To derive an identity for 2sin2θ using the cosine double-angle identity cos(2θ) = 1 − 2sin²θ, we can rearrange the equation and substitute the value of sin²θ to obtain the identity 2sin²θ = 1 - cos(2θ).
Explanation:To derive an identity for 2sin2θ using the cosine double-angle identity cos(2θ) = 1 − 2sin²θ, we can start by rearranging the equation to solve for sin²θ. This gives us sin²θ = (1 - cos(2θ))/2.
Next, we can substitute this value into the expression for 2sin²θ. This gives us 2sin²θ = 2(1 - cos(2θ))/2 = 1 - cos(2θ).
Therefore, the identity for 2sin²θ is 1 - cos(2θ).
Solve for x. 9x = 3x -36
If A game is played using one die. If the die is rolled and shows 2 the player wins 40. If the die shows any number other than 2 the player wins nothing
Planes Q and R are parallel. Lines a and b are shown on planes Q and R, respectively.
Which statement is true about lines a and b?
They are parallel lines.They are perpendicular lines.They are skew lines.They will intersect.Find the Slope A(4,6) B(-8,-3)
A direct variation function contains the points (2, 14) and (4, 28). Which equation represents the function?
Answer:
Sorry that I’m late, answer is C. y=7x
Step-by-step explanation:
14/2 = 7
28/ = 7
The quotient of a number and 0.2 is -2.6
Quotients are used to represent the division of numbers
The number is -0.52
Let the number be represented with x.
So, the quotient of a number (x) and 0.2 is -2.6 is represented as:
[tex]\frac x{0.2} = -2.6[/tex]
Multiply both sides of the equation by 0.2
[tex]0.2 * \frac x{0.2} = -2.6 * 0.2[/tex]
Evaluate the quotient, on the left hand side
x = -2.6 * 0.2
Evaluate the product on the right hand side
x = -0.52
Hence, the number is -0.52
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Simplify and determine the coefficient of (-2/3x)(3y)(-2x). -4 1 4
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50 POINTS
Identify the explicit function for the sequence in the table
How many solutions does this linear system have?
Amira sells balloon animals. She uses the same number of balloons for each animal she makes. The table compares the number of balloon animals sold and the remaining number of balloons on a certain day.
Animals Balloons
15 200
24 164
33 128
How many balloons does Amira use for each balloon animal?
Answer:
4 balloons for each balloon animal.
Step-by-step explanation:
Amira sells balloon animals. The given table compares the number of balloon animals sold and the remaining number of balloons on a certain day.
Animals Balloons
15 200
24 164
33 128
When she sold 15 animals the leftover balloons were 200.
then she sold 24 animals the number of leftover balloons were 164.
The difference of the number of items = 24 - 15 = 9 more animals sold
The difference of the leftover balloons = 200 - 164 = 36 balloons used
Now the number of her sold animals was = 33
and the left over balloons = 128
The number of sale was increased by 24 to 33 = 33 - 24 = 9 more animals
and the leftover balloons now 164 to 128 = 164 - 128 = 36 used
Therefore, we can see each sale of 9 balloon animals she used 36 balloons.
so Amira used for each balloon animal = [tex]\frac{36}{9}[/tex]
= 4 balloons.
Amira used 4 balloons for each balloon animal.
Answer:
The most she can sell is 65
Step-by-step explanation:
"How many balloon animals at most can Amira sell?"
how many times greater is the value of the digit 2 in the 23.876 than the value of the digit 2 in 3.254?
A painter made the table below to show the amount of paint it takes to cover a certain amount of square feet. The amount of paint varies directly with the area of each wall.
Answer:
Option A.
Step-by-step explanation:
The table made by the painter shows the amount of paint required to cover a certain amount of area of the wall.
Amount of paint is directly proportional to the area covered
In other words Amount of paint (y) ∝ Area covered (x)
y ∝ x
As we know when proportionality sign is removed, a constant known as proportionality constant is multiplied with the variable.
