x) = (1/2) ^ x

1. Find the value of f(-7):

y = f(-7) = (1/2)^-7

y = f(-7) = 2^7

y = f(-7) = 128

2. 1. Find the value of f(5):

y = f(5) = (1/2)^5

y = f(5) = 0.0313

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Answer:

The number that belongs in the green box is equal to 909.

General Formulas and Concepts:
Algebra I

Equality Properties

Trigonometry

[Right Triangles Only] Pythagorean Theorem:
[tex]\displaystyle a^2 + b^2 = c^2[/tex]

Step-by-step explanation:

Step 1: Define

Identify given variables.

a = 30

b = 3

c = x

Step 2: Find x

Let's solve for the general equation that allows us to find the hypotenuse:

Now that we have the formula to solve for the hypotenuse, let's figure out what x is equal to:

∴ the hypotenuse length x is equal to √909 and the number under the square root, our answer, is equal to 909.

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Topic: Trigonometry

Answer:

1

Step-by-step explanation:

The points go rise:1 up:1 so the slope is 1.

Answer:

hey!! we can taIk here

Step-by-step explanation:

Answer:

Step-by-step explanation:

(a). In set notation:

B. { x | x < 2 }

(b).

( - ∞ , 2 )

(c). See attachment.

Answer:

[tex]x = 2\\ \\x= -\dfrac 78[/tex]

Step-by-step explanation:

[tex]~~~~~~~8x^2 -9x -14=0\\\\\implies 8x^2+7x-16x-14=0\\\\\implies x(8x+7)-2(8x+7)=0\\\\\implies (x-2)(8x+7) = 0\\\\\implies x = 2,~ x= -\dfrac 78[/tex]

Cassie need to make the model is 2250 cm²

A square pyramid has a square base and four lateral faces in geometry. A pyramid is a polyhedron with a base and three or more triangular faces that meet above it .

The complete question is attached below:

We know,

A= a(a+ √(a²+4h²))

and, a =25cm, h=30 cm

So,

A= 25(25+ √(25²+4*30²))

=25(25+65)

= 2250 cm²

Hence, Cassie need to make the model is 2250 cm²

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Answer:

24

Step-by-step explanation:

Hello there!

To determine the amount (percentage) as from our question above, we have to divide the given percentage by 100% (original percentage) and then multiply by 120.

This is as shown below;

[tex] \frac{20\%}{100\%} \times 120 \\ \\ = 0.2 \times 12 \\ = 24[/tex]

. ° . = 24 <<<<<< Ans

I hope this helps.

Have a nice studies.

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

Explanation:

We want the relative frequency of the purple candies.

There are 16 purple out of 18+20+10+16 = 64 total

Divide the values to get 16/64 = 0.25

Exactly 25% of the candies are purple. This converts to the fraction 1/4.

The correct answer is 25%

First calculate the the total number of frequency

18+20+10+16

= 64

The total number of frequency is 64

Hence the relative frequency of purple is 16/64

Therefore , relative frequency of purple is 1/4 = 0.25

Therefore 25% of purple candies are there

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Answer:

longer leg: 20 in

shorter leg: 15 in

hypotenuse: 25 in

Formulas:

Pythagorean Theorem

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

c ... hypotenuse

a ... one leg

b ... another leg

Pythagorean theorem is used in right triangles (triangles in which one angle is 90°).

Step-by-step explanation:

longer leg: x

shorter leg: x - 5

hypotenuse: x + 5

To find x (longer leg), let's use Pythagorean theorem.

[tex]c^2 = a^2 + b^2\\(x+5)^2 = x^2 + (x-5)^2\\x^2 + 10x + 25= x^2 + x^2 -10x + 25\\x^2 + 10x = 2x^2 - 10x\\0 = x^2 - 20x[/tex]

Now let's factorize to get solutions for x.

[tex]0 = x(x-20)[/tex]

First solution:

[tex]x = 0[/tex]

Second solution:

[tex]x - 20 = 0\\x = 20[/tex]

Since a side of a triangle has to be a positive number, x is equal to 20.

Now let's just substitute x back to get side lengths. From the question the lengths are in inches.

longer leg: x = 20 in

shorter leg: x - 5 = 20 - 5 = 15 in

hypotenuse: x + 5 = 20 + 5 = 25 in

Answer:

6

Step-by-step explanation:

By the alternate exterior angles theorem,

120=15(x+2)

8=x+2

x=6

Answer:

900π

Step-by-step explanation:

Hello!

The radius is represented by the function [tex]r(t) = 15\sqrt{t +2}[/tex].

The radius of the ripple can be found when [tex]t=2[/tex]

Solve:

The radius of the ripple is 30 inches

The area of a circle is calculated by the formula [tex]A = \pi r ^2[/tex]

Given the radius is 30, we can find the area of the circle.

Solve:

Step-by-step explanation:

both halves of a diagonal must be equal.

so,

8b - 30 = 6b + 6

2b = 36

b = 18

and

9a - 13 = 5a + 5

4a = 18

a = 18/4 = 4.5

Answer:

y= -2/3x - 4

Step-by-step explanation:

the 2/3 comes from going up 2 spaces and over left 3 spaces. the b is negative 4 because of where it falls. the line is negative because of the direction it is in.

Answer:

The definition for the given piecewise-defined function is:   [tex]\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex].

Step-by-step explanation:

A piecewise-defined function represents specific rules over different intervals of the domain.  

