Before art class starts, Gordon makes sure that each table has a full bowl of paint. He has 1/2 of gallon of paint and it able to fill the bowls for all 7 tables. How many gallons of paint does each table have?

Answer:

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

- Divide through by cos²x ;

[tex]{ \tt{ \frac{ { \sin }^{2}x }{ \cos {}^{2} x} + \frac{ \cos {}^{2} x }{ { \cos }^{2} x} = \frac{1}{ { \cos }^{2}x } }} \\ \\ { \boxed{ \tt{ { \tan}^{2} x + 1 = { \sec }^{2} x}}}[/tex]

- Divide tgrough by sin²x;

[tex]{ \tt{ \frac{ { \sin }^{2}x }{ { \sin}^{2}x } + \frac{ \cos { }^{2}x }{ { \sin}^{2}x } = \frac{1}{ { \sin}^{2} x} }} \\ \\ { \boxed{ \tt{1 + { \cot }^{2}x = \csc {}^{2} x}}}[/tex]

Answer:

See below for proof.

Step-by-step explanation:

Given trigonometric identity:

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

To prove the identity 1 + cot²x = cosec²x :

[tex]\boxed{\begin{aligned}\sin^2x+\cos^2x &=1\\\textsf{Divide by $\sin^2x$}\implies \dfrac{\sin^2x}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x}& =\dfrac{1}{\sin^2x}\\1+\left(\dfrac{\cos x}{\sin x}\right)^2&=\left(\dfrac{1}{\sin x}\right)^2\\1+(\cot x)^2&=(\text{cosec}\: x)^2\\1+\cot^2x&=\text{cosec}\: ^2x\end{aligned}}[/tex]

To prove the identity tan²x + 1 = sec²x :

[tex]\boxed{\begin{aligned}\sin^2x+\cos^2x &=1\\\textsf{Divide by $\cos^2x$}\implies \dfrac{\sin^2x}{\cos^2x}+\dfrac{\cos^2x}{\cos^2x}& =\dfrac{1}{\cos^2x}\\\left(\dfrac{\sin x}{\cos x}\right)^2+1&=\left(\dfrac{1}{\cos x}\right)^2\\(\tan x)^2+1&=(\sec x)^2\\\tan^2x+1&=\sec ^2x\end{aligned}}[/tex]

Additional identities used:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Trigonometric Identities}\\\\$\tan \theta=\dfrac{\sin \theta}{\cos \theta}$\\\\$\cot \theta=\dfrac{\cos \theta}{\sin \theta}$\\\\$\csc \theta=\dfrac{1}{\sin \theta}$\\\\$\sec \theta=\dfrac{1}{\cos \theta}$\\\end{minipage}}[/tex]

Total annual expense= $8400

Saving= $622,834,634,198.44

Compound interest is addition of interest to the principal sum of a loan or deposit, or in other words, interest on principal plus interest. It is result of reinvesting interest, or adding it to the loaned capital rather than paying it out, or requiring payment from borrower, so that interest in the next period is then earned on principal sum plus previously accumulated interest. Compound interest is standard in finance and the economics.

Compound interest is contrasted with the simple interest, where previously accumulated interest is not added to the principal amount of the current period, so there is no compounding. The simple annual interest rate is interest amount per period, multiplied by the number of periods per year. The simple annual interest rate is also known as nominal interest rate (not to be confused with the interest rate not adjusted for inflation, which goes by the same name).

a)Total annual expense= total miles *average cost per mile

                                   = 30,000* 0.28= $8400

b)Amount deposited in 9 years= 8400*9= $75,600

total amount after 9 years= FV= PM[tex][(1+r)^{n}-1]/r[/tex]

                                           =8400* [tex][(1+8.5)^{9}-1]/8.5[/tex]

                                           =$622,834,709,798.44

Savings=622,834,709,798.44 -75600= $622,834,634,198.44

To know more about compound interest visit: https://brainly.com/question/14295570

#SPJ9

Solve the inequality for x.

