Proved [tex]Cos^{4}x - Sin^{4}x + 1[/tex] = 2Cos²x
We will use the Pythagorean Theorem:
Sin²x + Cos²x = 1
from which we can get:
Sin²x = 1 - Cos²x
Also, note that the difference in the similarity of squares can be written:
A² - B² = (A - B)
We can use this with A = Sin²x and B = Cos²x as follows:
[tex]Sin^{4}x - Cos^{4}x[/tex]
= (Sin²x)² - (Cos²x)²
= (Sin²x - Cos²x) (Sin²x + Cos²x)
= Sin²x - Cos²x
= (1 - Cos²x) - Cos²x
= 1 - 2Cos²x
Now move the 1 and negative symbols to the left.
[tex]Cos^{4}x - Sin^{4}x + 1[/tex] = 2Cos²x
The trigonometric ratios are related to each other by equations called trigonometric identities.
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three cards are drwan at random, without replacement, from a standard deck of 52 playing cards. compute the (conditional) probability that the first card drawn is spade, given that the second and third cards have spades.
If the second and third cards have a spade then the probability that the first card drawn is a spade is 21.2%.
Probability can be described as a field of mathematics that determines how likely a thing is to happen. When there is uncertainty about the outcome of an event we consider the probabilities of certain results or outcomes.
There are a total of 13 spades in a deck of 52 playing cards. So if the second and third cards have spades then there are 11 spades remaining.
The formula for probability can be given as;
probability = number of favorable outcomes or events ÷ total number of outcomes or events
probability = 11 / 52 = 0.212
Converting it into percentage;
probability = 0.212 × 100 = 21.2%
Therefore, the probability that the first card drawn is a spade is calculated to be 21.2%.
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Please help me please help me
two cards are drawn without replacement from a standard deck of 52 playing cards. what is the probability of choosing a jack and then, without replacement, a face card? express your answer as a fraction or a decimal number rounded to four decimal places.
write 5 to the power of -3 in its simplest form without any indices
lol this is annoying
5 to the power of -3 in its simplest form without any indices is 1/125 or 0.008.
Exponents
Exponent is defined as the method of expressing large numbers in terms of powers. That means, exponent refers to how many times a number multiplied by itself. For example, 6 is multiplied by itself 4 times, i.e. 6 × 6 × 6 × 6. This can be written as [tex]6^4[/tex]. Here, 4 is the exponent and 6 is the base. This can be read as 6 is raised to power 4.
We know that,
[tex]a^{-b}=\frac{1}{a^b}[/tex]
Hence, we can write,
[tex]5^{-3}=\frac{1}{5^3}[/tex]
We know that the cube of 5 is 125.
Hence, we can write,
[tex]\frac{1}{5^3}[/tex] = 1/125
1/125 is the simplest form of fraction and is equal to 0.008 in decimal form.
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Evaluate the expression. 12−{1+2[−1(3−8)]2} What is the value of the expression? Enter your answer in the box.
If the given mathematical expression 12−{1+2[−1(3−8)]2} is been evaluated , then it will become -18.
How can the expression be simplified?The mathematical expression can be simplified by first simplify the expression in the inner bracket one after the other.
The given mathematical expression is 12−{1+2[−1(3−8)]2}
then we can simplify one after the order by following the bracket one after the other, as :
12−{1+2[−1(3−8)]2}
12 - {1+2[-1(-5)]2}
Then we can further simplify the expression as :
12 - {3 [ 5 ] 2}
Then this can be further simplified as :
12 - {30}
This can be further simplified as :
-18
Therefore, the value of the expression given is -18
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Alyssa wants to play the game. Choose the statement below that is correct in all aspects. E(penny) = –0.33 and E(nickel) = 0.33, so she should play with the nickels. E(penny) = 0.5 and E(nickel) = 1.5, so she should play with the nickels. E(penny) = 1.75 and E(nickel) = 1.5, so she should play with the pennies. E(penny) = 2.38 and E(nickel) = 2, so she should play with the pennies
Using the expected value of a discrete distribution, the correct option is given as follows:
E(penny) = 1.75 and E(nickel) = 1.5, so she should play with the pennies.
What is the expected value of a discrete distribution?The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.
When 3 coins are tossed, the probabilities are given as follows:
All heads: (1/2)^3 = 0.125.All tails: (1/2)^3 = 0.125.At least one of each: 1 - 2(0.125) = 0.75.Hence the distribution for the penny points is:
P(X = -2) = 0.25.P(X = 3) = 0.75.The expected value for the penny points is:
-2 x 0.25 + 3 x 0.75 = 1.75.
