Answer:
[tex]\cos(2x)\sin(5x)=\dfrac{\sin (7x)+\sin (3x)}{2}[/tex]
Step-by-step explanation:
Given: [tex]\cos(2x)\sin(5x)[/tex]
Formula:
[tex]\sin A+\sin B=2\sin(\dfrac{A+B}{2})\cos(\dfrac{A-B}{2})[/tex]
[tex]\sin(\dfrac{A+B}{2})\cos(\dfrac{A-B}{2})=\dfrac{\sin A+\sin B}{2}[/tex]
Compare the given expression with formula
[tex]\cos(2x)\sin(5x)=\sin(\dfrac{A+B}{2})\cos(\dfrac{A-B}{2})=\dfrac{\sin A+\sin B}{2}[/tex]
Therefore,
[tex]\dfrac{A+B}{2}=5x\Rightarrow A+B=10x[/tex]
[tex]\dfrac{A-B}{2}=2x\Rightarrow A-B=4x[/tex]
Using two system of equation of A and B to solve for A and B
Add both equation to eliminate B
[tex]2A=14x[/tex]
[tex]A=7x[/tex]
Substitute A into A+B=10x
[tex]7x+B=10x[/tex]
[tex]B=3x[/tex]
Substitute A and B into formula
[tex]\cos(2x)\sin(5x)=\sin(\dfrac{7x+3x}{2})\cos(\dfrac{7x-3x}{2})=\dfrac{\sin (7x)+\sin (3x)}{2}[/tex]
Hence, Product as sum form [tex]\cos(2x)\sin(5x)=\dfrac{\sin (7x)+\sin (3x)}{2}[/tex]
Find the arc length of a central angle of pi/4 in a circle whose radius is 8 inches
A) pi/32 in
B) 2pi in
C) 360 in
D) 60 in
The arc length of a central angle of π/4 in a circle whose radius is 8 inches is 2π inches. Therefore, option B is the correct answer.
What is arc length of a circle?The arc length is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining the center to the end-points of the arc.
The formula to find the arc length of a circle is θ/360° ×2πr.
Given that, θ=π/4 =180/4 = 45° and radius = 8 inches.
Now, arc length = 45°/360° ×2×3.14×8
= 1/8×2×3.14×8
= 6.28 inches or 2π inches
Therefore, option B is the correct answer.
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what is 345,000 in expanded form
The Empire State Building is just over 1,450 feet tall. In anticipation of visiting this landmark on your vacation, you create a model of it using blocks.
a. Suppose you are making a model where one block represents 2 feet. About how many blocks tall is your model of the Empire State Building? What is the scale factor?
Answer:
725, 1 per every 2 ft.
Step-by-step explanation:
It gives the actual height for the building, 1,450, and it says one block is two feet. Divide 1,450 by 2 and you'd get 725. And we already know dat one block is two feet so it's 1:2
(why did I start singing ``You Reposted in the Wrong Hero Academia?`` UwUz)
(01.01) Evaluate −6 − (−3). Answers:18 −9 3 −3
The graph of f(x)=x^2 has been shifted into the form f(x)=(x-h)^2+k
What is the value of h?:
A. 1
B. -1
C. -3
D. 3
Answer:
D. 3
Step-by-step explanation:
Here h represents the horizontal shift. In the given graph, the graph shifted 3 units to the right from the origin.
Therefore, h = 3.
Answer: D. 3
Hope this will helpful.
Thank you.
A family has 8 girls and 4 boys. A total of 2 children must be chosen to speak on the behalf of the family at a local benefit. What is the probability that 2 girls and no boys are chosen?
A. 7/55
B. 14/33
C. 12/33
D. 1/6
Answer: The correct option is (B) [tex]\dfrac{14}{33}.[/tex]
Step-by-step explanation: Given that a family has 8 girls and 4 boys. A total of 2 children must be chosen to speak on the behalf of the family at a local benefit.
We are to find the probability that 2 girls and no boys are chosen.
Total number of children in the family = 8 + 4 =12.
Let S denote the sample space of choosing 2 children from the family of 12 children and A denote the event of choosing 2 girls and no boys.
Then, according to the given information, we have
[tex]n(S)=^{12}C_2=\dfrac{12!}{2!(12-2)!}=\dfrac{12\times11\times10!}{2\times1\times10!}=66,\\\\\\n(A)=^8C_2\times^4C_0=\dfrac{8!}{2!(8-2)!}\times1=\dfrac{8\times7\times6!}{2\times1\times6!}=28.[/tex]
Therefore, the probability of event A is given by
[tex]P(A)=\dfrac{n(A)}{n(S)}=\dfrac{28}{66}=\dfrac{14}{33}.[/tex]
Thus, the required probability is [tex]\dfrac{14}{33}.[/tex]
Option (B) is CORRECT.
