Proportions in Triangles

The coefficient of x needs to be 2.

The current coefficient is 2/5.

To eliminate the 5 in the denominator, we can multiply the equation by 5.

We get

[tex]2x + 30y = - 50[/tex]

Now, to solve the equations, all we need to do is add them.

Hope this helps!

Answer:

-1.9, -1.8, and -1.7 (answers will vary)

Step-by-step explanation:

Since you have to pick a rational number that is between -2 and -1 there is an infinite options you can choose from. A rational number is a number that can be written as a fraction so you could choose a number like -1.0000000000000000000000001 and 1.999999999999999999999999.

Answer:

0, 17

3 1/2, 5

-3 1/2, -12

0, 17

Answer:

17

Step-by-step explanation:

First label the 2 points.

A = (3, 1/2, 5), B = (3, 1/2, -12)

Then calculate the vector from A to B:

[tex]\vect{AB} = (0, 0, -17)[/tex]

And then calculate it's length by the formula:

[tex]||\vect{a}|| = \sqrt{x^2 + y^2 + z^2}[/tex], where x, y, z are the coordinates related to the vector.

[tex]||\vect{AB}|| = \sqrt{0^2 + 0^2 + (-17)^2} = \sqrt{(-17)^2} = |17| = 17[/tex]

Are you sure that one of the variables is v and not y?

v+4= -2(x-1)

Since you posted v, I will use v in place of y.

v + 4 = -2x + 2

v = -2x + 2 - 4

v = -2x - 2

Done!

Answer:

Step-by-step explanation:

It's a right triangle.

Use the Pythagorean theorem:

[tex]leg^2+leg^2=hypoyenuse^2[/tex]

We have:

[tex]leg=13.5\ cm,\ leg=x\ cm,\ hypotenuse=(x+8.45)\ cm[/tex]

Substitute:

[tex]13.5^2+x^2=(x+8.45)^2\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\182.25+x^2=x^2+2(x)(8.45)+8.45^2\qquad\text{subtract}\ x^2\ \text{from both sides}\\\\182.25=16.9x+71.4025\qquad\text{subtract 71.4025 from both sides}\\\\110.8475=16.9x\qquad\text{divide both sides by 16.9}\\\\\dfrac{110.8475}{16.9}=x\to x\approx6.6\ cm[/tex]

Answer:

First case The coefficient of the squared term is 4

Second case The coefficient of the squared term is 1/16

Step-by-step explanation:

I will analyze two cases

First case (vertical parabola open upward)

we know that

The equation of a vertical parabola in vertex form is equal to

[tex]y=a(x-h)^{2}+k[/tex]

where

a is the coefficient of the squared term

(h,k) is the vertex

we have

(h,k)=(-5,-2)

substitute

[tex]y=a(x+5)^{2}-2[/tex]

Find the value of a

Remember that

when the x-value is -4, the y-value is 2.

substitute

For x=-4, y=2

[tex]2=a(-4+5)^{2}-2[/tex]

[tex]2=a(1)-2[/tex]

[tex]a=2+2=4[/tex]

the equation is equal to

[tex]y=4(x+5)^{2}-2[/tex]

therefore

The coefficient of the squared term is 4

Second case (horizontal parabola open to the right)

we know that

The equation of a horizontal parabola in vertex form is equal to

[tex]x=a(y-k)^{2}+h[/tex]

where

a is the coefficient of the squared term

(h,k) is the vertex

we have

(h,k)=(-5,-2)

substitute

[tex]x=a(y+2)^{2}-5[/tex]

Find the value of a

Remember that

when the x-value is -4, the y-value is 2.

substitute

For x=-4, y=2

[tex]-4=a(2+2)^{2}-5[/tex]

[tex]-4=a(4)^{2}-5[/tex]

[tex]-4+5=a(16)[/tex]

[tex]a=1/16[/tex]

the equation is equal to

[tex]x=(1/16)(y+2)^{2}-5[/tex]

therefore

The coefficient of the squared term is 1/16

to better understand the problem see the attached figure

Answer:

