1) Find the minimum and maximum values for the function with the given domain interval.



minimum value = 7; maximum value = 8

minimum value = 0; maximum value = 7

minimum value = 0; maximum value = none

minimum value = none; maximum value = 8

minimum value = 0; maximum value = 8

The equation of parabola is expressed as: y² = -20x

What is the equation of the parabola?

A parabola refers to an equation of a curve, such that a point on the curve is equidistant from a fixed point, and a fixed line.

The fixed point is called the focus of the parabola, and the fixed line is called the directrix of the parabola.

Given, vertex = (0, 0)

Focus = (-5, 0)

We have to find the equation of the parabola.

The equation is of the form y = -ax²

Directrix x = 5.

As every point on parabola is equidistant from focus and directrix, the equation will be

y² + (x + 5)² = (x - 5)²

y² + x² + 10x + 25 = x² - 10x + 25

y² = - 10x - 10x

y² = -20x

Therefore, the equation of parabola is y² = -20x

y coordinate of midpoint of line AB is 6

Solution:

Given that endpoints of line AB is (10, 14) and (4, -2)

To find:  y-coordinate of the midpoint of line AB

The midpoint of line AB is given as:

For a containing [tex]A(x_1, y_1)[/tex] and [tex]B(x_2, y_2)[/tex] the midpoint is given as:

[tex]M(x, y)=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)[/tex]

Here in this question,

[tex]\left(x_{1}, y_{1}\right)=(10,14) \text { and }\left(x_{2}, y_{2}\right)=(4,-2)[/tex]

So midpoint is:

[tex]\begin{aligned}&M(x, y)=\left(\frac{10+4}{2}, \frac{14-2}{2}\right)\\\\&M(x, y)=\left(\frac{14}{2}, \frac{12}{2}\right)\\\\&M(x, y)=(7,6)\end{aligned}[/tex]

Therefore y coordinate of midpoint of line AB is 6

Answer:

The number of child tickets sold was 36

Step-by-step explanation:

The complete question is

At the city museum, child admission is $5.80 and adult admission is $9.30. On Monday, four times as many adult tickets as child tickets were sold, for a total sales of $1548.00. How many child tickets were sold that day?

Let

x ----> the number of child tickets sold

y ----> the number of adult tickets sold

we know that

[tex]5.80x+9.30y=1,548.00[/tex] ---> equation A

[tex]y=4x[/tex] ----> equation B

Solve by substitution

Substitute equation B in equation A    

[tex]5.80x+9.30(4x)=1,548.00[/tex]

solve for x

[tex]5.80x+37.2x=1,548.00[/tex]

[tex]43x=1,548.00[/tex]

[tex]x=36[/tex]

therefore

The number of child tickets sold was 36

Denzel took 1/40 of his earnings to the park but did not spend it on rides or popcorn.

Let's break down the information provided step by step to find the fraction of Denzel's earnings that he took to the park but did not spend on rides or popcorn.

Denzel put 1/2 of his earnings into savings. This means he kept 1/2 as his spending money for the amusement park.

Denzel spent 1/5 of the remaining amount on popcorn. This means he spent 1/5 * 1/2 = 1/10 of his earnings on popcorn.

Denzel also spent 3/4 of the remaining amount on rides. This means he spent 3/4 * 1/2 = 3/8 of his earnings on rides.

To find the fraction of his earnings that he took to the park but did not spend on rides or popcorn, we need to subtract the fractions spent on rides and popcorn from the fraction he took to the park.

Fraction taken to the park but not spent on rides or popcorn = 1/2 - (1/10 + 3/8)

To subtract fractions, we need a common denominator. The least common multiple of 10 and 8 is 40.

Converting the fractions to have a common denominator:

1/2 - (1/10 + 3/8) = 20/40 - (4/40 + 15/40) = 20/40 - 19/40 = 1/40

Therefore, Denzel took 1/40 of his earnings to the park but did not spend it on rides or popcorn.