Answer:

x > 2

Step-by-step explanation:

10x+5 > 25

Solve

Subtract 5 from each side

10x+5-5 > 25-5

10x > 20

Divide each side by 10

10x/10 > 20/10

x > 2

The coordinates along the directed line segment is (-1/8, 1/4)

The points are given as:

A  = (3, 4)

B = (-2, -2)

The ratio on the segment can be represented as;

m : n = 5 : 3

So, we have the coordinates of the point Q that lies along the directed line segment to be

Q = 1/(m + n) * (mx2 + nx1, my2 + ny1)

So, we have:

Q = 1/(5 + 3) * (5 * -2 + 3 * 3, 5 * -2 + 3 * 4)

Evaluate

Q = (-1/8, 1/4)

Hence, the coordinates of the point Q is (-1/8, 1/4)

Read more about line partitions at:

brainly.com/question/17374569

#SPJ1

The simplified  complex fraction is

120/11

This is further explained below.

Generally, the complex fraction  is

[tex]\frac{\left(\frac{16}{3}\right)}{\left(\frac{22}{45}\right)}[/tex]

Multiply the numerator by the reciprocal of the denominator.

[tex]\frac{16}{3} \cdot \frac{45}{22}[/tex]

Cancel the common factor of 2

[tex]$\frac{8}{3} \cdot \frac{45}{11}$[/tex]

Combine 8 and 15/11

=8*15/11

=120/11

In conclusion, The result can be shown as.

120/11

Read more about complex fractions

https://brainly.com/question/14693420

#SPJ1

Answer:

(-12, 21)

Explanation:

3 times a number decreased by 5 means 3x -5.

This number is between -41 and 58; therefore,

Now we solve for the unknown number x.

Adding 5 to the three sides gives

finally, dividing the inequality by 3 gives

the parenthesis indicates that the -12 and 21 are not included in the solution interval =.

The intersection point of the circle and the line is at-(2.68, 7.36).

For the given question;

The equation of the line is; y = 2x + 2   ......eq 1

The centre of the circle (h,k) = (0,2)

The radius of the circle = 6 units.

This is the equation's standard form.

(x - h)² + (y - k)² = r²

Put the values in the formula;

(x - 0)² + (y - 2)² = r²

(x)² + (y - 2)² = r²

Put the value of y from eq 1.

(x)² + (2x + 2 - 2)² = 6²

(x)² + (2x)² = 36

Simplifying;

x = ±6/√5

But the intersection must be in first quadrant.

x = 6/√5 = 2.68

Put the value of x in eq 1.

y = 2x + 2

y = 2×2.68 + 2

y = 7.36

Thus, the coordinates of the intersection of the line and the circle are (2.68, 7.36).

To know more about the equation of circle, here

https://brainly.com/question/1506955

#SPJ13

Answer:

x = 30

Step-by-step explanation:

2x + 15 = 3x - 15

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

2x + 15 = 3x - 15

-3x          -3x

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

-x + 15 = -15

    -15     -15

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

-x = -30

÷-1   ÷-1

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

x = 30

I hope this helps!

The vertical and horizontal asymptotes for the graphed function are given as follows:

a) The vertical asymptotes are given at  x = -1 and x = 2.

b) The horizontal asymptote is given at y = -1.

In the context of this problem, we have that the asymptotes can be extracted from the graph as follows:

More can be learned about asymptotes at https://brainly.com/question/4138300

#SPJ1

The property of a(b+(-b))=(b+(-b))a is called as Commutative Property of Multiplication.

The Commutative Property in mathematics is a rule that allows interchange the order of terms or factors without any change in the end result.

In multiplication, we can interchange the factors without affecting the end result of multiplication. For example, 2 x 3 = 3 x 2 = 6

In addition, we can interchange the terms without affecting the summation result. For example, 2+3 = 3+2 = 5

But Commutative Property does not apply to subtraction or division.

