Answer:
It is true that the x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx
Step-by-step explanation:
Given function is a trignometric one
y = tanx
we have tanx has values 0 for all multiples of pi.
i.e. tan x =0 whenever [tex]x = 2n\pi[/tex] for all integers n.
Also tanx has a period of pi.
It is a discontinuous graph extending in one period from -pi/2 to pi/2.
Hence the mid point of each period is the x intercept.
It is true that the x-intercepts of y=tanx are the same as the x-coordinates of the center points of y=tanx
Final answer:
The x-intercepts of the function y=tan(x) are at integer multiples of π. The concept of 'center points' for y=tan(x) is not well-defined due to the nature of the function having no central axis and being periodic.
Explanation:
The student is inquiring about the x-intercepts and center points of the function y=tan(x). The x-intercepts of the tangent function occur wherever y=0, which happens at values of x that are integer multiples of π, as the tangent function has a period of π.
On the other hand, the concept of center points is not well-defined for the tangent function since it does not have a central axis like an ellipse or a bounded pattern. The tangent function is periodic and continuous between its vertical asymptotes, which occur at odd multiples of π/2.
Diana is painting statues. She has \dfrac{7}{8} 8 7 start fraction, 7, divided by, 8, end fraction of a liter of paint remaining. Each statue requires \dfrac{1}{20} 20 1 start fraction, 1, divided by, 20, end fraction of a liter of paint. How many statues can she paint?
Answer:
Number of statues that can be painted are 17
Step-by-step explanation:
Initially Diana has [tex]\frac{7}{8}[/tex] liters of paint remaining.
Every statue requires [tex]\frac{1}{20}[/tex] liters of paint for painting.
We have to find how many statues we will be able to paint with this remaining paint.
To get the number of statues,
Number of statues = [tex]\frac{Paint remaining}{Paint required for 1 statue}[/tex]
number of statues = [tex]\frac{\frac{7}{8} }{\frac{1}{20} }[/tex]
= [tex]\frac{35}{2}[/tex] = 17.5
Since the number of statues is not an integer the maximum number of statues that can be painted are 17.
Answer: 35/2
Step-by-step explanation:
Lena ordered 12 copies of the same book for his book club members. The book cost $19 each and the other has a 15 shipping charge what is the total cost of Lena's order$
Answer:
Step-by-step explanation:
Total copies of books ordered by Lena for his book club members is 12. The cost of the book is $19 each. Since the books are the same, the total cost of the books will be
19 × 12 = $228
the order has a $15 shipping charge. It means that the total amount that Lena would pay for the 12 books is total cost of the books + shipping fee. it becomes
228 + 15 =
=$243
PLEASE ANSWER; MAY NOT BE HARD
Find the sum of all positive 3-digit numbers whose last digit is 2
Answer:
Step-by-step explanation:
102+202+302+402+502+602+702+802+902(4518)
+112+212+312+...+ 812+912(4608)
+122+222+322+...+822+922(4698)
+132+232+332+...+932(4788)
..........................................
+192+292+392+...+992(5328)
4518+4608+4698+...+5328
n=10
[tex]s=\frac{10}{2}(4518+5328)\\=5(9846)\\=49230[/tex]
Final answer:
To find the sum of all positive 3-digit numbers ending in 2, we calculate the total for each digit's place and sum them up, resulting in a total sum of 8280.
Explanation:
The problem requires finding the sum of all positive 3-digit numbers with a last digit of 2. To calculate this, we can identify that the first such number is 102 and the last is 992. There are 90 such numbers because they correspond to the tens digit going from 0 to 9 for each of the nine possible hundreds digits (1-9).
Since each number ends in 2, we can think of them as (100x + 10y + 2), where x is the hundreds digit (1 through 9) and y is the tens digit (0 through 9). To find the sum, we calculate the sum of the hundreds digits times their frequency, the sum of the tens digits times their frequency, and add 2 times the number of terms (90). The formula would be:
Sum = (Sum of hundreds values) * 10 * 9 + (Sum of tens values) * 1 * 90 + 2 * 90
The hundreds values are 1 through 9, whose sum is 45, and the tens values are 0 through 9, whose sum is 45 as well. Plugging these values into the formula, we get:
Sum = 45 * 10 * 9 + 45 * 1 * 90 + 2 * 90 = 4050 + 4050 + 180 = 8280.
Find the measure of each angle indicated.
Answer: the measure of the indicated angle is 100 degrees
Step-by-step explanation:
The sum of angles in a triangle is 180 degrees. Let x represent the unknown angle in the bigger triangle. Therefore,
x + 80 + 25 = 180 degrees
x + 105 = 180
x = 180 - 105 = 75 degrees.