Question: Denzel earned money after school. He put 1/2 of this month's earnings into savings. He took the rest to spend at the amusement park. He spent 1/5 of this amount on popcorn and 3/4 of it on rides. What fraction of his earnings did he take to the park but not spend on rides or popcorn?

Answer:the greatest number of pieces that can be cut is 2

Step-by-step explanation:

The total length of ribbon available is 3/4 meter. Sunday needs pieces measuring 1/3 meter for their project. This means that each length needed would be exactly 1/3 meter.

The number of pieces measuring 1/3 meter that can be cut from the ribbon would be

(3/4)/(1/3) = 3/4×3/1 = 9/4 = 2.25

Since the length needed is exactly 1/3 meter, the greatest number of pieces that can be cut will be 2

Using the values in the table you can easily use the provided x-values to plug into any equation to get a corresponding f(x) value. When applying this to the 4 functions, we see that only the second answer choice will give the exact same outputs when the inputs are plugged in from the table.

[tex]4x^2+3x-6[/tex]

[tex]x=-2\\f(-2)=4(-2)^2+3(-2)-6\\f(-2)=4[/tex]

[tex]x=0\\f(0)=4(0)^2+3(0)-6\\f(0)=-6[/tex]

[tex]x=4\\f(4)=4(4)^2+3(4)-6\\f(4)=70[/tex]

Answer: $116,000

Step-by-step explanation:

The net operating income will be the operation profit after deducting the expenses from the accrued revenue (reserve exclusive)

The revenue generated are $215,000 + $3,000

= $218,000

Expenses incurred;

$102,000

The net operating income = $218,000 - $102,000

= $116,000

Note that reserves is not used in business operation. Therefore it cannot be regarded either as revenue or expenses.

Answer:

Step-by-step explanation:

k = 6

To find the value of k, rationalize the denominator of 2/√3, and compare it with 2k/3 to find k = √3.

To rationalize the denominator of the fraction 2/√3, we need to make the denominator a rational number. We can do this by multiplying both the numerator and the denominator by √3.

So, after rationalizing, the fraction becomes 2√3/3. According to the problem statement, this is equivalent to 2k/3.

Therefore, we can equate 2k to 2√3:

2k = 2√3

k = √3

So, the value of k is √3.

The complete question is

When the denominator of 2/√3 is rationalized ,it becomes 2k/3​. Find k

Answer:

[tex]\frac{y}{x+y}[/tex]

Step-by-step explanation:

The required answer is the rate at  which Machine A  works when the two machines are combined.

Note: the rate of doing work is express as

[tex]rate=\frac{1}{time taken} \\[/tex]

Hence we can conclude that Machine A working rate is

[tex]machine A=\frac{1}{x} \\[/tex] and machine B working rate is

[tex]machine B=\frac{1}{y} \\[/tex]

When the two machine works together, the effective working rate is

[tex]\frac{1}{x}+\frac{1}{y}\\\frac{xy}{x+y}\\[/tex]

The fraction of the work that Machine B will not have complete because of Machine A help is the total work done by machine A

Hence the fraction of work done by A is expressed as

[tex]\frac{1}{x}*combine working rate[/tex]

[tex]\frac{1}{x}*\frac{xy}{x+y}\\\frac{y}{x+y} \\[/tex]

Hence the fraction of the work that Machine B will not have complete because of Machine A help is the total work done by machine A is [tex]\frac{y}{x+y} \\[/tex]

Answer: her interest in 3 years is $45

Step-by-step explanation:

For simple interest, the principal is not compounded. The interest is only on the original capital. The formula for simple interest is expressed as

I = PRT/100

Where

I represents the interest on the principal

P represents the initial amount

R represents the interest rate.

T represents the time in years.

From the information given

P = $300

R = 5%

T = 3 years

I = 300×5×3)/100

I = 4500/100 = 45

Answer:

  2 1/2 cups

Step-by-step explanation:

5 × (1/2 cup) = 5/2 cup = 2 1/2 cup

__

Or, you can add them up. You know from your study of fractions that two half-cups make 1 cup.