For example, 2-3 != 3-2

or, 2/3 != 3/2

Hence the answer to the question is Commutative Property of Multiplication.

To know more about "Commutative" click here

https://brainly.com/question/28747013

#SPJ1

The amount of time 26 students spend on their phones represent by a dot plot:

To learn more about dot plots, follow the below link:

https://brainly.com/question/15853311

Answer:

See attachment.

Step-by-step explanation:

A Dot Plot is a graphical display of quantitative data consisting of dots plotted on a simple scale.  

If a value occurs more than once, the dots are placed one above the other so that the height of the column of dots represents the frequency for that value.

Given data set:

{0, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12}

Label the increments on the given line to include the range of data.

Label them -2 to 16 increasing in 1 unit.

Label the dot plot with a suitable title describing what the dot plot represents.

Place "Time students spent on their phones on one given day (nearest hour)" under the line.

Count the number of data points for each discrete category:

[tex]\begin{array}{|c|c|}\cline{1-2} \sf Category & \sf Frequency\\\cline{1-2} 0 & 1\\\cline{1-2} 3 & 1\\\cline{1-2} 4 & 2\\\cline{1-2} 5 & 2\\\cline{1-2} 6 & 3\\\cline{1-2} 7 & 4 \\\cline{1-2} 8 & 4 \\\cline{1-2} 9 & 3\\\cline{1-2} 10 & 3\\\cline{1-2} 11 & 2\\\cline{1-2}12 & 1\\\cline{1-2}\end{array}[/tex]

Draw a stack of dots that number high for each category.

(See attachment).

[tex]{\Large \begin{array}{llll} w=t^4 e^{\frac{2-t}{5+5t}} \end{array}} \\\\[-0.35em] ~\dotfill\\\\ 4t^3e^{\frac{2-t}{5+5t}}~~ + ~~\stackrel{\textit{\LARGE product rule}}{t^4 \stackrel{\textit{\large chain rule}}{e^{\frac{2-t}{5+5t}}\cdot \stackrel{quotient~rule}{\cfrac{(-1)(5+5t)~~ - ~~(2-t)(5)}{(5+5t)^2}}}} \\\\\\ 4t^3e^{\frac{2-t}{5+5t}}~~ + ~~t^4 e^{\frac{2-t}{5+5t}}\cdot \cfrac{(-5-5t)~~ - ~~(10-5t)}{(5+5t)^2}[/tex]

[tex]4t^3e^{\frac{2-t}{5+5t}}~~ + ~~t^4 e^{\frac{2-t}{5+5t}}\cdot \cfrac{-15}{(5+5t)^2} \implies t^3 e^{\frac{2-t}{5+5t}}\left[4- \cfrac{15t}{(5+5t)^2} \right] \\\\\\ t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{(100t^2+200t+100)-15t}{(5+5t)^2} \right]\implies t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{100t^2+185t+100}{(5+5t)^2} \right] \\\\[-0.35em] ~\dotfill\\\\ ~\hfill {\Large \begin{array}{llll} \cfrac{dw}{dt}=t^3 e^{\frac{2-t}{5+5t}}\left[\cfrac{100t^2+185t+100}{(5+5t)^2} \right] \end{array}}~\hfill[/tex]

The most appropriate choice for decimal will be given by-

The number is 68.830

What is decimal?

The type of numbers which has a whole number part and a fractional part and the whole number part is seperated from the fractional part by a decimal point is known as decimal.

Decimal can be terminating as well as non terminating

Terminating decimal numbers are those numbers which has finite number of figures after the decimal point.

Non terminating decimal numbers are those numbers which has infinite number of figures after decimal points.

Here,

The required number is the same as sixty-eight and 830 thousandths.

So, the number is 68.830

To learn more about decimal, refer to the link:

https://brainly.com/question/703656

#SPJ13

Answer:

Step-by-step explanation:

You want the factors of 5y² +13y +6, and the solutions to 5y² +13y +6 = 0.