Let z represent the other unknown angle in the smaller triangle. Since the sum of the angles on a straight line is 180 degrees, therefore
75 + 55 + z = 180
130 +z = 180
z = 180 - 130 = 50 degrees
Let y represent the unknown angle that we are looking for. Therefore,
50 + y + 30 = 180
80 + y = 180
y = 180 - 80 = 100 degrees
Answer:
55
Step-by-step explanation:
Find the rate of change for x³. You need to work out the change in f(x)=x³ when x is increased by a small number h to x+h. So you will work out f(x+h)-f(x). Then do some algebra to simplify this. Then divide this by h to get the average rate of change of f(x) between x and x+h. The average rate of change of f(x) from x to x+h is:
Answer:
3x² +3xh +h²
Step-by-step explanation:
[tex]\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^3-x^3}{h}=\dfrac{(x^3+3x^2h+3xh^2+h^3)-x^3}{h}\\\\=\dfrac{3x^2h+3xh^2+h^3}{h}=3x^2+3xh+h^2[/tex]
One card is selected from a deck of cards. Find the probability of selecting a black card or a jack.
The probability of selecting a black card or a jack is 15/26.
Given that, one card is selected from a deck of cards.
What is the probability?Probability can be defined as the ratio of the number of favourable outcomes to the total number of outcomes of an event.
We know that, probability of an event = Number of favourable outcomes/Total number of outcomes
Total number of outcomes =52
The number of black cards in a deck =26
The number of jack cards in a deck =4
Probability of an event = 26/52 +4/52
= 30/52
= 15/26
Therefore, the probability of selecting a black card or a jack is 15/26.
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Final answer:
The probability of selecting a black card or a jack from a standard deck of 52 cards is 7/13.
Explanation:
To find the probability of selecting a black card or a jack from a standard deck of 52 cards, we need to consider the number of favorable outcomes and the total number of possible outcomes.
In a deck, there are 26 black cards (13 clubs and 13 spades) and a total of 4 jacks.
However, since two of the jacks are black, we must avoid counting them twice.
The probability becomes:
P(Black or Jack) = P(Black) + P(Jack) - P(Black and Jack)
P(Black) = 26/52, P(Jack) = 4/52, and P(Black and Jack) = 2/52
Thus, the probability is:
P(Black or Jack) = (26/52) + (4/52) - (2/52) = 28/52 = 7/13
Therefore, the probability of selecting a black card or a jack from a standard deck is 7/13.
Mr. Johnson currently has a square garden. It is in his garden and into a range of 5 feet shorter than three times shorter than times it width. He decides that the perimeter should be 70 feet. Determine the dimensions, in feet, of his new garden
Answer:
The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]
Step-by-step explanation:
Given:
Perimeter of new garden = 70 feet.
Let the length of the new garden be 'l'.
Also Let the width of the new garden be 'w'.
We need to find the dimension of new garden.
Now Given:
Length is 5 feet shorter than three times it width.
framing the equation we get;
[tex]l =3w-5 \ \ \ \ equation\ 1[/tex]
Now we know that;
Perimeter of rectangle is equal to twice the sum of length and width.
framing in equation form we get;
[tex]2(l+w)=70[/tex]
Now Diving both side by 2 using Division property of equality we get;
[tex]\frac{2(l+w)}2=\frac{70}{2}\\\\l+w =35[/tex]
Now Substituting equation 1 in above equation we get;
[tex]3w-5+w=35\\\\4w-5=35[/tex]
Adding both side by 5 Using Addition Property of equality we get'
[tex]4w-5+5=35+5\\\\4w=40[/tex]
Now Diving both side by 4 using Division property of equality we get;
[tex]\frac{4w}{4}=\frac{40}{4}\\\\w=10\ ft[/tex]
Now Substituting the value of 'w' in equation 1 we get;
[tex]l =3w-5\\\\l =3\times10-5\\\\l = 30-5\\\\l= 25\ ft[/tex]
Hence The Dimension of new garden is [tex]25 \ feet\ \times 10\ feet.[/tex]
The price of the dinner for the both of them was $30. They tipped their server 20% of that amount. How much did each person pay, if they shared the price of dinner and the tip equally?
Each person will pay 19.5 dollars.
Step-by-step explanation:
Given
Total bill for dinner = b=$30
First of all we will calculate the 30% of dinner bill to find the amount of tip
So,
[tex]Tip = t = 30\%\ of\ 30\\= 0.30*30\\=9[/tex]
the tip is $9
The total bill including tip will be:
[tex]= 30+9 = \$39[/tex]
Two persons have to divide the tip and dinner equally so,
Each person's share = [tex]\frac{39}{2} = 19.5[/tex]
Hence,
Each person will pay 19.5 dollars.