  (1/2 cup) + (1/2 cup) + (1/2 cup) + (1/2 cup) + (1/2 cup)

  = ((1/2 cup) +(1/2 cup)) +((1/2 cup) +(1/2 cup)) +(1/2 cup)

  = (1 cup) + (1 cup) + (1/2 cup)

  = (2 cup) + (1/2 cup)

  = 2 1/2 cup

Answer:

There were 440 Standard version of songs downloaded in Web music store.

Step-by-step explanation:

Given,

Total number of songs downloaded = 1310

Total size of the downloaded songs = 4752 MB

Size of standard version of song = 2.1 MB

Size of high quality version of song = 4.4 MB

Solution,

Let the number of standard version  of song be 'x'.

And also let the number of high quality version of song be 'y'.

Now, total number of songs is the sum of total number of standard version  of song and total number of high quality version of song.

On framing the above sentence in equation form, we get;

[tex]x+y=1310\ \ \ \ \ equation\ 1[/tex]

Now, Total size of the downloaded songs is the sum of total number of standard version of song multiplied with size of standard version  of song and total number of high quality version of song multiplied with size of high quality version of song.

On framing the above sentence in equation form, we get;

[tex]2.1x+4.4y=4752[/tex]

Multiplying with 10 on both side, we get;

[tex]10(2.1x+4.4y)=4752\times10\\\\21x+44y=47520\ \ \ \ equation\ 2[/tex]

Now multiplying equation 1 by 21, we get;

[tex]21(x+y)=1310\times21`\\\\21x+21y=27510\ \ \ \ equation\ 3[/tex]

Now subtract equation 3 from equation 2, we get;

[tex](21x+44y)-(21x+21y)=47520-27510\\\\21x+44y-21x-21y=20010\\\\23y=20010\\\\y=\frac{20010}{23}\\\\y=870[/tex]

On substituting the value of y in equation 1, we get the value of x;

[tex]x+y=1310\\\\x+870=1310\\\\x=1310-870=440[/tex]

Hence There were 440 Standard version of songs downloaded in Web music store.

Answer:

  36 square feet

Step-by-step explanation:

The area of one barrier is the product of the given dimensions:

  (9 ft)(2 ft) = 18 ft²

Two such barriers will have twice the area: 36 ft².

The combined area of the two barriers is calculated by multiplying the length by the height for each barrier to get an area for each one, and then those two areas are added together. Each barrier has an area of 18 square feet, so the total combined area is 36 square feet.

The question is asking for the combined area of two barriers, each being 9 feet long and 2 feet high. In order to find this, we must multiply the length by the height for each barrier, and then add these two areas together. The calculation would look like this:

Area of each barrier = Length x Height = 9 ft x 2 ft = 18 square feet

Now, since there are two barriers:

Combined Area = 2 x Area of each barrier = 2 x 18 square feet = 36 square feet

Therefore, the combined area of the two barriers is 36 square feet.

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Option A

Interest rate paid by Samantha to bank is 8.6 %

Solution:

Given that George has a credit score of 650

Samantha has a credit score of 520

The bank approved George’s loan application at 5.6% interest

To find: Interest rate paid by samantha to the bank

From given information in question,

Interest charged on Samantha’s loan is 3 percentage points higher than the interest rate on George’s loan

Thus we get,

Interest charged on Samantha’s loan = 3 percentage points higher than the interest rate on George’s loan

Interest charged on Samantha’s loan = 3 % higher than the interest rate on George’s loan

Interest charged on Samantha’s loan = 3 % + interest rate on George’s loan

Thus substituting the given George’s loan application at 5.6% interest,

Interest charged on Samantha’s loan = 3 % + 5.6 % = 8.6 %

Thus interest rate paid by samantha to bank is 8.6 %

Final answer:

Option A: 8.6%

Samantha will pay an interest rate of 8.6% on her loan from Westside Bank, which is 3 percentage points higher than George's rate of 5.6% due to her lower credit score.