The quadratic ax²+bx+c with a leading coefficient a≠1 can be factored by considering the factors of the product 'ac' that have a sum of 'b'.

  5·6 = 30 = (1)(30) = (2)(15) = (3)(10) = (5)(6)

The sums of these factor pairs are, respectively, 31, 17, 13, 11. The factor pair with a sum of 13 is (3)(10).

There are a couple of methods that can be used to apply this knowledge to the factorization of the quadratic. Perhaps the most straightforward is to write the quadratic as 4 terms, using the found values to rewrite the linear term as their sum:

  5y² +13y +6 = 5y² +3y +10y +6 . . . . . . . . . 13 ⇒ 3 +10

Now, pairs of terms can be factored:

  = y(5y +3) +2(5y +3)

  = (y +2)(5y +3) . . . . . . . fully factored expression

The solutions that makes the product of factors be zero will be the values of x that make the factors be zero. A product is only zero if one of the factors is.

  y +2 = 0   ⇒   y = -2

  5y +3 = 0   ⇒   y = -3/5

The solutions to the quadratic equation are y = {-2, -3/5}.

__

Additional comment

Another way to find the factors is this. For factors p, q that have a product of 'ac' and a sum of 'b', the factorization can be ...

  (ay +p)(ay +q)/a

For the values in this problem, this would be ...

  (5y +3)(5y +10)/5

The latter of these factors can be divided by 5, so this can be simplified to ...

  (5y +3)(y +2) . . . . as above

The solution of the system of equation is the point where the 2 lines cross each other.

By looking at the graph, we can see that they cross each other at (-3,-8)

Correct option: 2

a = 252, x = 7; y = 5925.13

Equation is given by the formula:

where:

yo = the initial population; r = growth rate, x = time

In your case, yo = 252. And the grow rate is 57%, therefore the r = 0.57. Finally, the exponential equation has this form:

Then, if we want to know what is the population after 7 days, we can evaluate this function:

The measure of the angle having the question mark, found using the law of sines is ? ≈ 140°

The law of sines states that the sine of an interior angle of a triangle is proportional to the the length of the side opposite to the angle.

The given parameters are;

The radius of circle A = 6.4

The radius of circle B = 4.0

The radius of circle C = 3.0

The measure of angle at C = 24°

The length of the segment AB = 6.4 + 4 = 10.4

According to the law of sines, we have;

[tex]\dfrac{sin(A)}{a} = \dfrac{sin(B)}{b} = \dfrac{sin(C)}{c}[/tex]

Where;

a = The length of the side opposite angle ∠A

b = The length of the side opposite angle ∠B

c = the length of the side opposite angle ∠C

From the diagram, the side opposite angle ∠C, is AB, which by comparison indicates that when side c = AB = 10.4

c = 10.4

Similarly, the side opposite angle ∠A = a = BC = 4 + 3 = 7

a = 7

The known information are therefore; ∠C = 24°, a = 7, c = 10.4

[tex]\dfrac{sin(A)}{a} = \dfrac{sin(C)}{c}[/tex]

[tex]\dfrac{sin(A)}{7} = \dfrac{sin(24^{\circ})}{10.4}[/tex]

[tex]sin(A) = \dfrac{sin(24^{\circ})}{10.4}\times 7[/tex]

[tex]\angle A = arcsine \left( \dfrac{sin(24^{\circ})}{10.4}\times 7\right)[/tex]

∠A + ∠B + ∠C = 180° (angle sum property of a triangle)

∠B = 180° - (∠A + ∠C)

Which gives;

[tex]\angle B = 180^{\circ} -\left(24^{\circ} + arcsine \left( \dfrac{sin(24^{\circ})}{10.4}\times 7\right)\right) \approx 140^{\circ}[/tex]

The measure of the angle is ∠B ≈ 140°

Learn more about the law of sines here:

https://brainly.com/question/15659539

#SPJ1

Since a sample is a subset of the population the sample mean varies around the mean of the population.