Keywords: Fractions, division
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(NOTE: this symbol "*" will be used to express variables that go under the number, like a reverse exponent.)Find the 4th term in the sequence with the following definition:
a*1=2
a*n=(a*n-1)^2
Answer:
256
Step-by-step explanation:
a₁ = 2
aₙ = (aₙ₋₁)²
a₂ = (a₂₋₁)²
a₂ = (a₁)²
a₂ = (2)²
a₂ = 4
a₃ = (a₃₋₁)²
a₃ = (a₂)²
a₃ = (4)²
a₃ = 16
a₄ = (a₄₋₁)²
a₄ = (a₃)²
a₄ = (16)²
a₄ = 256
HELP!!!!!!!!!
Your goal is to save at least $350.00 over the next 6 weeks. How much money must you save each week in order to meet that goal? Write and solve an inequality.
A) 6+x[tex]\geq[/tex]360;x[tex]\geq[/tex]354
B) 60x[tex]\leq[/tex]360;x[tex]\leq[/tex]10
C) x/6[tex]\leq[/tex]360;x[tex]\leq[/tex]2160
D) 6x[tex]\geq[/tex]360;x[tex]\geq[/tex]60
D) 6x≥360; x≥60
Step-by-step explanation:
The goal is to save at least $350 over the next 6 weeks.
Let the amount to save per week be x
x *6 should be equal or more than the goal.This is
6x ≥ 360
However, dividing the goal amount by number of weeks to get the amount to save per week gives;
360/6 =60
so x≥ 60
The inequality is thus : 6x ≥360;x≥60
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The total cost of producing a type of car is given by C(x)=12000−40x+0.04x2, where x is the number of cars produced. How many cars should be produced to incur minimum cost?
Answer:
Step-by-step explanation:
C'(x)=-40+0.08 x
C'(x)=0 gives
-40+0.08 x=0
x=40/0.08=500
C"(x)=0.08>0 at x=500
so C(x) is minimum if x=500
so 500 cars need to be produced for minimum cost.
or we can solve by completing the squares.
c(x)=12000+0.04(x²-1000 x+250000-250000)
=12000+0.04(x-500)²-0.04×250000
=0.04 (x-500)²+12000-10000
=0.04(x-500)²+2000
c(x) is minimum if x=500
To minimize the cost based on the provided quadratic cost function, 500 cars should be produced.
Explanation:This a problem of optimization in the arena of Calculus. The cost function C(x) = 12000-40x+0.04x2 is a quadratic function, and the minimum cost occurs at the vertex of the parabola described by this function.
For any quadratic function f(x)=ax2 +bx + c, minimum or maximum value occurs at x = -b/2a.
In this case, a = 0.04 and b = -40.
So minimum cost occurs when x = -(-40) / 2*0.04 = 500.
So, to incur minimum cost, 500 cars should be produced.
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Which equilibrium at one atmosphere pressure is correctly associated with Kelvin temperature at which it occur?a. ice-water equilibrium at 0 K
b. ice-water equilibrium at 32 K
c. steam-water equilibrium at 212 K
d. steam-water equilibrium at 373 K
The correct equilibrium at one atmosphere pressure associated with its Kelvin temperature is the steam-water equilibrium at 373 K.
Explanation:The equilibrium at one atmosphere pressure that is correctly associated with the Kelvin temperature at which it occurs is option d. steam-water equilibrium at 373 K. To explain, in the Kelvin temperature scale, the freezing point of water is 273.15 K and the boiling point is 373.15 K, both under standard atmospheric conditions (1 atmosphere pressure). So, at 373 K, the situation would be a steam-water equilibrium, not an ice-water equilibrium as in options a and b. The Kelvin temperature for ice-water equilibrium is 273.15 K and not 0 K and 32 K as stated in options a or b. Similarly, steam-water equilibrium does not occur at 212 K as suggested in option c.
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Option d. steam-water equilibrium at 373 K. The steam-water equilibrium at one atmosphere pressure occurs at 373 K.
The correct answer is:
Steam-water equilibrium at 373 KLet's break down the reasons:
Ice-water equilibrium: This occurs at 0°C, which is 273.15 K, not 0 K or 32 K.Steam-water equilibrium: The boiling point of water is 100°C, which is 373.15 K. At one atmosphere of pressure, water and steam are in equilibrium at this temperature.Thus, the correct association is the steam-water equilibrium occurring at 373 K at 1 atmosphere pressure.
If the following is a polynomial function, then state its degree and leading coefficient. If it is not, then state this fact.
f(x) = 10x5 + 7x4 + 5
A) Degree: 10; leading coefficient: 5
B) Not a polynomial function
C) Degree: 9; leading coefficient: 10
D) Degree: 5; leading coefficient: 10
Answer:
It's D; degree- 5; leading coefficient-10 :)
Step-by-step explanation:
I just took the test and got it right
The given polynomial function f(x) = 10x⁵ + 7x⁴ + 5 has Degree: 5; leading coefficient: 10 which is the correct answer would be option (D).