Option A 0.496

Step-by-step explanation:

we know that

The approximate average rate of change is equal to

[tex]\frac{f(8)-f(3)}{8-3}[/tex]

[tex]\frac{f(8)-f(3)}{5}[/tex]

we have

[tex]f(x)=0.01(2^{x})[/tex]

Find f(8)

For x=8

[tex]f(8)=0.01(2^{8})=2.56[/tex]

Find f(3)

For x=3

[tex]f(8)=0.01(2^{3})=0.08[/tex]

Find the approximate average rate of change

[tex]\frac{f(8)-f(3)}{5}[/tex]

substitute

[tex]\frac{2.56-0.08}{5}=0.496[/tex]

Answer: A, 0.496

Step-by-step explanation:

To find the difference, you need to raise the base, 2, to each number since x represents the days.

Raise 2 to the power of 3:

2^3 = 8

multiply by 0.01

0.01 * 8 = 0.08

That is the growth of day three.

Now do the same with the 8

Raise 2 to the power of 8

2^8 = 256

Now multiply that by 0.01

0.01 * 256 = 2.56

Now use this formula: f(b) - f(a)/b - a

2.56 - 0.08/8 - 3

Subtract 0.08 from 2.56

2.56 - 0.08 = 2.48

Subtract 3 from 8

8 - 3 = 5

Now divide: 2.48/5 = 0.496

0.496 is the average rate of change between day 3 and day 8. Also I got 100 on the test so I know the answer :))

Answer:

Answer is all real numbers.

<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>

---------(-3)---------------------(6)-------------

Step-by-step explanation:

6b<36

Divide both sides by 6:

b<6

or

2b+12>6

Subtract 12 on both sides:

2b>-6

Divide both sides by 2:

b>-3

So we want to graph b<6 or b>-3:

               o~~~~~~~~~~~~~~~~~~~~~~~~~~ b>-3

~~~~~~~~~~~~~~~~~~~~~~~~o                        b<6

_______(-3)____________(6)___________

So again "or" is a key word! Or means wherever you see shading for either inequality then that is a solution to the compound inequality.  You see shading everywhere so the answer is all real numbers.

<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>

---------(-3)---------------------(6)-------------

Answer:

All real numbers [tex](-\infty, \infty)[/tex]

Step-by-step explanation:

First we solve the following inequality

[tex]6b < 36[/tex]

Divide by 6 both sides of the inequality

[tex]b<\frac{36}{6}\\\\b<6[/tex]

The set of solutions is:

[tex](-\infty, 6)[/tex]

Now we solve the following inequality

[tex]2b + 12 > 6[/tex]

Subtract 12 on both sides of the inequality

[tex]2b + 12-12 > 6-12[/tex]

[tex]2b> -6[/tex]

Divide by 2 on both sides of the inequality

[tex]\frac{2}{2}b> -\frac{6}{2}[/tex]

[tex]b> -3[/tex]

The set of solutions is:

[tex](-3, \infty)[/tex]

Finally, the set of solutions for composite inequality is:

[tex](-\infty, 6)[/tex]  ∪ [tex](-3, \infty)[/tex]

This is: All real numbers [tex](-\infty, \infty)[/tex]

System of equations have infinitely many solution for x and y.

We have to given that,

System of equation are,

3x + 2y = 12

12x + 8y = 48

We can use the elimination method to solve,

System of equation are,

3x + 2y = 12  .. (i)

12x + 8y = 48  .. (ii)

Multiply by 4 in (i) and subtract from (ii);

12x + 8y - 12x - 8y  = 48 - 48

0 = 0

Hence, System of equations have infinitely many solution for x and y.

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Answer:

34.24 cm²

Step-by-step explanation:

You first need to find the area of the circle.

4 is radius. r*r*3.14=50.24

Now the triangle is 4*4=16.

50.24-16= 34.24

Answer:

B.

Step-by-step explanation:

The area of the shaded region will be the area of the circle minus the area of the triangle inside the circle. Then:

The circle has radius 4 cm (distance from the center to the edge of the circle), so the area of the circle is

[tex]A=\pi r^2 = 3.14(4cm)^2 = 3.14(16cm^2) = 50.24 cm^2.[/tex]

Now, the area of the triangle is

[tex]A_2 = \frac{b*h}{2}[/tex].

The base of the triangle is the diameter of the circle (as you can see in the image) and the height is the radius of the circle. Then the are of the triangle is

[tex]A_2 = \frac{8*4}{2} = \frac{32}{2} = 16cm^2[/tex].

Finally, the shaded area is [tex]A-A_2 = 50.24-16 = 34.24 cm^2[/tex].

Answer:

[tex]y-5=\frac{-1}{2}(x+2)[/tex] point-slope form

[tex]y=\frac{-1}{2}x+4[/tex]  slope-intercept form

Step-by-step explanation:

The slope-intercept form of a line is y=mx+b where m is the slope and b is the y-intercept.

The slopes of perpendicular lines are opposite reciprocals.

The slope of y=2x-5 is 2.

So we are looking for a line perpendicular to y=2x-5 which means we first to the take the opposite reciprocal of it's slope giving us:

opposite reciprocal of (2) is opposite (1/2)=-1/2.

So the slope of the line we are looking for is -1/2.

This means are equation for our line is in this form:

[tex]y=\frac{-1}{2}x+b[/tex]

To find b we will use a point (x,y) that is on our line.

We are given a point (x,y)=(-2,5).

Plug this into our equation:

[tex]5=\frac{-1}{2}(-2)+b[/tex]

[tex]5=1+b[/tex]

Subtract 1 on both sides:

[tex]4=b[/tex]

So the equation for our line that we are looking for is:

[tex]y=\frac{-1}{2}x+4[/tex]  (slope-intercept form).

You could also go for point-slope form [tex]y-y_1=m(x-x_1)[/tex] where m is the slope and [tex](x_1,y_1)[/tex] is a point on the line.

We have m=-1/2 and (x1,y1)=(-2,5) so our equation in point slope-form is:

 [tex]y-5=\frac{-1}{2}(x-(-2))[/tex]

Simplifying just a hair:

[tex]y-5=\frac{-1}{2}(x+2)[/tex].

Answer:

[tex]\large\boxed{x=-2\ or\ x=-\dfrac{1}{3}}[/tex]

Step-by-step explanation:

The quadratic formula of a quadratic equation

[tex]ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

We have the equation:

[tex]3p^2+7p+2=0\to a=3,\ b=7,\ c=2[/tex]