So y = k(x) where k is the proportionality constant
Now from the table to cover the area of 64 square feet, 4 pints of paint is required.
Now from the formula
4 = 64x
x = [tex]\frac{4}{64}=\frac{1}{16}[/tex]
Now we have to calculate the amount of paint required to cover the area of 256 square feet.
y = [tex]\frac{1}{16}\times 256[/tex]
y = 16 pints
Therefore, 16 pints of the paint will be required to cover 256 square feet.
Option A is correct.
Answer:
Just Pick 16 Pints
Step-by-step explanation:
Question Help Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 45 in. by 24 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume.
Answer:
The dimensions are [tex]35\times 14\times 5[/tex] and the volume is 2450 inches³.
Step-by-step explanation:
Given : The open rectangular box of maximum volume that can be made from a sheet of cardboard 45 in. by 24 in. by cutting congruent squares from the corners and folding up the sides.
To find : The dimensions and the volume of the open rectangular box ?
Solution :
Let the height be 'x'.
The length of the box is '45-2x'.
The breadth of the box is '24-2x'.
The volume of the box is [tex]V=L\times B\times H[/tex]
[tex]V=(45-2x)\times (24-2x)\times x[/tex]
[tex]V(x)=4x^3-138x^2+108x[/tex]
Derivate w.r.t x,
[tex]V'(x)=4(3x^2)-138(2x)+108[/tex]
[tex]V'(x)=12x^2-276x+108[/tex]
[tex]V'(x)=12(x^2-23x+90)[/tex]
The critical point when V'(x)=0
[tex]12(x^2-23x+90)=0[/tex]
[tex]x^2-23x+90=0[/tex]
[tex](x-18)(x-5)=0[/tex]
[tex]x=18,5[/tex]
18 is not possible we reject.
So, the height is 5 inches.
Derivate again w.r.t x,
[tex]V''(x)=24x-276[/tex]
[tex]V''(5)=24(5)-276[/tex]
[tex]V''(5)=120-276[/tex]
[tex]V''(5)=-156<0[/tex]
i.e. V(x) is maximum at x=5.
The dimensions are
Height = 5 inches
Length = 45-2(5)=35 inches
Breadth = 24-2(5)=14 inches.
The maximum volume is
[tex]V=35\times 14\times 5[/tex]
[tex]V=2450[/tex]
So, The dimensions are [tex]35\times 14\times 5[/tex] and the volume is 2450 inches³.
What dose 0.07x1.22 eaqual
Final answer:
When multiplying 0.07 by 1.22, the product is 0.0854. This result is calculated by first multiplying the numbers as whole numbers and then placing the decimal point so that the total number of decimal places in the product matches the sum of the decimal places in the factors.
Explanation:
To calculate the product of 0.07 and 1.22, you follow the standard multiplication procedure for decimals:Multiply as if the numbers were whole numbers; ignore the decimal points for a moment.Count the total number of decimal places in both of the numbers you're multiplying. In 0.07, there are two decimal places. In 1.22, there are two decimal places as well.Add the number of decimal places together. In this case, 2 (from 0.07) plus 2 (from 1.22) equals 4 decimal places in total.Perform the multiplication without considering the decimal, which gives you 7 * 122 = 854.Place the decimal point in the answer (854) so that there are 4 decimal places, giving you 0.0854.Therefore, 0.07 multiplied by 1.22 equals 0.0854.
Final answer:
When multiplying 0.07 by 1.22, we first ignore the decimals and multiply 7 by 122 to get 854, and then we place the decimal point so the result has the same number of decimal places as the sum of the factors, resulting in 0.0854.
Explanation:
To find the product of two decimal numbers, you multiply them just as you would with whole numbers, taking care to place the decimal point in the correct position in the final answer. For the multiplication of 0.07 by 1.22, you perform the multiplication ignoring the decimals first, and then place the decimal in the final answer so that the total number of decimal places equals the sum of the decimal places in the factors.