Open interval: This means that the endpoint is not included in the interval.

We can use the following symbols to indicate the exclusion of endpoints in the interval:

Closed interval: This implies the inclusion of endpoints in the interval.

We can use the following symbols to indicate the inclusion of endpoints in the interval:

The best way to determine which piecewise-defined function represents the graph is by observing the endpoints and orientation of both partial lines.

Based on our observations from the previous step, we can infer that x > -1 or x ≤ -1 apply to piecewise-defined functions A or D. However, only one of those two options represent the graph.

    [tex]\boxed{\displaystyle\sf Option\:A)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ 2x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex]

We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.

Substitute x = -2 into f(x) = 2x + 2:  

⇒ The output value of f(-2) = -2 is not included in the graph of the partial line whose endpoint is at (-1, 1).

We must choose a domain value that falls within the interval of x > -1 whose output is included in the graph of the partial line with an open dot.

Substitute x = 0 into  f(x) = x + 4:

⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).

Conclusion for Option A:

Option A is not the correct piecewise-defined function because one of the pieces, f(x) = 2x + 2, does not specify the interval (-∞, -1].

    [tex]\boxed{\displaystyle\sf Option\:D)\:\:\:f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex]

We must choose a domain value that falls within the interval of x ≤ -1 whose output is included is included in the graph of the partial line with a closed dot.

Substitute x = -2 into f(x) = x + 2:

⇒ The output value of f(-2) = 0 is included the graph of the partial line whose endpoint is at (-1, 1).

We must choose a domain value that falls within the interval of x > -1 whose output is included is included in the graph of the partial line with an open dot.

Substitute x = 0 into f(x) = 2x + 4:

⇒ The output value of f(0) = 4 is included in the graph of the partial line whose endpoint is at (-1, 2).  

We can infer that the piecewise-defined function that represents the graph is:

[tex]\boxed{\displaystyle\sf\ Option\:D:\:\: f(x) = \begin{cases}\displaystyle\sf\ x + 2 & \sf\:{if\:\:x \leq -1} \\\displaystyle\sf\ 2x + 4 & \sf\:{if\:\:x > -1}\end{cases}}[/tex].

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The system of equations has the solution:

x = 111/42

y = 27/14

z = -15/6

We start with 3 linear equations:

2x+y-z=8

x+4y+z=7

-3x+2y-3z=21

To solve this, first we need to isolate one of the variables. I will isolate x on the second one:

x = 7 - 4y - z

Now we can replace that in the other two:

2*( 7 - 4y - z) + y - z = 8

-3*(7 - 4y - z) + 2y - 3z = 21

Now we need to isolate other variable, let's isolate y on the above one:

14 - 8y - 2z + y - z = 8

-7y - 3z = 8 - 14 = -6

z = (-6 + 7y)/-3 = 2 - (7/3)*y

Now we replace this on the last equation:

-21 + 12y -3z + 2y - 3z = 21

14y - 6z = 42

14y - 6*(2 - (7/3)*y) = 42

14y - 12 + 14y = 42

28y = 42 + 12 = 54

y = 54/28 = 27/14

Now that we know the value of y, we can find the value of z:

z = 2 - (7/3)*y = 2 - (7/3)*27/14 = 2 - 27/6 = -15/6

And the value of x:

x = 7 - 4y - z = 7 - 4*(27/14) + 15/6 = 7 - 48/7 + 15/6 = 111/42

So the solution is:

x = 111/42

y = 27/14

z = -15/6

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Answer:

126.5 years

Step-by-step explanation:

N(t) = 384(1.13)^t

2,000,000,000 = 384(1.13)^t

(1.13)^t = 2,000,000,000/384

log [(1.13)^t) = log (2,000,000,000/384)

t × log 1.13 = log (2,000,000,000/384)

t = [log (2,000,000,000/384)]/(log 1.13)

t = 126.5

Answer:

4

Step-by-step explanation:

35-10=25

25÷5.75= 4.3

The total time for the trip is:

T =  3mi*( 3/2x).

Where x is the rate at which she comes home.

Remember the relation:

distance = rate*time.

We know that the distance is 3 miles (done twice).

First, the rate is 2x and then the rate is x, then the time it took the first half is:

t = 3mi/2x

And for the coming back:

t = 3mi/x

Then the total time for the trip is:

T =  3mi/2x +  3mi/x = 3mi( 1/2x + 1/x) = 3mi*( 3/2x).

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The rule for the linear function that describes the length of the vine is L = 1.3 + 0.2d

The number of days it would take the vine to grow to be 10.3 inches long is 45 days

From the question, we are to determine a linear function that describes the situation

From the given information,

Then vine is 1.3 inches long now

and it grows 0.2 inches a day

∴ After the days, the vine will increase by 0.2d in length

Then,

If L represents the length of the vine, d represent the number of days

The length of the vine after d number of days will be

L = 1.3 + 0.2d

Hence, the rule for the linear function that describes the length of the vine is L = 1.3 + 0.2d

Now, we are to determining the number of days it will take the vine to grow to be 10.3 inches long

Using the above linear function

L = 1.3 + 0.2d

10.3 = 1.3 + 0.2d  

0.2d = 10.3 - 1.3

0.2d = 9

d = 9 / 0.2

d = 45

Hence, the number of days it would take the vine to grow to be 10.3 inches long is 45 days

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