Explanation:

The question involves calculating the interest rate Samantha will pay to the bank for a personal loan.

Given that George has a credit score of 650 and was approved for a loan at 5.6% interest, and Samantha has a lower credit score of 520, her interest rate will be 3 percentage points higher than George's.

To find Samantha's interest rate, we simply add 3 percentage points to George's rate of 5.6%.

Samantha's interest rate = George's interest rate + 3%
Samantha's interest rate = 5.6% + 3%
Samantha's interest rate = 8.6%

Answer:

Currency= $291 and check= $581.25.

Step-by-step explanation:

Given: Cash received= $75

          Total deposit= $872.25

Lets assume currency deposited be dollar "x"

∴  As given check deposited will be "2x"  

Now, calculating amount of currency deposited.

We know that, [tex]currency\ deposit + check\ deposit= \$872.25[/tex]

∴ [tex]x+2x= \$872.25[/tex]

⇒[tex]3x=\$872.25[/tex]

Cross multiplying

∴[tex]x= \$290.75 \approx \$291 \textrm{ as Kenneth John have not deposited any coins}[/tex]

∴  Amount of currency deposited is $291.

Next, computing to get amount deposited through check.

As we know check deposited is double of currency.

Check deposited= [tex]2\times \$291= \$ 582[/tex]

∵ No coins were deposited and there is total deposit is $872.25.

∴ We will consider amount deposited through check is $581.25.

Kenneth John deposited $290.75 in currency and $581.50 in checks.

To determine the amounts of currency and checks deposited by Kenneth John, we will define the variables for clarity. Let C represent the amount in currency deposited and CH represent the amount in checks deposited.

Given:

The total deposit after adding checks and currency is $872.25.

The checks deposited totaled twice the currency deposited (CH = 2C).

We can set up the following equation based on the given information:

[tex]C + CH = 872.25[/tex]

Since CH = 2C, we substitute CH:

[tex]C + 2C = 872.25[/tex]

This simplifies to:

[tex]3C = 872.25[/tex]

Solving for C, we divide both sides of the equation by 3:

[tex]C = \[\frac{872.25}{3} = 290.75[/tex]

Kenneth deposited $290.75 in currency.

Then, we calculate the amount in checks:

[tex]CH = 2 \times 290.75 = 581.50[/tex]

Answer:

After 4 miles driven by cab the amount would be same in both cities.

Step-by-step explanation:

Let the number of miles be 'x'.

Given:

In NYC

Flat fee of cab = $4

Per mile charge = $1.25

Total cab charges is equal to sum of Flat fee of cab and per mile charge multiplied by number of miles.

Framing in equation form we get;

Total cab charges in NYC = [tex]4+1.25x[/tex]

In Chicago

Flat fee of cab = $6

Per mile charge = $0.75

Total cab charges is equal to sum of Flat fee of cab and per mile charge multiplied by number of miles.

Framing in equation form we get;

Total cab charges in Chicago = [tex]6+0.75x[/tex]

Now we need to find number of miles driven so that the amount could same in both cities.

Total cab charges in NYC = Total cab charges in Chicago

[tex]4+1.25x=6+0.75x[/tex]

Combining like terms we get;

[tex]1.25x-0.75x=6-4\\\\0.5x=2[/tex]

using Division Property we will divide both side by 0.5 we get;

[tex]\frac{0.5x}{0.5} =\frac{2}{0.5} \\\\x=4[/tex]

Hence After 4 miles driven by cab the amount would be same in both cities.

Answer:

24 times

Step-by-step explanation:

Given:

Number of cups required = 6 cups

Measuring cup capacity = [tex]\frac{1}{4}=0.25[/tex] of a cup.

Now, each time the measuring cup fills 0.25 of a cup.

So, we use unitary method to find the number of times the measuring cup has to be used to get a total of 6 cups.