Substitute:

[tex]b^2-4ac=7^2-4(3)(2)=49-24=25\\\\\sqrt{b^2-4ac}=\sqrt{25}=5\\\\x_1=\dfrac{-7-5}{2(3)}=\dfrac{-12}{6}=-2\\\\x_2=\dfrac{-7+5}{2(3)}=\dfrac{-2}{6}=-\dfrac{1}{3}[/tex]

Answer:

-1/3,-2

Step-by-step explanation:

The quadratic formula is [-b±√(b^2-4ac)]/2a.  In this equations 3p^2 is a, 7p is b, and 2 is c.

Then just plug in the numbers.

[-7±√((-7^2)-4(3)(2)]/2(3)

[-7±√(25)]/6

(-7+5)/6  and  (-7-5)/6

-1/3 and -2 are the answers, if you plug these numbers into the original equation, you find that they equal 0 which means that they work.

Answer:

1st and 2nd quadrants

Step-by-step explanation:

You have sine is positive since it is 4/5.

Sine is the y-coordinate.

On the coordinate plane y is positive in the 1st and 2nd quadrants.

The values of θ that could terminate when sin θ = 4/5 are in the first and second quadrants. In the first quadrant, both the sine and cosine values are positive, while in the second quadrant, only the sine value is positive.

To find the quadrants where θ could terminate when sin θ = 4/5, we need to examine the values of sin θ in each quadrant. The sine function is positive in the first and second quadrants so that θ could terminate in either of these quadrants. In the first quadrant, the sine and cosine values are positive, while in the second quadrant, only the sine value is positive.

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let's recall Cavalieri's Principle, solids with equal altitudes and cross-sectional areas at each height have the same volume, so even though this cylinder is slanted with a height = x and a radius = 8, the cross-sectional areas from the bottom to top are the same thickness and thus the same area, so its volume will be the same as a cylinder with the same height and radius that is not slanted.

[tex]\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\ \cline{1-1} r=8\\ h=x\\ V=768\pi \end{cases}\implies 768\pi =\pi (8)^2(x)\implies 768\pi =64\pi x \\\\\\ \cfrac{768\pi }{64\pi }=x\implies 12=x[/tex]

The value of the product (3 - 21) * (3 + 21) is -432, obtained by applying the distributive property.

To find the value of the product (3 - 21) * (3 + 21), we can use the distributive property or the difference of squares identity. Here's how it works:

(3 - 21) * (3 + 21) = (3 - 21) * [(3) + (21)]  (Apply the distributive property)

Now, let's calculate each part:

1. (3 - 21) = -18

2. (3 + 21) = 24

Now, we multiply these results together:

-18 * 24 = -432

So, the value of the product (3 - 21) * (3 + 21) is -432.

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Final answer:

The product of (3-21) and (3 + 21) is -324.

Explanation:

The value of the product (3-21) (3 + 21) is -324.

To find the product, first calculate the values within the parentheses:

3 - 21 = -18

3 + 21 = 24

Multiply these two values: -18 * 24 = -324.

Answer:

Inverse of a function

Step-by-step explanation:

If the x- and y-values in each pair of a set of ordered pairs are interchanged, the resulting set of ordered pairs is known as the inverse of a function.

For example, given the following function:

y = 2x

If x=0 → y= 0

If x=1 → y= 2

If x=2 → y= 4

Now, if we find the inverse of the function:  

y = 2x → x = 2y → y = x/2

Now:

If x=0 → y= 0

If x=2 → y= 1

If x=4 → y= 2

Comparing both cases, you will notice that the ordered pairs are effectively interchanged.

The resulting set of ordered pairs formed by interchanging the x- and y-values in each pair is known as the inverse of the original set.

If the x- and y-values in each pair of a set of ordered pairs are interchanged, the resulting set of ordered pairs is known as the inverse of the original set. In mathematics, an ordered pair is a pair of objects written in a specific order, typically as (x,y). If you switch the positions of the elements to form (y,x), you create an ordered pair that represents the inverse relationship. This is particularly relevant in the context of functions and relations on a Cartesian coordinate system. The concept of ordered pairs is fundamental to understanding mappings and the domain and range of relations.

