What is the domain of the piecewise function?
A. x<0, x>4
B. -∞ 4
C. all real numbers

[tex]\text{L.H.S}\\\\=1+ \cos^4 x - \sin^4 x\\\\=1+ (\cos^2 x)^2 - (\sin^2 x)^2\\\\=1+(\cos^2 x + \sin^2 x)(\cos^2 x- \sin^2 x)\\\\=1+1\cdot(\cos 2x)\\\\=1+\cos 2x\\\\=2\cos^2 x\\\\=\text{R.H.S}\\\\\text{Proved.}[/tex]

Answer:

a) 1/2, b) = - sqrt(3) / 2

Step-by-step explanation:

a) Let A = 120 degree, B = 180 degree

Using cosine sum is cos(120+180) = cos(120)*cos(180) - sin(120)* sin(180)

which will give you 1/2

b) Let A = 360, B = 60 degree

Using sine difference: sin(360-60) = sin(360) cos(60) - cos(360) sin (60)

gives you -sqrt(3)/2

Cos (300°) using the cosine sum identity is 1/2.

sin (300°) using the sine difference identity is √3/2.

Trigonometric ratios are the ratios which is described for two sides of a right angled triangle.

The main three trigonometric ratios are sine, cosine and tangent functions.

Part A :

The cosine sum identity is,

cos (A + B) = cos (A) cos (B) - sin (A) sin (B)

We can write cos (300°) as,

cos (300°) = cos (180° + 120°)

                  = cos (180°) cos (120°) - sin (180°) sin (120°)

sin (180°) = 0 and cos (180°) = -1

So,  cos (300°) = -cos (120°) = -(-1/2) = 1/2

Part B :

Sine difference identity is,

Sin (A - B) = sin (A) cos (B) - cos(A) sin (B)

We can write sin (300°) as,

sin (300°) = sin (360° - 60°)

                = sin (360°) cos(60°) - cos(360°) sin (60°)

                = - sin (60°)

                = -√3 / 2

Hence the value of Cos (300°) using the cosine sum identity is 1/2 and the sin (300°) using the sine difference identity is √3/2.

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On the basis of least variation from the expected value is:

   Sample A ⇒ Sample D ⇒ Sample B ⇒ Sample C

Variance is the expectation of the squared deviation of a random variable from its population mean or sample mean.

Here, on the basis of least variation,

we had arranged the given sample as per their expected value.

Thus, On the basis of least variation from the expected value is:

   Sample A ⇒ Sample D ⇒ Sample B ⇒ Sample C

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Step-by-step explanation:

quarter circle means that you just find the are and then divide it by 4.

hence,

(3.14 × 8²) ÷ 4 = answer

the answer is 50.24 cm²

hope this helps.

Answer: 6.28

Step-by-step explanation: In order to get your answer, the equation that you need to do is 8 X 3.14 / 4 to get your answer.

The numbers that can be pulled based on the probability of the calls will be 90, 110, and 50.

From the information given, the bag has 9 blue cubes, 11 red cubes, and 5 green cubes and when one draws a cube and replace it in the bag 250 times, the number of blue balls that can be gotten will be:

= 9/(9+ 11 + 5) × 250

= 9/25 × 250

= 90

The number of red balls that can be gotten will be:

= 11/25 × 250

= 110

The number of green balls that can be gotten will be:

= 5/25 × 250

= 50

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The average value of f over the region D is 243/4

To answer the question, we need to know what the average value of a function is

The average value of a function f(x) over an interval [a,b] is given by

[tex]\frac{1}{b - a} \int\limits^b_a {f(x)} \, dx[/tex]

Now, given that we require the average value of f(x,y) = 3xy over the region D where D is the triangle with vertices (0, 0), (1, 0), and (1, 9).

x is intergrated from x = 0 to 1 and the interval is [0,1] and y is integrated from y = 0 to y = 9