∵ 0.25 cups = 1 time the measuring cup being used.

∴ 1 cup = [tex]\frac{1}{0.25}=4[/tex] times the measuring cup being used.

So, 6 cups = [tex]4\times 6=24[/tex] times the measuring cup being used.

Hence, the number of times the measuring cup has to be used to get 6 cups of flour is 24 times.

A. x²-14x+49; is a polynomial

Step-by-step explanation:

(x-7)² can be written as (x-7)(x-7)

Expanding the expression

x(x-7)-7(x-7)

x²-7x-7x+49

x²-14x+49 ⇒⇒A quadratic function, which is a polynomial of degree 2

This function demonstrates the closer property of multiplication in that the change in order of multiplication does not change the product. This is called commutative property.

(x-7)(x-7)

-7(x-7)+x(x-7)

-7x+49+x²-7x

x²-14x+49

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Keywords : product, closure property of multiplication,

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Answer:

A is correct

Step-by-step explanation:

I took the test and got it right

Answer:

The Amount draw from the account after 10 years is $109,555 .

Step-by-step explanation:

Given as :

The principal deposited in account = p = $50,000

The rate of interest = 8% semiannually

The time period for the amount will be in account = t = 10 years

Let The Amount draw from the account after 10 years = $A

Now, From Compound Interest method

Amount = principal × [tex](1+\dfrac{\texrm rate}{2\times 100})^{\textrm 2\times time}[/tex]

A = p × [tex](1+\dfrac{\texrm r}{2\times 100})^{\textrm 2\times t}[/tex]

Or, A = $50,000 × [tex](1+\dfrac{\texrm 8}{2\times 100})^{\textrm 2\times 10}[/tex]

Or, A = $50,000 × [tex](1.04)^{20}[/tex]

Or, A = $50,000 × 2.1911

Or, A = $109,555

So, The Amount draw from the account after 10 years = A = $109,555

Hence,The Amount draw from the account after 10 years is $109,555 . Answer

I think 68 or56

It can't be 34 that would be less, 146 would be over quarter

Good evening ,

Answer:

Step-by-step explanation:

measure arc AB = 2×(m∠AXB) = 2×(34) = 68°.

:)

Answer:

The length of each side is 17 in, 24 in, 24 in.

Step-by-step explanation:

Given,

Perimeter of the triangle = [tex]25\ in[/tex]

Length of 1st side = [tex]2w+1[/tex]

Length of 2nd side = [tex]3w[/tex]

Length of 3rd side = [tex]3w[/tex]

The perimeter of a triangle is equal to the sum of the length of all the three sides of the triangle.

Perimeter of the triangle = Length of 1st side + Length of 2nd side + Length of 3rd side

Now substituting the given values, we get;

[tex]2w+1+3w+3w=25\\\\8w+1=25\\\\8w=25-1\\\\8w=24\\\\w=\frac{24}{8}=3[/tex]

Now we have the value of w so we can calculate the length of each side.

Length of 1st side = [tex]2w+1=2\times8+1=16+1=17\ in[/tex]

Length of 2nd side = [tex]3w=3\times8=24\ in[/tex]

Length of 3rd side = [tex]3w=3\times8=24\ in[/tex]

Thus the length of each side is 17 in, 24 in, 24 in.

Answer: the amount of money invested at the 5% rate is $15000

Step-by-step explanation:

Let x represent the amount of money invested at the rate of 5%.

Let y represent the amount of money invested at the rate of 8%.