When 2 coins are tossed, the probabilities are given as follows:
All heads: (1/2)² = 0.25.All tails: (1/2)² = 0.25.At least one of each: 1 - 2(0.25) = 0.50.Hence the distribution for the nickel points is:
P(X = -2) = 0.5.P(X = 5) = 0.5.The expected value of nickel points is:
E(X) = -2 x 0.5 + 5 x 0.5 = 1.5.
Due to the higher expected value, she should play with the pennies and the third option is correct.
What is the missing information?
The complete problem is given by the image at the end of the answer;
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Answer:
The answer is C
Step-by-step explanation:
edge
simplify 3x + 5y - x + 2y
Answer:
2x + 7y
Step-by-step explanation:
3x + 5y - x + 2y ← collect like terms
= (3x - x) + (5y + 2y)
= 2x + 7y
By Simplifying 3x + 5y - x + 2y, we get 2x + 7y
More Explanation
The simplification here follows the basic method of addition and subtraction in algebra.
Given : 3x + 5y - x + 2y
Step 1: align the numbers with similar variables.
Therefore, we get, 3x - x + 5y + 2y
Step 2: Solve by addition and subtraction.
Therefore, we get, 2x + 7y
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Three students were scheduled to give book reports in 1 hour.
After the first report, 2/3 hour remained (HINT: this means the first persons' report was 1/3 of an hour)
The next report took 1/6 hour. What fraction remained?
The fraction of hour that remained is 1/2 hour.
It is given in the question that three students were scheduled to give book reports in 1 hour.
After the first report, 2/3 hour remained. The next report took 1/6 hour.
We have to find the fraction of hour that remained.
We know that,
Total time for reports = time taken for first report + time taken for second report + time remained for third report
Hence, we can write,
Time taken for first report = 1-2/3 = 1/3 hour
Time taken for second report = 1/6 hour
1 = 1/3 + 1/6 + time remained for third report
1 = 2/6 + 1/6 + time remained for third report
1 = 3/6 + time remained for third report
1-3/6 = time remained for third report
3/6 = 1/2 = time remained for third report
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assume the cost of a gallon of milk is $2.70. with continuous compounding, find the time it would take the cost to be 4 times as much (to the nearest tenth of a year), at an annual inflation rate of 6%.
The time it would take the cost to be 4 times the current cost of a gallon of milk at an annual inflation rate of 6% is calculated to be 23.8 years.
To determine the time for the cost to be 4 times the current cost of a gallon of milk, we use the compound interest formula as follows;
A(t) = A(0) (1 + r)^t
Here A(t) is the cost of 4 times the current cost at time t
A(0) is the current cost
r is the interest rate
t is the time period
Therefore;
2.70 × 4 = 2.70( 1 + 0.06)^t
2.70 × 4 / 2.70 = (1.06)^t
4 = (1.06)^t
Taking log on both sides;
t × log(1.06) = log 4
t = log 4 / log(1.06)
t = 23.791 = 23.8 years
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17. The area of the rectangle shown exceeds the area of the square by
2 cm². Find x.
y
X
x-1
18. The area of the square exceeds the area of the rectangle by 13 m².
Find y.
X
x + 2
y
X
y + 1
19. The area of the square is half the area of the rectangle. Find x.
y-3
2(x + 4)
(x - 2)
The values of the missing variables from the areas of rectangle and square are;
17) x = 4
18) y = 5
19) x = 4
What is the area of the rectangle and square?17) The formula for area of rectangle is;
A = Length * Width
Formula for area of square is;
A = side * side
Thus;
A_rect = (x - 1)(x + 2) = x² + x - 2
A_sq = x²
The area of the rectangle shown exceeds the area of the square by
2 cm². Thus;
x² + x - 2 - x² = 2
x - 2 = 2
x = 4
18) Area of the square; A_sq = y²
Area of the rectangle; A_rect = (y - 3)(y + 1) = y² - 2y - 3
The area of the square exceeds the area of the rectangle by 13 m². Thus;
y² - y² + 2y + 3 = 13
2y = 10
y = 5
19) The area of the square is half the area of the rectangle. Thus;
x² = ¹/₂(2(x² + 2x - 8))
x² = x² + 2x - 8
2x - 8 = 0
x = 4
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Which expression is equivalent to quantity y raised to the negative second power times z raised to the seventh power end quantity over quantity z raised to the negative third power times y raised to the eighth power end quantity all raised to the negative second power? z raised to the tenth power over y raised to the tenth power y raised to the tenth power over z raised to the tenth power z raised to the twentieth power over y raised to the twentieth power y raised to the twentieth power over z raised to the twentieth power
The expression z raised to the twentieth power over y raised to the twentieth power is equivalent to the expression given in the question.