So, [tex]\frac{1}{b - a} \int\limits^b_a {f(x,y)} \, dA = \frac{1}{1 - 0} \int\limits^1_0 \int\limits^9_0 {3xy} \, dxdy \\= \frac{3}{1} \int\limits^1_0 {x} \,dx\int\limits^9_0 {y} \,dy\\ = \frac{3}{1} [\frac{x^{2} }{2} ]^{1}_{0}[\frac{y^{2} }{2} ]^{9}_{0} \\= 3[\frac{1^{2} }{2} - \frac{0^{2}}{2} ] [\frac{9^{2} }{2} - \frac{0^{2}}{2} ] \\= 3[\frac{1}{2} - 0 ][\frac{81}{2} - 0 ]\\= \frac{81}{2} X3 X \frac{1}{2} \\= \frac{243}{4}[/tex]

So, the average value of f over the region D is 243/4

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Step-by-step explanation:

The wording is slightly confusing so I will just solve the equation.

2 (y + 6) = 4y

We expand the bracket

2y + 12 = 4y

We subtract 2y from both sides

12 = 2y

We divide both sides by 2

6 = y

Answer:

320 cents per pound

Step-by-step explanation:

so we do 9.60 divided by 3, which is 3.20. So, 3.20 for each pound. Now we need in cents. 3 dollars is 300 cents, plus another 20, so 320 cents. I hope this helped.

By weighted average, we need 75 liters of 24 % iodine solution and 25 liters of 40 % iodine solution to obtain 100 liters of 28 % iodine solution.

Physically speaking, concentration is equal to the amount of solute divided by the volume of solution. We have two solutions with same solute and different concentration and can find the right proportion between the 24 % solution and the 40 % solution by concept of weighted average:

x · 24 + (1 - x) · 40 = 28

40 - 16 · x = 28

16 · x = 40 - 28

16 · x = 12

x = 3/4

By weighted average, we need 75 liters of 24 % iodine solution and 25 liters of 40 % iodine solution to obtain 100 liters of 28 % iodine solution.

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Step-by-step explanation:

the missing side is given by sqrt(16² - 8²)

hence,

the number that belongs in the green box is 192

Answer:

Architects use geometry to study and divide space as well as draft detailed building plans. Builders and engineers rely on geometric principles to create structures safely. Designers apply geometry (along with color and scale) to make the aesthetically pleasing spaces inside. Applying geometry in design is unavoidable.

Answer: 2,111 soccer balls

Step-by-step explanation:

         We will divide the total amount of money by the price per ball to find how many balls the league can buy.

        $6,333 / $3 = 2,111

        The league can buy 2,111 soccer balls.

Answer:

B) [tex]8\sqrt[4]{4}[/tex]

Step-by-step explanation:

[tex](2^{8} *2^{6} )^\frac{1}{4} = (2^{8+6} )^\frac{1}{4} =(2^{14} )^\frac{1}{4} =2^{\frac{7}{2} }=\sqrt{2^{7} } =2^{3} \sqrt{2} =8\sqrt{2} = 8\sqrt[4]{2^{2} } = 8\sqrt[4]{4}[/tex]

The solution to the absolute value inequality is (-3,-2)

On the absolute value inequality, we have:

Interval = -3 to 2

The intervals are represented with open circles.

This means that -3 and 2 are exclusive of the values of the inequality.

So, we have:

-3 < x < 2

As an interval notation, we have:

(-3,-2)

Hence, the solution to the absolute value inequality is (-3,-2)

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[tex]f(x) = 3(x+5)\\\\f(a+2) = 3(a+2 +5) \\\\~~~~~~~~~~~~=3(a+7)\\\\~~~~~~~~~~~~=3a+21[/tex]

Answer:

x = 11

Step-by-step explanation:

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

3(2x-3) = 6x - 9

3/(6x-9) = 2/(3x+5)

3(3x+5) = 2(6x-9)

9x + 15 = 12x - 18

12x - 9x = 15 + 18

3x = 33

x = 11

Answer:

[tex]\mathrm x= \dfrac{87}{10} \quad or \quad 8.7[/tex]

Explanation:

[tex]\longrightarrow \sf \dfrac{3}{4} (2x-3)=\:\dfrac{2}{3}\:x+5[/tex]

[tex]\longrightarrow \sf \dfrac{3}{2} x-\dfrac{9}{4} =\:\dfrac{2}{3}\:x+5[/tex]

[tex]\longrightarrow \sf \dfrac{3}{2} x-\dfrac{2}{3}x=5+\dfrac{9}{4}[/tex]