Multiplying 7 by 122 equals 854. Since 0.07 has two decimal places and 1.22 has two decimal places, the product must have four decimal places. Therefore, the correct placement of the decimal point in 854 would be after four digits from the right, which gives us a final answer of 0.0854.
The circumference of a circular field is
260.62 yards. What is the diameter of the field? Use 3.14 for π and do not round your answer.
The diameter of the field is 83 yards.
Explanation:To find the diameter of a circular field, we can use the formula for the circumference of a circle, which is C = 2πr, where C is the circumference and r is the radius. Given that the circumference of the field is 260.62 yards, we can solve for the radius by dividing the circumference by 2π.
Using the value of π as 3.14, we can calculate:
260.62 / (2 * 3.14) = 41.50 yards
Since the diameter of a circle is twice the radius, the diameter of the field is 2 * 41.50 = 83 yards.
Using the rule of 72, mc003-1.jpg, how long will it take for the principal to double with an annual compound interest rate of 6%?
6 years
9 years
12 years
15 years
Answer:
Option C. 12 years
Step-by-step explanation:
Let principal amount taken = x
We know the formula of compound interest
Final amount = Principal amount×[tex](1+\frac{r}{n})^{nt}[/tex]
Where r = rate of interest (per year)
n = number of times compounded (annually)
t = time in years (years)
Here we have to find the time in which principal amount is doubled.
From the question r = 6% = .06
n = 1
Principal amount = P
Final amount = 2P
Now we put these values in the formula
[tex]2P=P(1+.06)^{t}[/tex]
[tex]2=(1.06)^{t}[/tex]
Now we take log on both the sides
[tex]log(2)=log(1.06)^{t}[/tex]
0.301 = tlog(1.06)
0.301 = t×(.025)
[tex]\frac{0.301}{0.25}=t[/tex]
t = 12 years.
Option C. 12 years is the answer.
If the equation y = 15x + 2 is changed to y = 15x - 4, how will the graph of the line change?
The graph of the line change is, It will shift down 6 units.
The given function is,
y = 15x + 2
Now, We want to find how the graph of this function will change if the function is transformed to
y = 15x -4
We rewrite this function in terms of the first one to get:
y = 15x + 2 + (-6)
Hence, the function with shift down 6 units.
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write the following equation in standard form. x^5+2x^3+6x+1/5
The standard form is x⁵ + 2x³ + 6x + [tex]\frac{1}{5}[/tex].
To write the given equation x⁵ + 2x³ + 6x + [tex]\frac{1}{5} \\[/tex] in standard form, follow these steps:
Identify and list all terms in the polynomial: x⁵, 2x³, 6x, and [tex]\frac{1}{5}[/tex].Arrange the terms in descending order based on the exponent of x:x⁵ + 2x³ + 6x + [tex]\frac{1}{5}[/tex]Ensure all coefficients and constants are in standard form:The polynomial in standard form is: x⁵ + 2x³ + 6x + [tex]\frac{1}{5}[/tex].An orchard has 864 orange trees. The number of rows exceeds the number of trees per row by 5 . How many trees are there in each row?
Final answer:
Using a quadratic equation, we found that there are 24 orange trees in each row of the orchard.
Explanation:
To determine how many orange trees are there in each row, we will set up an equation based on the information provided. Let's let x represent the number of trees per row. According to the problem, the number of rows is x + 5. Since the total number of trees in the orchard is 864, we can create the following equation:
x * (x + 5) = 864
To solve this quadratic equation, we need to find the value of x that satisfies the equation:
First, expand the left side of the equation: [tex]x^{2}[/tex] + 5x.
Write the equation in standard form: [tex]x^{2}[/tex] + 5x - 864 = 0.
Factor the quadratic equation: (x + 36)(x - 24) = 0.
Solve for x: x = -36 or x = 24.
Since we cannot have a negative number of trees per row, we disregard x = -36.
Therefore, there are 24 trees in each row.
1/4 of 12 bottles of water
Prakrit bought a pack of paper for $5.69 and printer toner for $9.76. He paid with a $20 bill. What was his change?
hello i need help in this, someone pls!!!