Mrs. Mary Moolah invested $20,000 in two different types of bonds. This means that

x + y = 20000

The formula for simple interest is expressed as

I = PRT/100

Where

P represents the principal

R represents interest rate

T represents time

Considering the investment at the rate of 5%,

P = x

R = 5

T = 1

I = (x × 5 × 1)/100 = 0.05x

Considering the investment at the rate of 8%,

P = y

R = 8

T = 1

I = (y × 8 × 1)/100 = 0.08y

If Mrs. Moolah's combined profit from both investments was $1,150, it means that

0.05x + 0.08y = 1150 - - - - - -1

Substituting x = 20000 - y into equation 1, it becomes

0.05(20000 - y) + 0.08y = 1150

1000 - 0.05y + 0.08y = 1150

- 0.05y + 0.08y = 1150 - 1000

0.03y = 150

y = 150/0.03 = 5000

Substituting y = 5000 into x = 20000 - y, it becomes

x = 20000 - 5000

x = 15000

Answer:

the answer is 1:12

Step-by-step explanation:

hope it helps!

Answer:

Correct answer: False

Step-by-step explanation:

coeff c= 3/2 = 1,5        coeff c₁ = 18/8 = 2,25

c ≠ c₁

God is with you!!!

Answer: False

Step-by-step explanation: When we are asked to determine whether two ratios form a proportion, what we are really being asked to do is to determine whether the ratios are equal because if the ratios are equal, then we know they form a proportion.

So in this problem, we need to determine whether 3/2 = 18/8. The easiest way to determine whether 3/2 = 18/8 is to use cross products. If the cross products are equal, then the ratios are equal.

The cross products for these two ratios are 3 x 8 and 2 x 18.

Since 3 x 8 is 24 and 2 x 18 is 36, we can easily see that 24 ≠ 36 so the cross products are not equal which means that the ratios are not equal and since the ratios are not equal, we know that they do not form a proportion.

So the answer is false. 3/2 and 18/8 do not form a proportion.

a) The recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s is [tex]\(a_n = 2a_{n-1} + a_{n-2}\).[/tex]

b) The initial conditions for the recurrence relation are [tex]\(a_1 = 3\) and \(a_2 = 9\).[/tex]

c) There are 21 ternary strings of length six that do not contain two consecutive 0s.

a) To derive the recurrence relation, consider the possibilities for the last digit in the string. If the last digit is 1 or 2, it doesn't affect the constraint of avoiding consecutive 0s. Hence, for strings of length n that end in 1 or 2, there are[tex]\(a_{n-1}\)[/tex]possibilities. However, if the last digit is 0, the previous digit cannot be 0 to satisfy the constraint. Therefore, for strings of length n that end in 0, there are \(a_{n-2}\) possibilities. This results in the recurrence relation[tex]\(a_n = 2a_{n-1} + a_{n-2}\).[/tex]

b) The initial conditions are established by considering strings of length 1 and 2. For strings of length 1, there are three possibilities (0, 1, or 2). For strings of length 2, there are nine possibilities (00, 01, 02, 10, 11, 12, 20, 21, 22), but among these, 00 is excluded to avoid consecutive 0s, leaving a total of nine valid strings. Therefore, the initial conditions are[tex]\(a_1 = 3\) and \(a_2 = 9\).[/tex]

c) To find the number of ternary strings of length six that do not contain two consecutive 0s, utilize the recurrence relation. Starting from the initial conditions, compute[tex]\(a_6 = 2a_5 + a_4\)[/tex] using the relation, which results in [tex]\(a_6 = 21\).[/tex]

Thus, there are 21 ternary strings of length six that satisfy the condition of not having two consecutive 0s.

"In summary, the recurrence relation [tex]\(a_n = 2a_{n-1} + a_{n-2}\)[/tex]governs the number of ternary strings of length n without consecutive 0s, with initial conditions[tex]\(a_1 = 3\) and \(a_2 = 9\)[/tex]. Computing[tex]\(a_6\)[/tex]using the relation yields 21 valid ternary strings of length six that do not contain two consecutive 0s."

The recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s is a_n = 2a_n-1 + 2a_n-2. The initial conditions are a_1 = 3 and a_2 = 8. Using these, we calculate that there are 448 such ternary strings of length six.

Ternary Strings without Consecutive 0s

Let's define a ternary string as a string composed of the digits 0, 1, and 2. We need to find a recurrence relation for the number of such strings of length n that do not contain two consecutive 0s.

