Please help with problem 11

Answer:

[tex]\frac{\sqrt{3}}{2}[/tex]

Step-by-step explanation:

[tex]\sin 2x=2\sin x \cos x \\ \\ \sin 120^{\circ}=2\sin 60^{\circ} \cos 60^{\circ} \\ \\ =2(\sqrt{3}/2)(1/2) \\ \\ =\frac{\sqrt{3}}{2}[/tex]

The largest prime divisor of the expression [tex]15^{6} - 7^{6}[/tex]  is 379.

Here, we are given an expression: 15^6 - 746

This can be written as-

[tex][15^{3}] ^{2} - [7^{3}] ^{2}[/tex]

now, using the property [tex]a^{2} -b^{2}[/tex] = (a + b)(a - b), we get-

[tex][15^{3}] ^{2} - [7^{3}] ^{2}[/tex]

= [tex](15^{3} +7^{3}) (15^{3} -7^{3})[/tex]

= (3375 + 343) (3375 - 343)

= 3718 × 3032

Now, here the factors of 3718 will be

3718 = 2 x 11 x 13 x 13

Here, the largest prime factor we find is 13

Similarly, the factors of 3032 will be

3302 = 2 * 2 * 2 * 379

Here, the largest prime factor we find is 379

Thus, the largest prime divisor (factor) of the given expression is 379.

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Answer:

X = 40

Step-by-step explanation:

X/5 = 8

Multiply each side by 5 to solve for x

X/5 * 5 = 8*5

X = 40

The amount invested in the account with 2.3% interest is $1650 whereas $150 was invested in the account whose interest rate is 3.5%

What is Hattie's annual earnings on the investment of $1,800?

The earnings that Hattie would receive every year on the investment is $1,800, the investment multiplied by the expected rate of return of 2.4%

annual earnings=$1,800*2.4%

annual earnings=$43.20

Let the amount invested at 2.3% to be x and the amount invested at 3.5% would be (1800-x), hence, the interest earnings are shown thus:

interest on 2.3% investment=x*2.3%=0.023x

interest on 3.5% investment=(1800-x)*3.5%

interest on 3.5% investment=63-0.035x

total interest=0.023x+63-0.035x

total interest=63-0.012x

43.20=63-0.012x

0.012x=63-43.20

0.012x=19.80

x=19.80/0.012

x=amount invested at 2.3%=$1650

amount invested at 3.5%=$1800-$1650

amount invested at 3.5%=$150

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The result obtained for the logarithms expression 2 log 5+ log 40 is 3.

The logarithm of a number is the exponent by which a fixed value recognized as the base must be raised in attempt to calculate that number. Because 1000 is ten to the third power, this same log of 1000 to base ten is three.

Some rules of the logarithmic functions are-

now, the given expression to be evaluated is;

= 2 log 5+ log 40

Apply the Power Rule in the first term;

= log 5²+ log 40

= log 25+ log 40

Now, using the Product Rule, the expression becomes.

= log (25×40)

= log (1000)

If the base is taken as 10, then the value of the log (1000) = 3.

Therefore, the obtained value of the value of the logarithms expression 2 log 5+ log 40 is 3.

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Transformation of y= bx into y=7(x-5) will take place by by horizontal translation by 5 units to the right, vertical dilation by a scale factor of 7 .

We have given two function y=b x   and  y=7 (x-5) which means that first y changes into (x -5 ). i.e., horizontal translation by 5 units to the right. Then it changes into 7 (x - 5). i.e., vertical dilation by a scale factor of 7.

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The  total cost to the customer is $8.86.

Taxes are mandatory contributions levied on individuals or corporations by a government entity—whether local, regional, or national.

Given:

Logo shirts at a cost of $10 per shirt.

Every month that a shirt doesn’t sell, the store reduces the selling price by 25%.

There is a 5% sales tax on the selling price.

According to given question we have

Cost of shirt = $10

Cost of the shirt after 100% increase

=100% of 10

= $10

Price of shirt after 100% increase

= $10 + $10

= $20

if shirt does not sell, price reduces by 25%. It means remaining value is 75% after every reduction.

So after 3 reductions,  remaining value is 75% after every reduction

So, final price will be  

$20 * 0.75 * 0.75 * 0.75

= $8.4375

There is a sales tax of 5% which customer will pay on the final price.

So, price for customer will be

$8.4375 + (5% of 8.4375)

= 8.4375 + 0.421875

= $8.859375≈ $8.86

Therefore, the  total cost to the customer is $8.86.

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The required simplification of rational expression is 3(x + 5) / (x + 3).

What is a rational number?

Any real number that can be represented in the form of p/q, where q is not equal to 0 is called a rational number. The decimal expansion of rational numbers is terminating and repeating.