At your birthday party, 5/8 of the 24 guests were relatives. How many relatives were at your party?
You have a gift card for your favorite clothing store for the amount of $60. You have found a shirt you want to buy that costs $15. If you don't want to spend more than the amount of the gift card, which of the following inequalities could be used to determine the amount you have left to spend?
Answer:
D. x + 15 ≤ 60
Step-by-step explanation:
Let x be the amount you have left to spend after you purchase the shirt.
We must add to x the cost of the shirt, $15. This gives us x+15.
We only have $60 to spend; this means we can spend anything up to and including $60. This means we want "less than or equal to." This gives us the inequality
x+15 ≤ 60
The inequality of the form x + 15 ≤ 60 is the correct choice.
What is an inequality ?An inequality is a comparison between two non equal expression or terms or numbers.
According to the given question You have a gift card for your favorite clothing store for the amount of $60.
You have found a shirt you want to buy that costs $15 but you don't want to spend more than the amount of the gift card.
Assuming the amount you will have is x dollars.
∴ x + 15 ≤ 60
As x ≤ 60 - 15
x ≤ 45.
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The function f(x) = x2 + 5x – 6 is shifted 4 units to the left to create g(x). What is g(x)?
Answer:
g(x)=(x+4)^2+5(x+4)-6
The use of the normal probability distribution as an approximation of the sampling distribution of p̄ is based on the condition that both np and n(1 – p) equal or exceed _____.
a. .05
b. 5
c. 15
d. 30
The use of the normal distribution as an approximation for the sampling distribution of[tex]\( \bar{p} \)[/tex] requires both [tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] to be at least 5. So, the answer is (b) 5.
The normal approximation for the sampling distribution of [tex]\( \bar{p} \),[/tex] the sample proportion, relies on the conditions that both[tex]\( np \)[/tex] and [tex]\( n(1 - p) \)[/tex] are greater than or equal to 5. Here, [tex]\( n \)[/tex] represents the sample size, and [tex]\( p \)[/tex] is the probability of success in a binary outcome. These conditions ensure that the distribution of[tex]\( \bar{p} \)[/tex] is approximately normal, allowing for the application of statistical techniques based on the normal distribution. When[tex]\( np \)[/tex]and [tex]\( n(1 - p) \)[/tex]are both at least 5, the central limit theorem asserts that the distribution of [tex]\( \bar{p} \)[/tex] is sufficiently close to normal, facilitating the use of standard normal tables and z-scores for making probability calculations and inferences about the population proportion. This approximation is pivotal in hypothesis testing and confidence interval construction for proportions in statistical analyses.
Yikes mctugg bought 1/2 pounds of potato salad. He ate 2/3 of it for lunch. How much potato salad was left for an afternoon snack
A text message plan costs $3 per month pulse $0.32 per text. find the monthly cost for x text messages
Does anyone know a rule that work for all of these
Jackson is flying at a constant rate of 500 mph from New York, New York, to Austin, Texas, which is a distance of 1,750 miles. If the distance Jackson has traveled in miles is a function of time in hours he has been flying, determine an appropriate domain and range for this problem situation.
The domain of the function is [tex][0,3.5][/tex] and the range of the function is [tex][0,1750][/tex].
Speed [tex]= 500[/tex] mph
Distance [tex]= 1750[/tex] miles
Distance [tex]=[/tex] Speed [tex]\times[/tex] Time
The distance Jackson has traveled in miles is a function of time in hours he has been flying.
[tex]f(t)=500t.[/tex]
For maximum distance, [tex]f(t)=1750[/tex] miles.
[tex]1750=500t.[/tex]
[tex]t=\frac{1750}{500}[/tex]
[tex]t=3.5[/tex] hours
So, domain [tex]= [0,3.5][/tex]
Range [tex]= [0,1750][/tex]
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Find counter example to disprove the conjecture:
if the quotient of two numbers is positive, then the two numbers must be positive