For example, if you have an ordered pair representing a function, such as (3,4), its inverse would be (4,3). This reflects a new relationship where the original output is now the input and vice versa. The set of all such inverted pairs from a function forms the inverse function, provided that each y value is associated with only one x value (the function is one-to-one).

A relation where the ordering of elements matters contrasts with a set where the order does not affect the identity of the set. Hence, achieving an inverse relationship by switching the ordered pairs is a useful tool in mathematical problem-solving and analysis.

Answer:

A plane figure with 4 sides is called a quadrilateral.

A quadrilateral is a plane figure bounded by four straight lines in mathematics.

Plane figure bounded by four straight lines: In mathematics, a quadrilateral is a plane figure bounded by four straight lines. Examples of quadrilaterals include squares, rectangles, parallelograms, and trapezoids.

Answer:

he would have 10 roses left

Step-by-step explanation:

12-5=7-2=5

5+9=14-4=10

Answer:

The red point (-4 , -6)

Step-by-step explanation:

* Lets revise the complex number

- The complex number z = a + bi, where a is the real part and b is the

  imaginary part

- The real part represented by the x-axis and the imaginary part

  represented by the y-axis

- The value of i is √(-1)

- The complex conjugate of a complex number is the number with an

 equal real part and an imaginary part equal in magnitude but opposite

 in sign

- Ex: the conjugate of a + bi is a - bi

* Lets solve the problem

∵ A is an complex number

∵ The x-coordinate of A is -4 and the y-coordinate of it is 6

∵ The x-axis is the real axis and y-axis is the imaginary axis

∴ A = -4 + 6i

∵ The conjugate numbers are equal in real part and the imaginary

  part equal in magnitude and different in sign

∴ The conjugate of A = -4 - 6i

- From the graph The red point (-4 , -6) represents the complex

 conjugate of point A

Answer:

[tex](6,17)[/tex]

Step-by-step explanation:

we know that

In this problem

The gate hinges must be placed at the mid-point of the fence.

The formula to calculate the midpoint between two points is

[tex]M(\frac{x1+x2}{2},\frac{y1+y2}{2})[/tex]

we have

[tex]A(4,12),B(8,22)[/tex]

substitute the values in the formula

[tex]M(\frac{4+8}{2},\frac{12+22}{2})[/tex]

[tex]M(6,17)[/tex]

Answer: (6,17)

Step-by-step explanation: Plato and Edmentum

Answer:

-[tex]\sqrt{5}[/tex]

Step-by-step explanation:

A root with square root or under root is only obtained when we take the square root of both sides. Remember that when we take a square root, there are two possible answers:

For example, for the equation:

[tex]x^{2}=3[/tex]

If we take the square root of both sides, the answers will be:

[tex]x=\sqrt{3} \text{ or } x= -\sqrt{3}[/tex]

Only getting one solution with square root is not possible. Solutions with square root always occur in pairs.

For given case, the roots are 6 and [tex]\sqrt{5}[/tex]. Therefore, the 3rd root of the polynomial function f(x) had to be -[tex]\sqrt{5}[/tex]

It seems you made error while writing option B, it should be - square root of 5.

Answer:

B

Step-by-step explanation:

Answer:

x = [tex]\frac{7 - r}{a}[/tex]

Step-by-step explanation:

ax+r=7

ax = 7 - r

x = [tex]\frac{7 - r}{a}[/tex]

Answer:

Option B, C and D are correct choices.

Step-by-step explanation:

We have been given an equation [tex]3.5+1.2(6.3-7x)=9.38[/tex]. We are asked to choose the steps that are involved in solving the equation.

Let us solve the equation.

Using distributive property [tex]a(b+c)=ab+ac[/tex], we will distribute 1.2 to 6.3 and [tex]-7x[/tex].

[tex]3.5+1.2*6.3-1.2*7x=9.38[/tex]

[tex]3.5+7.56-8.4x=9.38[/tex]

Therefore, option B is the correct choice.

Now, we will combine like terms.

[tex]11.06-8.4x=9.38[/tex]

Therefore, option C is the correct choice.

Now, we will subtract 11.06 from both sides of our equation.

[tex]11.06-11.06-8.4x=9.38-11.06[/tex]