y raised to the negative second power = [tex]y^{-2}[/tex]
z raised to the seventh power = [tex]z^{7}[/tex]
Thus, in the numerator we have the expression- [tex]y^{-2} z^{7}[/tex]
In the denominator, we will have-
z raised to the negative third power = [tex]z^{-3}[/tex]
y raised to the eighth power = [tex]y^{8}[/tex]
denominator will be = [tex]z^{-3} y^{8}[/tex]
Thus, the expression becomes-
[tex](\frac{y^{-2 }z^{7} }{z^{-3} y^{8} })^{-2}[/tex]
Now, that we've decoded the expression, let us solve it-
[tex]({y^{-2 }y^{-8} z^{7} z^{3} )^{-2}[/tex]
= [tex]({y^{(-2-8) } z^{(7+3)})^{-2}[/tex]
= [tex]({y^{(-10) } z^{(10)})^{-2}[/tex]
= [tex]{y^{(-10*-2) } z^{(10*-2)}[/tex]
= [tex]{y^{(20) } z^{(-20)}[/tex]
= [tex]\frac{z^{20} }{y^{20} }[/tex]
= [tex](\frac{z}{y} )^{20}[/tex]
Thus, the expression z raised to the twentieth power over y raised to the twentieth power is equivalent to the expression given in the question above.
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Write the formula for a meal that has a total cost of C
that is equal to the price of the food F plus 20% (for tip) of
the price of the food.
Answer:
C = F(0.20) + F
Step-by-step explanation:
C is the total cost. F(0.20) will reduce the value of the food to the tip amount required, and then adds the tip value to the price of the food, F.
Example inputs: let F be equal to 10
C = F(0.20) + F turns into C = 10(0.20) + 10
10(0.20) will cause 10 to be multiplied by the tip value, 20%, and 20% is equivalent to 0.20. Once you multiply 10 by 0.20, you get 2. After that, you add 2 to the price of food, which is 10, and you get a total cost of 12.
12 = 10(0.20) + 10
A woman decides to start a small business making monogrammed cocktail napkins. She can set aside $1870 for monthly costs. Fixed costs are $1329.50 per month and variable costs are $3.70 per set of napkins. How many sets of napkins can she afford to make per month?
[tex] \: \huge \tt{\pink{A}{\orange{N}S{\green{W}{\blue{E}{\purple{R}}}}}} \\ \\ \: \: \: \tt \rightarrow {\blue{\fbox{147 \: sets}}} (approx)[/tex]
Step-by-step explanation:
Fixed costs = $ 1329.50 per month
Variable cost are $3.70 per set of napkins
Total cost she set aside = $1870 for monthly costs
Amount left for napkins = Total cost - Fixed Cost
= 1870 - 1329.50
= $540.50
$540.50 she can spend on napkinsCost for one set = $3.70
Let set of napkins = x
Total cost for napkins = $540.50
x × 3.7 = 540.50 x = 540.50 / 3.70x = 146.08 (approx) or 147 setsTherefore , she can afford to make 147 (approx) sets of napkins per month.
a vendor sells hot dogs and bags of potato chips. a customer buys 3 hot dogs and 3 bags of potatochips for $8.25. another customer buys 5 hot dogs and 3 bags of potato chips for $11.25. find thecost of each item.
Cost of one hot dog is and potato chips is
As per the given problem a customer buys 3 hot dogs and 3 bags of potato chips for $8.25. we have to find the cost of each item.
let price of one hot dog is x nd price of potato chips is y.
Formulating the Equation we get
3x+3y=$8.25
another customer buys 5 hot dogs and 3 bags of potato chips for $11.25.
Formulting the Equation we get
5x+3y=$11.25.
we get two linear equation with two unknowns.
solving both equation we get $8.25-3x=$11.25-5x
or 2x=$3
x=$1.5
y=$1.25
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Sort the following transformations according to whether or not they preserve distances.
Preserves Distance
Does NOT Preserve Distance
A. Counter-clockwise rotation about the origin
B. A dilation of 1/5
C. A translation that moves a figure up 5 units
D. A reflection over the y-axis
E. A dilation of -4
The truth statements about the transformations are:
A. Counter-clockwise rotation about the origin ⇒ PreserveB. A dilation of 1/5 ⇒ Do not preserveC. A translation that moves a figure up 5 units ⇒ PreserveD. A reflection over the y-axis ⇒ PreserveE. A dilation of -4 ⇒ Do not preserveHow to determine the category of the transformation?There are four types of transformation, and they are:
DilationRotationReflectionTranslationThe representation of the types of transformation are:
Dilation: k(x, y)Rotation: rotation by anglesReflection: reflection across linesTranslation: (x + h, y + k)Of the above transformation, only the dilation transformation do not preserve side lengths or distance
Hence, the categories of the transformations are
(a), (c) and (d) ⇒ Preserve
(b) and (e) ⇒ Do not preserve
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( Please help + Will mark brainliest )
Answer the question in photo.