What is a polynomial?A polynomial is defined as a mathematical expression that has a minimum of two terms containing variables or numbers. A polynomial can have more than one term.
The degree of a polynomial is the highest exponent of the variables in the polynomial. In this case, the highest exponent is 5, so the degree of the polynomial is 5.
The leading coefficient is the coefficient of the term with the highest degree.
In this case, the coefficient of the term with the highest degree (x⁵) is 10, so the leading coefficient is 10. Therefore, the correct answer is D).
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Company X sells leather to company Y for $60,000. Company Y uses the leather to make shoes, selling them to consumers for $180,000. The total contribution to gross domestic product (GDP) is
Answer: $180,000
Step-by-step explanation:
Gross Domestic Product (GDP) is the total monetary value of all finished goods and services made within a country during a specific period. It can be used to estimate the size and growth rate of the country's economy.
In the case above Company X sell leather which is not a finished good to Company Y, so it will not contribute to the gross domestic product (GDP). Company Y sells leather shoes which is a finished good to the consumers, which will contribute to the GDP.
Therefore the total contribution to GDP is $180,000
Really need help with this .
Answer:
Step-by-step explanation:
The attached photo shows the diagram of quadrilateral QRST with more illustrations.
Line RT divides the quadrilateral into 2 congruent triangles QRT and SRT. The sum of the angles in each triangle is 180 degrees(98 + 50 + 32)
The area of the quadrilateral = 2 × area of triangle QRT = 2 × area of triangle SRT
Using sine rule,
q/SinQ = t/SinT = r/SinR
24/sin98 = QT/sin50
QT = r = sin50 × 24.24 = 18.57
Also
24/sin98 = QR/sin32
QR = t = sin32 × 24.24 = 12.84
Let us find area of triangle QRT
Area of a triangle
= 1/2 abSinC = 1/2 rtSinQ
Area of triangle QRT
= 1/2 × 18.57 × 12.84Sin98
= 118.06
Therefore, area of quadrilateral QRST = 2 × 118.06 = 236.12
Answer:
216 square units
Step-by-step explanation:
Apparently, we're supposed to ignore the fact that the given geometry cannot exist. The short diagonal is too short to reach between the angles marked 98°. If Q and S are 98°, then R needs to be 110.13° or more for the diagonals to connect as described.
__
The equal opposite angles of 98° suggests that the figure is symmetrical about the diagonal RT. That being the case, diagonal RT will meet diagonal QS at right angles. Then the area is half the product of the lengths of the diagonals:
(1/2)×18×24 = 216 . . . . square units
_____
In a quadrilateral, the area can be computed as half the product of the diagonals and the sine of the angle between them. Here, we have assumed the angle to be 90°, so the area is simply half the product of diagonal measures.
On a coordinate plane, a dashed straight line has a negative slope and goes through (0, 2) and (4, 0). Everything below and to the left of the line is shaded. Which point is a solution to the linear inequality y < Negative one-halfx + 2? (2, 3) (2, 1) (3, –2) (–1, 3)
Answer:
Option C.
Step-by-step explanation:
A dashed straight line has a negative slope and goes through (0, 2) and (4, 0).
The given inequality is
[tex]y<-\dfrac{1}{2}x+2[/tex]
We need find the point which is a solution to the given linear inequality.
Check the given inequality for point (2, 3).
[tex]3<-\dfrac{1}{2}(2)+2[/tex]
[tex]3<1[/tex]
This statement is false. Option 1 is incorrect.
Check the given inequality for point (2, 1).
[tex]1<-\dfrac{1}{2}(2)+2[/tex]
[tex]1<1[/tex]
This statement is false. Option 2 is incorrect.
Check the given inequality for point (3, -2).
[tex]-2<-\dfrac{1}{2}(3)+2[/tex]
[tex]-2<0.5[/tex]
This statement is false. Option 3 is correct.
Check the given inequality for point (-1,3).
[tex]3<-\dfrac{1}{2}(1)+2[/tex]
[tex]3<1.5[/tex]
This statement is false. Option 4 is incorrect.
Therefore, the correct option is C.
Answer:
C
Step-by-step explanation:
(2,1)
The cost of a peanut butter bar is $0.07 more than the cost of a chocolate bar. If you buy 5 peanut butter bars and 6 chocolate bars, the total cost is $6.40. How much does the chocolate bar cost?
$0.61
$0.55
$0.54
$0.62
Hello!
To be quick and simple, your answer would be $0.55
The cost of the chocolate bar in the given scenario is $0.55. This was determined by solving a two-variable system of linear equations from the information provided.