Part (a)

Let a_n represent the number of ternary strings of length n that do not contain consecutive 0s. Consider the possibilities for the first digit of the string:

Thus, we have the recurrence relation: a_n = 2a_{n-1} + 2a_{n-2}

Part (b)

The initial conditions can be determined as follows:

Part (c)

Using the recurrence relation and initial conditions:

Therefore, the number of ternary strings of length six that do not contain two consecutive 0s is 448.

Answer:

0.64

Step-by-step explanation:

Given: Craig has a box of chocolate with 5 rows in it.

           Each row has 20 chocolate.

           Craig and his friends eat 64 chocolate.

As given, we understand that there is box of chocolate with 5 rows and each row have 20 chocolate, therefore we can find total number of chocolate.

Total number of chocolate=[tex]5\ rows \times 20\ chocolates = 100\ chocolates[/tex]

∴ Total number of chocolates in box= 100.

Now, we know craig and his friends eat 64 chocolate.

∴ To find decimal of number chocolate eaten out of 100 chocolate, we need to  put numbers in fraction first then convert it in decimal.

Number of chocolate ate by craig and his friends is [tex]\frac{64}{100} = 0.64[/tex]

∴ Craig and his friends eat 0.64 chocolates.

Answer:

cos(3x) --> cos³(x) - 3sin²(x)cos(x)

Step-by-step explanation:

The text in pink are the trig identities I used to convert cos(2x) and sin(2x) into their other equivalent forms.

This question is pretty much asking if you know how to use your trig identities if i understand it correctly.

Final answer:

To rewrite cos 3x using only sin x and cos x, we can use the trigonometric identity: cos 3x = 4(cos x)^3 - 3(cos x). This identity allows us to express cos 3x in terms of cos x. However, if we want to rewrite it using only sin x and cos x, we can use the Pythagorean identity: (cos x)^2 = 1 - (sin x)^2.

Explanation:

To rewrite cos 3x using only sin x and cos x, we can use the trigonometric identity: cos 3x = 4(cos x)^3 - 3(cos x). This identity allows us to express cos 3x in terms of cos x. However, if we want to rewrite it using only sin x and cos x, we can use the Pythagorean identity: (cos x)^2 = 1 - (sin x)^2. So, we can substitute this identity into the previous equation to get: cos 3x = 4(1 - (sin x)^2)^3 - 3(1 - (sin x)^2).

Answer:

Part 12) [tex]Center\ (2,-3),r=2\ units, (x-2)^2+(y+3)^2=4[/tex]

Part 13) [tex]m\angle ABC=47^o[/tex]

Step-by-step explanation:

Part 12) we know that

The equation of a circle in center-radius form is equal to

[tex](x-h)^2+(y-k)^2=r^2[/tex]

where

(h,k) is the center of the circle

r is the radius of the circle

In this problem

Looking at the graph

The center is the [tex]point\ (2,-3)[/tex]

The radius is [tex]r=2\ units[/tex]

substitute in the expression above

[tex](x-2)^2+(y+3)^2=2^2[/tex]

[tex](x-2)^2+(y+3)^2=4[/tex]

Part 13) we know that

The measure of the external angle is the semi-difference of the arcs it covers.

so

[tex]m\angle ABC=\frac{1}{2}[arc\ DE-arc\ AC][/tex]

we have

[tex]arc\ DE=142^o[/tex]

[tex]arc\ AC=48^o[/tex]

[tex]m\angle ABC=\frac{1}{2}[142^o-48^o][/tex]

[tex]m\angle ABC=47^o[/tex]

Answer:

Answer C: Becky is 27; John is 31

Step-by-step explanation:

1. John is 4 years older than Becky--27(Becky's age)+ 4=31(John's age)

2. Sum of their ages is 58--27(Becky's age)+31(John's age)=58

So, the correct answer is Answer C.

Answer:

C. Becky is 27; John is 31