Simplifying the rational expression:  x² + 3x - 10 / x² + 4x - 12 . 3x + 18 / x + 3

Given a rational expression,

x² + 3x - 10 / x² + 4x - 12 . 3x + 18 / x + 3

On simplifying, we get,

(x² + 3x - 10).(3x + 18) / (x² + 4x - 12).(x + 3)

For numerator, (x² + 3x - 10).(3x + 18), we have,

= 3(x + 5)(x - 2)(x + 6)

For denominator,  (x² + 4x - 12).(x + 3)

= (x + 6)(x - 2)(x + 3)

Now, (x² + 3x - 10).(3x + 18) / (x² + 4x - 12).(x + 3) = 3(x + 5)(x - 2)(x + 6) / (x + 6)(x - 2)(x + 3)

= 3(x + 5) / (x + 3)

The restrictionto the given rational expression is that the denominator,  x + 3 should not be 0.

Hence, the required simplification of rational expression is 3(x + 5) / (x + 3).

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The logarithmic form of the given equation is log₃(1) = 0.

A logarithm is the degree to which a number must be raised in order to produce another number. For instance, the logarithm of 100 in base ten is 2, since ten multiplied by two equals 100: log 100 = 2, since 10² = 100.

The given equation is 1 = 3⁰

To convert this exponential form into logarithmic form, we are required.

Consider the given exponential form:1 = 3⁰

Taking logarithm on both sides with the base as 3, we get,⇒ log₃(1) = log₃(3⁰)

Now using the logarithmic rule logₐ(aⁿ) = n × logₐ(a) in the RHS of the above equation, we get

⇒ log₃(1) = 0 × log₃(3)

⇒ log₃(1) = 0 (Because 0 multiplied by any number equals zero,)

Therefore, the value of 1 = 3⁰ is given by log₃(1) = 0.

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The value of radical expression 5√0.00032 x⁵ is (6.4e-5)x.

By considering the given radical expression

5√0.00032 x⁵

=(0.00032 x⁵ )^1/5

=(6.4e-5)x

A y-intercept, a horizontal asymptote, a domain (the range of x values at which the function exists), and a constant growth factor, b, are the four main characteristics of exponential functions.

An expression with a square root is referred to as a radical expression. Radicand: A value or phrase contained within the radical symbol. Equation with radical expressions and variables as radicands is referred to as a radical equation.

The 'V'-shaped portion of the radical symbol will have a superscript number when it is used to represent any root other than a square root. For instance, the expression 3(8) denotes the cube root of 8.

The radical symbol, which resembles a check mark, the index, a small number written outside the radical symbol, and the radicand, a quantity written beneath the horizontal bar of the radical symbol, are the three main components of the radical expression a^b.

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√2 is 1 approximately less than √6. (option A).

The square root of a number can be described as the factor of a number that when multiplied by itself would yield the number. For example, the square root of 16 is 4. This is because 4 multiplied by 4 is equal to 16. The sign used to represent square root is √.

The square root of 2 is 1.414

Th square root of 6 is 2.449

The difference in the square roots of 6 and 2 is : 2.449 - 1.414 = 1.04

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Answer:

y=x+9

Step-by-step explanation:

The general equation for a straight line is y=mx+c

1. first we need to find the gradient (m) to do this we do

4-5/-5-(-4)

which gives the answer of 1. This means or gradient (m) is 1. we can substitute this in our general formula y=mx+c which gives

y=1x+c this can also be written as y=x+c

2. we need to find the value of c. to do this we can sub in any co-ordinates from the question in this case we will be using (-4,5)

-4 is the x co-ordinate

5 is the y co-ordinate

5= -4+c

to get c we add 4 on each side we get

9=c

we can sub this into the equation we produced with the gradient giving us

y=x+9

3. the final answer is y=x+9

Answer:

Step-by-step explanation:

The factors of given equation are -3/4 and -2.

Rewriting the equation -

4x² + 11x + 6 = 0

The factors for the given expression will be calculated based on two factors -

1. The sum of factors should be equal to sum of numeral value of b (11 in this case)

2. The product of factors should be equal to product of numeral values of a and c (24 in this case)

As per the above rules, the appropriate factors should be 8 and 3.

Rewriting the given equation with possible factors -

4x² + 8x + 3x + 6 = 0

Taking common values

4x(x + 2) +3(x + 2) = 0

(x + 2) (4x + 3) = 0

x = -3/4, -2

Thus, the factors of given expression are -3/4 and -2.

To check the correctness of factors, we will keep each value of x in equation.

When x = -3/4

4x² + 11x + 6 = 0

4(-3/4) + 11(-3/4) + 6 = 0

9/4 -33/4 + 6 = 0

9/4 = 2.25 and 33/4 = 8.25

2.25 - 8.25 + 6 = 0

-6 + 6 = 0

0 = 0

When x = -2

4x² + 11x + 6 = 0

4(-2)² + 11(-2) + 6 = 0

16 - 22 + 6 = 0

22 - 22 = 0

0 = 0

Since both values of x give 0 on both sides of equation, the factors of equation are correct.