Explanation:This problem is a classic example of a system of linear equations, specifically two-variable linear equations. Here, we need to find the cost of one chocolate bar and one peanut butter bar, and we have two pieces of information that can be translated into equations. The first information is that a peanut butter bar costs $0.07 more than a chocolate bar. The second is that 5 peanut butter bars and 6 chocolate bars total $6.40. We'll use these equations to solve for the variables.
Let's denote the cost of the chocolate bar as x and the cost of the peanut butter bar as y. Then, from the information given, we can form two equations:
y = x + $0.075y + 6x = $6.40Substitute the first equation into the second to solve for x:
5(x + $0.07) + 6x = $6.405x + $0.35 + 6x = $6.4011x + $0.35 = $6.4011x = $6.05x = $0.55So the cost of the chocolate bar is $0.55.
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Howdy! Id love to have these questions answered asap! Thank you for the help!
1) Which angle is not coterminal to 120 degrees?
A. 840
B. -180
C. 480
2) Use the unit circle and the reference angle to determine which of the following trigonometric values is correct when theta = -90
A. Cos theta = undefined
B. Sin theta = -1
C. Tan = 0
Answer:
1. B.
2. B.
Step-by-step explanation:
Trigonometry
1) Coterminal angles can be found by adding or subtracting 360° (or 2\pi radians) to a given angle. If we have 120°, adding 360° gives 480°, adding again 360° gives 840°. There is no way to get -180°, so this option is not a coterminal angle to 120°
2)
A. [tex]Cos (-90^o)=0[/tex], and not undefined
B. [tex]Sin (-90^o)=-1[/tex]. This is correct
C. [tex]Tan (-90^o)[/tex] is undefined, not zero
Thus the only correct option is B.
James and Lucas competed in climbing a mountain and coming back. James climbed at an average rate of 3 mph and ran back at 4 mph. Lucas climbed at an average rate of 2 mph and ran back at 5 mph. If the path from the bottom to the top of the mountain was 2 miles, how much time did each take to finish and who was faster? Answer:
It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.
Step-by-step explanation:
Given,
Climbing rate of James = 3 mph
Running back rate of James = 4 mph
Climbing rate of Lucas = 2 mph
Running back rate = 5 mph
Total distance = 2 miles
We know that;
Distance = Speed * Time
As we have to find, we will rearrange the formula in terms of time
[tex]Time=\frac{Distance}{Speed}[/tex]
Time took by James for climbing = [tex]\frac{2}{3}\ hours[/tex]
Time took for running back = [tex]\frac{2}{4}\ hours[/tex]
Total time = [tex]\frac{2}{3}+\frac{2}{4}=\frac{8+6}{12}=\frac{14}{12}[/tex]
Total time taken by James = [tex]\frac{7}{6}\ hours[/tex]
1 hour = 60
Total time taken by James = [tex]\frac{7}{6}*60=70\ minutes[/tex]
Time took by Lucas for climbing = [tex]\frac{2}{2}=\ 1\ hour[/tex]
Time took by Lucas for climbing = 60 minutes
Time took on return = [tex]\frac{2}{5} of\ an\ hour=\frac{2}{5}*60=24\ minutes[/tex]
Total time taken by Lucas = 60+24 = 84 minutes
Therefore,
It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.
Keywords: distance, speed
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Answer:
Step-by-step explanation:
It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.
Step-by-step explanation:
Given,
Climbing rate of James = 3 mph
Running back rate of James = 4 mph
Climbing rate of Lucas = 2 mph
Running back rate = 5 mph
Total distance = 2 miles
We know that;
Distance = Speed * Time
As we have to find, we will rearrange the formula in terms of time
Time took by James for climbing =
Time took for running back =
Total time =
Total time taken by James =
1 hour = 60
Total time taken by James =
Time took by Lucas for climbing =
Time took by Lucas for climbing = 60 minutes
Time took on return =
Total time taken by Lucas = 60+24 = 84 minutes
Therefore,
It took 70 minutes for James to finish and 84 minutes for Lucas to finish and James was faster.®
The third term in a geometric sequence is -81. The common ratio is 1/3
What is the second term of the sequence?
If you answer, can you explain it?
Answer:
Step-by-step explanation:
The formula for the nth term of a geometric sequence is expressed as follows
Tn = ar^(n - 1)
Where
Tn represents the value of the nth term of the sequence
a represents the first term of the sequence.
n represents the number of terms.
From the information given,
r = 1/3
T3 = - 81
n = 3
Therefore,
- 81 = a× 1/3^(3 - 1)
-81 = a × (1/3)^2
-81 = a/9
a = -81 × 9 = - 729
The exponential equation for this sequence is written as
Tn = - 729 * (1/3)^(n-1)
Therefore, to find the second term,T2, n = 2. It becomes
T2 = - 729 * (1/3)^(2-1)
T2 = - 729 * (1/3)^1
T2 = - 729 * (1/3)
T2 = - 243
What does the fundamental theorem of algebra illustrate?