A computer company makes a rectangular screen with a diagonal of 20 inches. The width of the screen is 4 inches less than its length. The dimensions of the computer screen are modeled by the equation x2 + (x – 4)2 = 202. What is the value of x, the length of the screen?
A.) x = –16
B.) x = –12
C.) x = 12
D.) x = 16
*The answer is D, x=16. If anyone wants to answer with an explanation of how this is the answer, I will reward them with brainliest.*
x^2 +(x-4)^2 =20^2
x^2 +(x-4)^2 =400
x^2 +(x-4)^2 -400=0
factor:
2x^2-8x-384=0
Factor the 2 out of 2X^2
2(x-16) (x+12) =0
Divide both sides by 2
(x-16) (x+12) =0
X=16 , x=-12
Number cannot be negative so x=16
Answer: D.) x = 16
Step-by-step explanation:
Given: The dimensions of the computer screen are modeled by the equation :
[tex]x^2+(x-4)^2=20^2[/tex]
To find the value of x , we need to solve the equation.
Now, the above equation can be written as :
[tex]x^2+x^2+16-8x=400\\\\\Rightarrow\ 2x^2-8x+16=400\\\\\Rightarrow2x^2-8x+384=0\\\\\Rightarrow\ 2(x^2-4x+192)=0\\\\\Rightarrow\ x^2-4x+192=0\\\\\Rightarrow(x-16)(x+12)=0\\\\\Rightarrow\ x=16\ or\ x=-12[/tex]
Since side cannot be negative therefore, x= 16
How do I find the coordinates of angels lab.
A-town and b-ville are connected by a straight, 420-mile road. at noon, atu left a-town for b-ville, and brek left b-ville for a-town. if atu travels at 56 miles per hour and brek travels at 49 miles per hour, how many miles apart will atu and brek be 1 hour before they meet?
Two mechanics worked on a car. the first mechanic worked for 20 hours, and the second mechanic worked for 5 hours. together they charged a total of $1900 . what was the rate charged per hour by each mechanic if the sum of the two rates was $155 per hour?
The length of a rectangle is 9 feet longer than the width. if the perimeter is 78 feet, find the length and width of the rectangle.
Which number line represents the solutions to |x – 2| = 6?
Answer: D IS THE ANSWER
Ariel has a plastic ice cream cone in her food playset. The ice cream cone is a half-sphere sitting on top of a cone. What is the approximate volume of the toy ice cream cone? Use 3.14 for ππ .
the approximate volume of the toy ice cream cone is around [tex]\( \frac{32}{3}π \)[/tex]cubic inches, considering the half-sphere as[tex]\( \frac{16}{3}π \)[/tex] and the cone as [tex]\( \frac{16}{3}π \).[/tex]
let's calculate the approximate volume of the toy ice cream cone step by step.
First, we need to determine the volume of the half-sphere, which is the ice cream part.
The formula for the volume of a sphere is:
[tex]\[ V_{\text{sphere}} = \frac{4}{3}πr^3 \][/tex]
But since we only have a half-sphere, we'll divide the result by 2.
Now, we need to find the radius ((r)) of the half-sphere. We can do this by measuring the radius of the ice cream cone's base.
Let's say the radius of the ice cream cone's base is (r = 2) inches.
Substituting the value of (r) into the formula:
[tex]\[ V_{\text{half-sphere}} = \frac{1}{2} \times \frac{4}{3}π(2)^3 \][/tex]
[tex]\[ V_{\text{half-sphere}} = \frac{1}{2} \times \frac{4}{3}π(8) \][/tex]
[tex]\[ V_{\text{half-sphere}} = \frac{1}{2} \times \frac{32}{3}π \][/tex]
[tex]\[ V_{\text{half-sphere}} = \frac{16}{3}π \][/tex]
Now, let's find the volume of the cone part.
The formula for the volume of a cone is:
[tex]\[ V_{\text{cone}} = \frac{1}{3}πr^2h \][/tex]
Where (h) is the height of the cone.
Let's say the height of the cone is (h = 4) inches.
Substituting the values into the formula:
[tex]\[ V_{\text{cone}} = \frac{1}{3}π(2)^2(4) \][/tex]
[tex]\[ V_{\text{cone}} = \frac{1}{3}π(16) \][/tex]
[tex]\[ V_{\text{cone}} = \frac{16}{3}π \][/tex]
Now, to find the total volume of the ice cream cone, we'll add the volumes of the half-sphere and the cone together:
[tex]\[ V_{\text{total}} = V_{\text{half-sphere}} + V_{\text{cone}} \][/tex]
[tex]\[ V_{\text{total}} = \frac{16}{3}π + \frac{16}{3}π \][/tex]
[tex]\[ V_{\text{total}} = \frac{32}{3}π \][/tex]
So, the approximate volume of the toy ice cream cone is [tex]\( \frac{32}{3}π \)[/tex]cubic inches.
what is the slope of the line that passes through (1,4) and (1,-3)?