[tex]\longrightarrow \sf \dfrac{5}{6} x=\dfrac{29}{4}[/tex]

[tex]\longrightarrow \sf x=\dfrac{29(6)}{4(5)}[/tex]

[tex]\longrightarrow \sf x=\dfrac{87}{10} \quad or \quad 8.7[/tex]

Answer : 11 and 13

Step-by-step explanation:

Let the numbers be X and Y then,

By question

X= Y - 2 .....................(i)

2Y = 3X - 7 ...............(ii)

putting the value of X from (i) on equation (ii) we get,

2Y = 3(Y - 2) - 7

2Y = 3Y - 6 - 7

2Y - 3y = - 13

-Y = - 13

Y = 13

now putting value of Y in equation (i)

X = 13 - 2= 11

hence those numbers are 11 and 13 respectively

Answer:

See below

Step-by-step explanation:

If the whole rectangle were shaded (all 25 parts) that would be 100%

So, we can set up a proportion.

25 = 100%

11 = x

Then, cross-multiply to get,

25x = 1100

Divide both sides by 25 to solve for x

x = 44.

The x value represents the percentage shaded.

44%

Answer:

O The value of f(2) is smaller than the value of f(1).

Step-by-step explanation:

First, let's solve for both. When the problem says f(1) or f(2), this just means that the x value is equal to that. So:
f(1) = -5(1)^2 + 2(1) + 9 = 6
f(2) = -5(2)^2 + 2(2) + 9 = -7
Since f(1) = 6 and f(2) = -7, we know that f(1) is greater than f(2). Therefore, the value of f(2) is smaller than the value of f(1)

Answer:$249895.46

Step-by-step explanation:

Well, if 2250 dollars increase by 17% every year for 30 years, then it will be 2250*(1.17)^30

which would be $249895.46

so if you could increase money by 17% every year, 2250 dollars now will be worth a lot in 30 years!

Answer:

Step-by-step explanation:

Harry buys a ream of computer paper priced at $9.99 before a 6.0%

sales tax. He uses a coupon at checkout for a 10% discount. After the

purchase he mails in a form

A coterminal angle of θ such that 0 ≤ θ ≤ 2π is equal to 7π/4 and the exact value of Sin(θ) = -1/√2.

A coterminal angle can be defined as an angle that share the terminal side of an angle which occupies the standard position i.e it has the same initial side.

In this scenario, all angles that are multiples of 2π and added to the given angle (-9π/4) would be coterminal. For the range [0, 2π], 8π should be added to the given angle as follows:

Coterminal angle = -9π/4 + 4π

Coterminal angle = -9π/4 + 16π/4

Coterminal angle = 7π/4.

The reference angle is given by:

7π/4 -π = 3π/4.

Therefore, the exact values of all six (6) trigonometric functions evaluated at θ are:

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Answer:

2

Step-by-step explanation:

Reason:

Vertical angles are equal to each other. Call them x. They must add to 180 degrees because they are supplementary (they are able to form a straight line).

x+x = 180

2x = 180

x = 180/2

x = 90

Answer: 1.125 pounds of blueberries or  [tex]1 \frac{1}{8}[/tex] or [tex]\frac{9}{8}[/tex]

Step-by-step explanation:

Since she will eat it for 9 days, just multiply or add the pounds.

Answer:

[tex]0.95*10^8[/tex]

Step-by-step explanation:

Hello!

Rules for scientific notation format:

[tex]\bold{0.95*10^8}[/tex]

There is a multiplication to a power of 10, but the other factor is less than 1.

This is NOT in Scientific Notation.

All the other options have a multiplication operation to a power of 10, and all the other factors are above 1 and less than 10.

Considering the given function, the ordered pair (5,8) means that the signed up for 5 weeks of yoga classes and 8 weeks of kickboxing classes.

The function that represents the relationship between the number x of yoga classes that Donovan signs up for and the number y of kickboxing classes is given by:

8x + 12y = 136.

Hence the ordered pair (5,8) means that the signed up for 5 weeks of yoga classes and 8 weeks of kickboxing classes.

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Answer:

the measure of each angle is 75degrees