Answer:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
Step-by-step explanation:
The fundamental theorem of algebra guarantees that a polynomial equation has the same number of complex roots as its degree.
We have to find the roots of this given equation.
If a quadratic equation is of the form [tex]ax^{2}+bx +c=0[/tex]
Its roots are [tex]\frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex] and [tex]\frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]
Here the given equation is [tex]2x^{2}-4x-1[/tex] = 0
a = 2
b = -4
c = -1
If the roots are [tex]x_{1} and x_{2}[/tex], then
[tex]x_{1}[/tex] = [tex]\frac{-2+\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{24}}{4}[/tex]
= [tex]\frac{2+\sqrt{6} }{2}[/tex]
[tex]x_{2}[/tex] = [tex]\frac{-2-\sqrt{(-4)^{2}-4\times 2\times (-1)}}{2\times 2}[/tex]
= [tex]\frac{4 +\sqrt{8}}{4}[/tex]
= [tex]\frac{2-\sqrt{6} }{2}[/tex]
These are the two roots of the equation.
a set of cards includes 15 yellow cards, 10 green cards and 10 blue cards. find the probability of each event when a card is chosen at random not yeallow or green
Answer:
P(not yellow or green)=\frac{2}{7}[/tex]
Step-by-step explanation:
a set of cards includes 15 yellow cards, 10 green cards and 10 blue cards
Total cards= 15 yellow + 10 green + 10 blue = 35 cards
Probability of an event = number of outcomes divide by total outcomes
number of outcomes that are not yellow or green are 10 blue cards
So number of outcomes = 10
P(not yellow or green)= [tex]\frac{10}{35} =\frac{2}{7}[/tex]
The probability of choosing a card that is neither yellow nor green from the set is 2/7, as there are 10 blue cards and a total of 35 cards.
Explanation:The question asks for the probability of choosing a card that is neither yellow nor green from a set containing 15 yellow cards, 10 green cards, and 10 blue cards. To find this probability, we must consider only the blue cards, as they are not yellow or green. The total number of blue cards is 10, and the total number of cards is 35 (since 15 + 10 + 10 = 35).
To calculate the probability, we use the formula:
P(Blue card) = Number of blue cards / Total number of cards = 10 / 35 = 2/7
Thus, the probability of randomly choosing a card that is not yellow or green (i.e., a blue card) is 2/7.
On a certain sight-seeing tour, the ratio of the number of women to the number of children was 5 to 2. What was the number of men on the sight-seeing tour?
(1) On the sight-seeing tour, the ratio of the number of children to the number of men was 5 to 11.
(2) The number of women on the sight-seeing tour was less than 30.
Answer:
There were 22 men in the sight-seeing tour
Step-by-step explanation:
If the number f women on the sight-seeing tour was less than 30, the closest number that can be divided by 5 is 25, so we suppose that there were 25 women, so with the ratio women(5): children(2) we can apply a rule of three, so we multiply 25*2 divided by 2, now we know that there were 10 children, now with the ratio of men(11): children(5) we apply another rule of three, and multiply 10*11 and then divide it by 5, and now we know that there were 22 men in the sight-seeing tour.