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the slope would be -8/3. you go 8 down and 3 to the right.

The transformation rule f(x) → - 4 · f(x) is the consequence of combining a reflection about the x-axis and a vertical stretch.

According to the statement, we must conclude what kind of rigid transformations are behind the transformation rule f(x) → - 4 · f(x). Rigid transformations are transformations applied on a function such that its form is not altered. The most common transformations are listed below:

Then, we find that the transformation rule f(x) → - 4 · f(x) is the consequence of combining a reflection about the x-axis and a vertical stretch.

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The amount of nuts used in trail mix that had 10 cups of raisins will be 7.5 nuts.

From the information, Ken makes trail mix. the ratio of nuts to raisins is 3 to 4.

The amount of nuts used in trail mix that had 10 cups of raisins will be:

= 10 × 3/4

= 30/4

= 7.5

Therefore, the amount of nuts used in trail mix that had 10 cups of raisins will be 7.5 nuts.

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The cost in dollars and cents for each dog pillow is $17.95

What is the total cost of  3 giant dog pillows?

The total cost of the  3 giant dog pillows is determined as the amount Amy gave the cashier minus the change the cashier handed to Amy, in essence, the cost in dollars and cents for each dog pillow can now be computed the total cost of the  3 giant dog pillows divided by the number of dog pillows purchased

total cost for  3 giant dog pillows=total cost of amount given to cashier change received

total cost of amount given to cashier=$60

change received=$6.15

total cost for  3 giant dog pillows=$60-$6.15

total cost for  3 giant dog pillows=$53.85

cost per dog pillow=total cost for  3 giant dog pillows/number of dog pillows purchased

cost per dog pillow=$53.85/3

cost per dog pillow=$17.95

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Step-by-step explanation:

the answer to what question ?

is the request to find (solve for) x ?

if so, we have

(-3(2x - 6) + 5x)/6 = 7 + (2(-4x + 1) - 6)/3

now, I usually remove the fractions first by multiplying both sides with both denominators :

1. multiply by 6

-3(2x - 6) + 5x = 42 + 6(2(-4x + 1) - 6)/3

2. I can skip the general multiplication by 3, because the denominator 3 goes away in interaction with the new factor 6

-3(2x - 6) + 5x = 42 + 2(2(-4x + 1) - 6)

now, we do all the multiplications and see what we get :

-6x + 18 + 5x = 42 + 2(-8x + 2 - 6) =

= 42 - 16x + 4 - 12

-x + 18 = -16x + 34

15x = 16

x = 16/15

Answer:

Step-by-step explanation:

2x

The equation of the line in point-slope form is:

y - 8 = 3(x + 6) or y - 2 = 3(x + 8).

The equation, y - b = m(x - a), represents the equation of a line in point-slope form, where:

Given two points, (-8, 2) and (-6, 8), find the slope (m) of the line that passes through the two points:

Slope (m) = change in y / change in x = (8 - 2)/(-6 -(-8))

Slope (m) = 6/2

Slope (m) = 3

Substitute m = 3 and (-6, 8) into y - b = m(x - a):

y - 8 = 3(x - (-6))

y - 8 = 3(x + 6)

Or substitute m = 3 and (-8, 2) into y - b = m(x - a):

y - 2 = 3(x - (-8))

y - 2 = 3(x + 8)

Therefore, the equation of the line in point-slope form is:

y - 8 = 3(x + 6) or y - 2 = 3(x + 8).

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Answer:

25%

Step-by-step explanation:

Based on the given conditions, formulate

Calculate the sum or difference:

 Multiply a number to both the numerator and the denominator

: Write as a single fraction: 

Calculate the product or quotient: 

Rewrite a fraction with denominator equals 100 to a percentage: 

Answer:

B

Step-by-step explanation:

Solve the exponent first and thats the only remotely close answer

If Annie can mow 1/6 of an acre in 50 minutes, then in an hour, she can mow 1/5 of an acre.

Given the data in the question;

Let x represent the number of acres that Annie can mow in hour.

Also convert 1 hour to minutes

1 hour = (1 × 60)min = 60min

Annie can mow 1/6 of an acre in 50 minutes

Annie can mow x of an acre in 60 minutes

Hence;

1/6 × 60 = x × 50

60/6 = 50x

50x = 60/6

50x = 10

x = 10/50

x = 1/5

If Annie can mow 1/6 of an acre in 50 minutes, then in an hour, she can mow 1/5 of an acre.

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After the solution, the value of x for the equation √x+2 = x is 2.

So,

Given the equation: √x+2 = x

Then,

To get a quadratic, square it:

Quadratic formula:

However, when we square in our work, we must ensure that we haven't accidentally introduced extra roots.

In this case, -1 is not a solution to the original equation, whereas 2 is.

Therefore, after the solution, the value of x for the equation √x+2 = x is 2.

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