Answer:
3/2
Step-by-step explanation:
6/4 = 1.5
3/2 = 1.5
So we know 1.5 is the scale and it asks for it in fraction form which would be 3/2.
A truck can be rented from Company A for $40 a day plus $0.40 per mile. Company B charges $20 a day plus $0.50 per mile to rent the same truck. Find the number of miles in a day at which the rental costs for Company A and Company B are the same.
The number of miles in day at which the rental cost for the company A and Company B are the same is 200 miles
The Company A rented the truck for $40 a day plus $0.40 per miles
The expression will be
40+0.40x
Where x is the number of miles
The company B rented the truck for $20 a day plus $0.50 per miles
20+0.50x
To find the number of miles in a day at which the rental costs for Company A and Company B are the same, the linear equation will be
40+0.40x = 20+0.50x
40-20 = 0.50x-0.40x
0.1x = 20
x = 200 miles
Hence, the number of miles in day at which the rental cost for the company A and Company B are the same is 200 miles
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You want to hang three equally-suzed travel posters on a wall so that posters on the ends are each 3 feet from the end of the wall. You want the sapacing between posters to be equal. Wrte and somve an equation to determine how much space you should leave between the posters.
The equation is:
You Should leave ______ feet of space.
plss help me
Answer:
[tex]\frac{15-6-6}{2}[/tex] = x
You should have 1.5 feet of space between each picture.
Step-by-step explanation:
Let x be the spacing between the pictures
[tex]\frac{15-6-6}{2}[/tex] = x The total length - end spacing - the width of the posters. What is left is the blank spaces between the posters. Since there are 2 blank spaces, we will divide by 2 to find out the spacing between the two pictures. Which is 1.5 feet.
each side of a square is increasing at a rate of 2 cm/s. at what rate (in cm2/s) is the area of the square increasing when the area of the square is 64 cm2?
At 32 cm^2/s rate the area of the square is increasing when the area of the square is 64 [tex]cm^{2}[/tex]
As per given problem each side of a square is increasing at a rate of 2 cm/s. let side of square is x.so the area of it is A= [tex]x^{2}[/tex].now differentiating w.r.t time A
[tex]\frac{dA}{dt}[/tex]= 2x[tex]\frac{dx}{dt}[/tex]
when A=64 [tex]cm^{2}[/tex] x=8 cm
also given [tex]\frac{dx}{dt}[/tex]=2cm/s
so [tex]\frac{dA}{dt}[/tex]=2×2×8=32
So at 32 cm^2/s rate the area of the square is increasing when the area of the square is 64 [tex]cm^{2}[/tex]
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The weather forecast for the weekend is a 35% chance of rain on Saturday and a 65% chance of rain on Sunday. Your friend says that means there is a 100% chance of rain this weekend. What error did your friend make?
Answer:
Because these probabilities are independent, the friend was supposed to multiply them, not add them. So the probability of rain this weekend is
.35 × .65, or .2275, meaning that the probability of rain this weekend is 22.75%
What is 40÷8175 PLEASE HELP ME H E L P
Answer:
Step-by-step explanation: 0.00489296636 if you divided by 40 to 8175 but if you wanted 8175 divided 40 it would equal 204.375 just letting you know i used a calculator so if these aren't it then i'm not sure
Which trigonometric ratio can be used to find the measurement of F using only the lengths shown.
The trigonometric ratio which can be used to find the measurement of F using only the lengths is: D. tangent only.
What is a tangent?In geometry, a tangent is also known as a tangent line and it can be defined as a straight line that touches a plane curve at a specific point.
Mathematically, the tangent of an angle is given by this expression:
Cosθ = Adj/Hyp
Where:
Hyp represents the hypotenuse of a right-angled triangle.Adj represents the adjacent side of a right-angled triangle.θ is the angle.Since both the opposite side and adjacent side of this right-angled triangle are given, cosine is the trigonometric ratio which can be used.
For this right-angled triangle, the cosine of the angle (θ) is given by:
Cosθ = 9/F
F = 9/Cosθ
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1/4 Marks
a) Complete the prime factor trees for 16 and 40:
16
b)
2
2
8
2
4
17/31 Marks
2
2
40
What is the Lowest Common Multiple of 16 and 40?