Find the imaginary part of\[(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)^6.\]
Answer:
The imaginary part is 0
Step-by-step explanation:
The number given is:
[tex]x=(\cos(12)+i\sin(12)+ \cos(48)+ i\sin(48))^6[/tex]
First, we can expand this power using the binomial theorem:
[tex](a+b)^k=\sum_{j=0}^{k}\binom{k}{j}a^{k-j}b^{j}[/tex]
After that, we can apply De Moivre's theorem to expand each summand:[tex](\cos(a)+i\sin(a))^k=\cos(ka)+i\sin(ka)[/tex]
The final step is to find the common factor of i in the last expansion. Now:
[tex]x^6=((\cos(12)+i\sin(12))+(\cos(48)+ i\sin(48)))^6[/tex]
[tex]=\binom{6}{0}(\cos(12)+i\sin(12))^6(\cos(48)+ i\sin(48))^0+\binom{6}{1}(\cos(12)+i\sin(12))^5(\cos(48)+ i\sin(48))^1+\binom{6}{2}(\cos(12)+i\sin(12))^4(\cos(48)+ i\sin(48))^2+\binom{6}{3}(\cos(12)+i\sin(12))^3(\cos(48)+ i\sin(48))^3+\binom{6}{4}(\cos(12)+i\sin(12))^2(\cos(48)+ i\sin(48))^4+\binom{6}{5}(\cos(12)+i\sin(12))^1(\cos(48)+ i\sin(48))^5+\binom{6}{6}(\cos(12)+i\sin(12))^0(\cos(48)+ i\sin(48))^6[/tex]
[tex]=(\cos(72)+i\sin(72))+6(\cos(60)+i\sin(60))(\cos(48)+ i\sin(48))+15(\cos(48)+i\sin(48))(\cos(96)+ i\sin(96))+20(\cos(36)+i\sin(36))(\cos(144)+ i\sin(144))+15(\cos(24)+i\sin(24))(\cos(192)+ i\sin(192))+6(\cos(12)+i\sin(12))(\cos(240)+ i\sin(240))+(\cos(288)+ i\sin(288))[/tex]
The last part is to multiply these factors and extract the imaginary part. This computation gives:
[tex]Re x^6=\cos 72+6cos 60\cos 48-6\sin 60\sin 48+15\cos 96\cos 48-15\sin 96\sin 48+20\cos 36\cos 144-20\sin 36\sin 144+15\cos 24\cos 192-15\sin 24\sin 192+6\cos 12\cos 240-6\sin 12\sin 240+\cos 288[/tex]
[tex]Im x^6=\sin 72+6cos 60\sin 48+6\sin 60\cos 48+15\cos 96\sin 48+15\sin 96\cos 48+20\cos 36\sin 144+20\sin 36\cos 144+15\cos 24\sin 192+15\sin 24\cos 192+6\cos 12\sin 240+6\sin 12\cos 240+\sin 288[/tex]
(It is not necessary to do a lengthy computation: the summands of the imaginary part are the products sin(a)cos(b) and cos(a)sin(b) as they involve exactly one i factor)
A calculator simplifies the imaginary part Im(x⁶) to 0
Two school buses leave the same school at the same time, but are headed in opposite directions. One bus traveled 45 miles per hour and the other bus traveled 40 miles per hour.
How long until the buses are 170 miles apart?
Answer:
2 hours
Step-by-step explanation:
45x + 40x = 170
85x = 170
x = 2
Answer:
40 mph: 4 hours 15 minutes
45 mph: 3 hours 46 minutes 40 seconds
Step-by-step explanation:
The rate of separation = 45 + 40 = 85 mph
speed = distance / time
85 = 170 / t
t= 170/85 hour
Consider the vector field ????(x,y,z)=(5z+y)????+(4z+x)????+(4y+5x)????.
a. Find a function ???? such that ????=∇???? and ????(0,0,0)=0.
b. Suppose C is any curve from (0,0,0) to (1,1,1). Use part a. to compute the line integral ∫????????⋅????????.
Answer:
a) [tex]5xz + xy + 4yz[/tex]
b) 10
Step-by-step explanation:
a) Here [tex]F(x,y,z)=(5z+y)i+(4z+x)j+(4y+5x)k[/tex]
Since the case [tex]F[/tex] = ∇[tex]f[/tex] holds, then
∇[tex]f = f_xi+f_yj+f_zk[/tex] = [tex](5z+y)i+(4z+x)j+(4y+5x)k[/tex]
So, [tex]f_x = 5z + y[/tex]
If we integrate [tex]f_x[/tex] with respect to x, we will get an integration constant C which is also a function that depends to y and z.
Hence,
[tex]f = \int f_xdx = 5xz + xy + g(y,z)[/tex]
Now we need to find g(y,z).
So first let's take the derivative of g(y,z) with respect to y.
[tex]f_y = x + g_y(y,z) = 4z + x[/tex]
Hence, [tex]g_y(y,z) = 4z[/tex]
So now, if we integrate [tex]g_y[/tex] with respect to y to find g(y,z)
[tex]g = \int g_ydy = 4yz + C[/tex]
Thus,
[tex]f = 5xz + xy + g(y,z) = 5xz + xy + 4yz + C[/tex]
And since [tex]f(0,0,0)=0[/tex], then [tex]C=0[/tex]
Thus,
[tex]f = f(x,y,z) = 5xz + xy + 4yz[/tex]
b) By the Fundamental Theorem of Line Integrals, we know that
[tex]\int\limits^a_b F. dr = F[r(b)]-F[r(a)][/tex]
Hence,
[tex]\int\limits^a_b F. dr = F(1,1,1)-F(0,0,0) =[(5+1+4)-(0+0+0)]=10[/tex]
To find ????, solve the system of partial differential equations. Use the function ???? from part a to compute the line integral.
Explanation:To find a function ???? such that ????=∇???? and ????(0,0,0)=0, we can solve the system of partial differential equations. Let ????=????????+????????+????????, then compute the partial derivatives of ???? with respect to each variable. Equating these partial derivatives to the given vector field components, we can solve for the unknown function ???? and find its value at the point (0,0,0).