An average of a group of data is called a sample mean. A data set's central tendency, standard deviation, and variance can all be determined using the sample mean. A population average can be determined using the sample mean, among other things.

The statistic known as the sample mean is created by arithmetically averaging the values of a variable in a sample. The sample mean is an estimator of the expected value if the sample is taken from probability distributions with a common expected value.

Learn more about sample on:

https://brainly.com/question/12892403

#SPJ1

Answer:

(-∞, -12), (2, ∞)

or

(-∞, -12) U (2, ∞)

Step-by-step explanation:

|3x + 15| > 21, |3x + 15| < -21

3x + 15 > 21,  3x + 15 < - 21

     -15    -15         -15     -15

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

3x > 6               3x < - 36

÷3  ÷3               ÷3     ÷3

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

x > 2                  x < -12

(-∞, -12), (2, ∞)

or

(-∞, -12) U (2, ∞)

I hope this helps!

-2

The range of the function is a set of all the values of y on the graph

The lowest point of the graph on the y-axis is -2

Therefore, the lowest value of the range of the function shown on the graph is -2

The image of the transformation are (2, -9), (-9, -7), (-7, -4)

The coordinates of the triangle are given as

(-2, -9), (9, -7), (7, -4)

The transformation is given as

Reflection across the y-axis

This is represented as

(x, y) ⇒ (-x, y)

This means that the transformation is (-x, y)

When this rule is applied on the coordinates, we have

(2, -9), (-9, -7), (-7, -4)

Next, we plot the image of the transformation on a coordinate plane

See attachment for the image of the transformation

Read more about transformation at

brainly.com/question/27224272

#SPJ1

Answer:

r = 5%

Step-by-step explanation:

Formula : I=PRT

GIVEN:

I = 125

P = 5000

R = R

T = 6

125= (5000)(R)(0.5)

125=2500R

0.05=R

Interest rate = 0.05(100)

                    = 5%

After simplified the equation we get
a=54
b= 540
c=1800
d=2000

What is simplified expression?

Solving a math problem is the same thing as simplifying an expression. You basically try to write an expression as simply as you can when you simplify it. At the conclusion, there shouldn't be any more multiplying, dividing, adding, or removing to do. Take this formula, for instance: 4 + 6 + 5. To simplify an expression, create an equivalent expression without any terms that are similar. The expression will then be rewritten using the fewest terms possible. When confronted with the phrase 4x+5(3x12),

Sol-

2(3x+10)^3

=2(Bx)^3+3.(3x)^2.10+3.3x.10^2+10^3

=2(27x^3+270x^2+900x+1000)

=54x^3+540x^2+1800x+2000

So,

a=54

b=540

c=1800

d=2000

To know more about simplified expression click-
https://brainly.com/question/723406

#SPJ9

Answer:

I think its 64 please let me know if its wrong

Answer:Explanation:

C. 75

n cthe oomparison of variables, a negative correlation means that as the value of one of the variables increases, the value of the other variable decreases.

Given the variables k and l.

• L=150 when k = 7

• L=? when k=14

The variable k has increased.

Thus, to satisfy the condition for a negative correlation, the variable, l must decrease.

Thus, the only likely possible value of l is 75.

Option C is correct.

If a jet engine close up produces sound at 152 decibels then the decibel reading of a pair of nearby jet engines is 168.62 decibel

A jet engine close up produces sound at 152 decibels

The the relative intensity of one jet engine = [tex]126^{152}[/tex]

The relative intensity of a pair of jet engines = 2× [tex]126^{152}[/tex]

The reading of a pair of nearby jet engine in decibel = 10log(The relative intensity of a pair of jet engines)

Substitute the values in equation

The reading of a pair of nearby jet engine in decibel = 10×log(2× [tex]126^{152}[/tex])

= 168.62 decibel

Hence, if a jet engine close up produces sound at 152 decibels then the decibel reading of a pair of nearby jet engines is 168.62 decibel

Learn more about relative intensity here

brainly.com/question/13783394

#SPJ1