82%
5
Answer:
Answer in photo!!! :))
Step-by-step explanation:
Q20. in an assorted colored lego box, 5/6 of the legos were red, and 2/3 of the remaining legos were green. if there were 18 legos, how many of the legos were green?
Answer:
2 legos were green
Step-by-step explanation:
Given- in an assorted colored lego box, 5/6 of the legos were red, and 2/3 of the remaining legos were green. if there were 18 legos,
let->
x number of the red ball then probability= x/18=5/6
=>x=15
then, 3 balls are other in which the green +remaining ball is present according to the question.
2/3 times of 3 are green legos==>
(2/3)*3=2 are green legos;
and the last ball 1 lego is remain.
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The large of two numbers is less than 3 times the small number if the sum of the numbers is 35 find the numbers
The numbers are 26 and 9 respectively.
What is an algebraic expression?An algebraic expression can best be defined as an arithmetic or mathematical expression that is known to consist of variables, coefficients, constants, factors as well as terms.
These expressions said to contain numerous arithmetic operations, which includes;
SubtractionBracketParenthesesMultiplicationAdditionDivision, with others inclusive.From the information given, we have that
Let the small number be x
The large number be y
y < 3x (1)
Also,
x + y = 35 equation 2
x = 35 - y
Substitute the value of x in equation 1
y < 3(35 - y)
expand the bracket
y < 105 - 3y
collect like terms
y + 3y < 105
4y < 105
Make 'y' the subject
y < 105/4
y < 26. 25
y = 26
substitute the value of y
x + y = 35
x + 26 = 35
x = 9
Hence, the values are 26 and 9
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56x raise to power minos 4 divided by 14x raise to power minos 8
56x raise to power minus 4 divided by 14x raise to power minus 8 has its answer calculated as 4[tex]x^2[/tex] by solving through formulas of exponents and powers.
What are exponents?A mark placed exactly or right above a mathematical equation or a numeral that helps to denote the action of raising to a power of something.
Exponent is also defined as one of the methods of expressing many large numbers in terms of powers. This means that exponent refers to how many times a single number or an equation is multiplied by itself.
The statement 56x raise to power minus 4 divided by 14x raise to power minus 8 can be simplified in equational terms as
[tex]\frac{56x^-{4}}{14x^{-8}}[/tex]
which can be further simplified as [tex]4x^{4}[/tex]
What does the term Exponentiation stand for?Exponentiation is a type of mathematical operation that can be written as bⁿ, involving two numbers. The base b and then the exponent or power n is included in it and pronounced as "b to the power n".
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8. Игральную кость кидают 6 раз. Какова вероятность того, что «шестерка» выпадет а) не менее трех раз; б) не более трех раз?ъ
Answer:
б) не более трех раз
find the values of k for which the line y = 2x + k + 2 cuts the curve y = 2x^2 + (k + 2)x + 8 at two distinct points.
The values of k for which the line y = 2x + k + 2 cuts the curve y = 2x² + (k + 2) + 8 at two distinct points are; k > -12 or k < 4
What is the point of Intersection of the line and the curve?We are given the equation of the line as;
y = 2x + k + 2
Equation of the curve is;
y = 2x² + (k + 2) + 8
Now, to find the the values of k for which the line y = 2x + k + 2 cuts the curve y = 2x² + (k + 2) + 8 at two distinct points, we will just equate both equations to get;
2x + k + 2 = 2x² + (k + 2) + 8
2x + k + 2 = 2x² + kx + 2x + 8
2x² + kx + (-k + 6) = 0
Now, Two distinct solutions is the same as having a positive discriminant.
b² - 4ac > 0
k² - 4(2 * (-k + 6)) > 0
k² + 8k - 48 > 0
Solving using quadratic equation calculator gives;
k > -12 or k < 4
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2. Find the parabola with equation y = ax² + bx whose tangent line at (1, 1) has equation y = 3x - 2.
Answer:
Step-by-step explanation:
write an equation for the line will give brainliest
The equation of the line graph in slope intercept form is gotten as;
y = x + 3
What is the equation of the Line in Slope Intercept Form?The general formula for the equation of a line in slope intercept form is;
y = mx + c
where;
m is slope
c is y-intercept
Now, from the given straight line equation graph, we can see that the line crosses the x-intercept at x = -3 and crosses the y-intercept at y = 3. Thus;
When x = 0, y = 3
When y = 0, x = -3
Slope is gotten by using two points and their coordinates to get;
m = (3 - 0)/(0 - (-3))
m = 3/3
m = 1
Thus, the equation of the line is y = x + 3
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