To compute the line integral ∫????????⋅????????, we can use the fundamental theorem of calculus for line integrals. Since ????(x,y,z) is the gradient of ????, the line integral is equal to the change in ???? along the curve C from (0,0,0) to (1,1,1). We can use the function ???? found in part a to evaluate this change in ????.
Learn more about Partial differential equations here:https://brainly.com/question/36613189
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Alice sleeps an average of 9 hours per night. A cat can sleep up to 20 hours per day. About how many more hours does a cat sleep in 1 month that Alice?
Answer:
Cat sleeps 420 hours than Alice in a month
Step-by-step explanation:
Given:
Number of hours Alice sleeps per night = 9 hours
Number of hours Cat sleeps per night = 20 hours
To Find:
How many more hours does a cat sleep in 1 month that Alice
Solution:
Let
The total number of hours for which Alice Sleeps in one month be x
The total number of hours for which Cat sleeps in one month be y
Step 1: Number of hours for which Alice Sleeps in one month
X = number of days in a month X Number of hours Alice sleeps per night
X = 30 X 9
X = 180 Hours
Step 2: Number of hours for which Cat Sleeps in one month
y= number of days in a month X Number of hours cat sleeps per night
y = 30 X 20
y = 600 Hours
Now ,
=> y – x
=>600 – 180
=>420 hours
13. Write an equation for the given function given the amplitude, period, phase shift, and vertical shift.
amplitude: 4, period 4 phase shift = vertical shift = -2
Answer:
[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]
Step-by-step explanation:
Let's start with the original function.
[tex]y=a sin\frac{2\pi t}{T}[/tex]
We can immediately fill in the amplitude 'a' and period 'T' , as the question defines these for us, and provides values for 'a' and 'T', 4 and 4[tex]\pi[/tex] respectively.
[tex]y=4sin(\frac{2\pi t}{4\pi } )[/tex]
Now we only have phase shift and vertical shift to do. Vertical shift is very easy, you can just add it to the end of the right side of the expression. A positive value will shift the graph up, while a negative value will move shift the graph down. We have '-2' as our value for vertical shift, so we can add that on as so:
[tex]y=4sin(\frac{2\pit }{4\pi } )-2[/tex]
Now phase shift the most complicated of the transformations. Basically, it is just movement left or right. A negative phase shift moves the graph right, a positive phase shift moves the graph left (I know, confusing!). Phase shift applies directly to the x variable, or in this case the t variable. To achieve a -4/3 pi phase shift, we need to input +4/3 pi into the function, because of the aforementioned negative positive rule. Here is what the function looks like with the correct phase shift:
[tex]y=4sin(\frac{2\pi(t+\frac{4}{3}\pi ) }{4\pi } )-2[/tex]
This function has vertical shift -2, phase shift -4/3 [tex]\pi[/tex], amplitude 4, and period 4[tex]\pi[/tex].
Desmos.com/calculator is a great tool for learning about how various parts of an equation affect the graph of the function, If you want you can input each step of this problem into desmos and watch the graph change to match the criteria.
Jacob and Ayden work at a dry cleaners ironing shirts. Jacob can iron 25 shirts per hour, and Ayden can iron 35 shirts per hour. Ayden worked twice as many hours as Jacob and they ironed 380 shirts between them. Determine the number of hours Jacob worked and the number of hours Ayden worked.
Answer:
Step-by-step explanation:
Start with the unknown, which is the number of hours J worked and the number of hours A worked. If A worked twice as many hours as J, then J worked x hours and A worked 2x hours. If J can iron 25 shirts per hour, x, then the number of shirts he can iron in his shift is 25x. If A can iron 35 shirts per hour, x, then the number of shirts he can iron in his shift is 35(2x). The number of shirts they iron together in x hours is
25x + 35(2x) = 380 and
25x + 70x = 380 and
95x = 380 so
x = 4
This means that J worked 4 hours and A worked 8 hours.
A baker uses 2 1/3 cups of cookie dough and 1/4 cup of chocolate chips to make 10 cookies. If the baker has 3 cups of chocolate chips, how much dough will he need
Answer: he will need 28 cups of dough for 3 cups of chocolate.
Step-by-step explanation:
The baker uses 2 1/3 cups of cookie dough and 1/4 cup of chocolate chips to make 10 cookies. Converting 2 1/3 cups of cookie dough into improper fraction, it becomes 7/3 cups of cookie dough.
It means that for every 1/4 cup of chocolate, 7/3 cups of cookie dough is needed.
Let x represent the amount of cookie dough needed for 3 cups of chocolate chips.
If 7/3 dough = 1/4 cup of chocolate
x dough = 3 cups of chocolate
x × 1/4 =7/3 × 3
x/4 = 7
x = 7×4 = 28 cups of dough