A. 7
B. -7
C. undefined
D. 0
Use Gauss-Jordan elimination to solve the following linear system. –4x + 4y + 5z = 9 –2x + 2y + z = 3 –4x + 5y = 4
Jerome earns $200 per month plus $12 for bike he sells at the bike shop.This month, he wants t6o earn at least $400. Whats the least bikes that jerome can sell to reach his goal.
how do i solve this??
area for the hexagon = 1/2*p*a
p = perimeter = 18*6=108
a = apothem = 15.59
108*15.59 = 1683.72/2 = 841.86 sq units
Area of rectangle = 18 x 21 = 378
841.86-378 = 463.86 rounded to 464 square units
A coin is tossed 20 times. A person, who claims to have extrasensory perception, is asked to predict the outcome of each flip in advance. She predicts correctly on 14 tosses. What is the probability of being correct 14 or more times by guessing? Does this probability seem to verify her claim?
To calculate the probability of correctly predicting 14 or more coin tosses out of 20, we can use the binomial probability formula.
Explanation:To calculate the probability of correctly predicting 14 or more coin tosses out of 20, we can use the binomial probability formula. The probability of getting exactly k successes in n independent trials with a probability p of success in each trial is given by:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where C(n, k) is the binomial coefficient, n! / (k! * (n - k)!), p is the probability of success (0.5 for a fair coin), and n is the number of trials (20 tosses).
To find the probability of getting 14 or more successes, we need to sum the probabilities of getting 14, 15, 16, ..., 20 successes.
P(X >= 14) = P(X = 14) + P(X = 15) + P(X = 16) + ... + P(X = 20)
We can calculate these probabilities and then add them together to get the overall probability.
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Why are techniques like cluster sampling and multistage sampling just as externally valid as simple random sampling?
Due to the fact that they all contain elements of random selection, techniques like cluster sampling and multistage sampling are just as externally valid as simple random sampling due to the fact that they all contain elements of random selection.
Cluster sampling are more or less alike, each heterogeneous and resembling the overall population. The population is split into representative clusters, then a selection is done one or few from the clusters, then lastly, a census is performed with those selected clusters.
In multistage sampling, several methods are combined in sampling schemes and occasionally utilized the same method or different methods or more than once. Though they have different steps in being done, they both have random selection elements.
Tables of values for f(x) and g(x) are given below. Write g(x) as a transformation of f(x).
Order of operation:3[10-(27 divided by 9)] sorry I don't have the division sign
What is the length of a diagonal of a cube with a side length of 3 cm? (ALL OF THE ANSWERS ARE SQUARE ROOTED) 18,27,33,38
Answer:
27
Step-by-step explanation:
i just took the test
if the reserve rate is 3% and a bank receives a deposit of 16,000 how much of the 16,000 is the bank free to loan out?
that means the bank keeps 3% and can loan out 97% of the total
16000*0.97 = $15,520 can be loaned out
solve the inequality 9x-11 > 4x +12
What is the simplified form of 30 times x to the sixth power over 14 times y to the fifth power times the fraction 7 times y-squared over 6 times x to the fourth power ?
Which of the following are geometric sequences? Check all that apply. A. 10, 5, 2.5, 1.25, 0.625, 0.3125 B. 5, 10, 20, 40, 80, 160 C. 3, 6, 9, 12 ,15, 18 D. 1, 3, 9, 27, 81
The options that are geometric progressions are:
D. 1, 3, 9, 27, 81A. 10, 5, 2.5, 1.25, 0.625, 0.3125B. 5, 10, 20, 40, 80, 160 What is a geometric progression?This is a progression of numbers which usually have a defined ratio that is between them.
The ratio in the options above are:
[tex]\frac{1}{2} , \frac{3}{1} , \frac{2}{1}[/tex]
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Answer:it’s a,c,d
Step-by-step explanation:
Two horses are ready to return to their barn after a long workout session at the track. the horses are at coordinates H(1,10) and Z(10,1). Their barns are located in the same building, which is at coordinates B(-3,-9). Each unit/grid on the coordinate plane represents 100 meters. Which horse is closer to the barn? Justify your answer?
#23 please!!! Thanks
2x+7=12x-6 looking for x
Find the volume of each figure to the nearest tenth. Show your work. please