[tex]-8.4x=-1.68[/tex]

Therefore, option D is the correct choice.

Now, to solve for x, we need to divide both sides of our equation by [tex]-8.4[/tex]

[tex]\frac{-8.4x}{-8.4}=\frac{-1.68}{-8.4}[/tex]

[tex]x=0.2[/tex]

Answer:

Distribute 1.2 to 6.3 and –7x;

Combine 3.5 and 7.56.

Subtract 11.06 from both sides.

Step-by-step explanation:

To answer this expression. Let's follow P.E.M.D.A. order, the acronym for  PArenthesis, Exponents, Multiplication, Division and Addends.  So distributing the factor 1.2 to the parenthesis content:

[tex]3.5+1.2(6.3-7x)=9.38 \\3.5+7.56-8.4x=9.38[/tex]

Then adding the 3.5 to 7.56 to simplify it:

[tex]3.5+7.56-8.4x=9.38\\\11.06-8.4x=9.38[/tex]

The next step in order to isolate is to subtract 11.06 from both sides

[tex]11.06-8.4x-11.06=9.38-11.06[/tex]

Then it goes on

[tex]-8.4x=-1.68\:\:*(-1)\\8.4x=1.68\Rightarrow x=\frac{1.68}{8.4}\Rightarrow x=\frac{1}{5}\\S=\{ {\frac{1}{5}\}[/tex]

The correct solution is,

⇒ 2 / (1 - tan²x)

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ tan (2x) / tan x

Now, We know that;

tan 2x = (2 tanx / 1 - tan²x)

Hence, We can simplify as;

⇒ tan (2x) / tan x

⇒ (2tan (x) /1 - tan²x) / tan x

⇒ 2 / (1 - tan²x)

Thus, The correct solution is,

⇒ 2 / (1 - tan²x)

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The expression tan(2x)/tan(x) simplifies using the double-angle identity for tangent to 2/(1 - tan^2(x)), which corresponds to Option A.

To simplify the trigonometric expression tan(2x)/tan(x) using double-angle identities, we can use the double-angle identity for tangent:

tan(2x) = 2tan(x)/(1 - tan2(x)).

Now, if we divide tan(2x) by tan(x), we get:

tan(2x)/tan(x) = (2tan(x)/(1 - tan2(x)/tan(x).

With tan(x) in the numerator and denominator, it cancels out, leaving:

2/(1 - tan2(x)).

Therefore, the correct answer is Option A: 2/(1 - tan2(x)).

Answer:

See below in bold,.

Step-by-step explanation:

There are 30-17 = 13 boys in the class.

a.  Prob(First is a boy ) = 13/30 and Prob( second is a boy = 12/29).

As these 2 events are independent:

Prob( 2 boys being picked) = 13/30 * 12/29 =  26/145 or 0.179 to the nearest thousandth.

b. By a similar method to a:

Prob ( 2 girls being picked) =  17/30 * 16/29 = 136/435 = 0.313 to the nearest thousandth.

c.  Prob (First student is a boy and second is a girl) =  13/30 * 17/29 = 221/870.

Prob ( first student is a girl and second is a boy)  = 17/30 * 13/29 = 221/870

These 2 events are not independent so they are added:

Prob( one of the students is a boy)  = 2 (221/870 =  221/435 = 0.508 to the nearest thousandth.

Answer:

The correct option is D) (5x − 2)(2x − 3).

Step-by-step explanation:

Consider the provided expression.

[tex]10x^2-19x+6[/tex]

Where x is time in minutes.

We need to find the appropriate form of the expression that would reveal the time in minutes when the trough is empty.

When the trough is empty the whole expression becomes equal to 0.

Substitute the whole expression equal to 0 and solve for x that will gives us the required expression.

[tex]10x^2-19x+6=0[/tex]

[tex]10x^2-15x-4x+6=0[/tex]

[tex]5x(2x-3)-2(2x-3)=0[/tex]

[tex](5x-2)(2x-3)=0[/tex]

Now consider the provided option.

By comparison the required expression is D) (5x − 2)(2x − 3).

Hence, the correct option is D) (5x − 2)(2x − 3).

Answer:

The correct answer is D

Step-by-step explanation:

Answer:

6

Step-by-step explanation:

I took the same test

Answer:

56.4 miles

Step-by-step explanation:

4228/75 = 56.37333333

Answer:

C. 3 times

Step-by-step explanation:

If cube a has an edge length of 2 and cube b has an edge length of 6,the volume of cube b than cube a is 3 times greater.

For this case we have that by definition, the volume of a cube is given by:

[tex]V = l ^ 3[/tex]

Where:

l: It's the side of the cube

Cube A:

[tex]l = 2\\V = 2 ^ 3 = 8 \ units ^ 3[/tex]

Cube B:

[tex]l = 6\\V = 6 ^ 3 = 216 \ units ^ 3[/tex]

We divide:

[tex]\frac {216} {8} = 27[/tex]

Thus, the volume of cube B is 27 times larger than that of cube A.

Answer:

Option A