Answers
Trigonometric Equations
Solve:
[tex]9\tan ^3x=3\tan x[/tex]In the interval [0,2pi)
We have to find all the values of x that make equality stand. First, divide by 3:
[tex]3\tan ^3x=\tan x[/tex]Subtract tan x
[tex]3\tan ^3x-\tan x=0[/tex]Factor tan x out:
[tex]\tan x(3\tan ^2x-1)=0[/tex]One solution comes immediately:
tan x = 0
There are two angles whose tangent is 0:
[tex]x=0\text{ , x=}\pi[/tex]The other solutions come when equating:
[tex]3\tan ^2x-1=0[/tex]Adding 1, and dividing by 3:
[tex]\tan ^2x=\frac{1}{3}[/tex]Taking the square root:
[tex]\tan x=\sqrt[\square]{\frac{1}{3}}=\pm\frac{\sqrt[]{3}}{3}[/tex]The positive answer gives us two solutions:
[tex]\tan x=\frac{\sqrt[]{3}}{3}[/tex]x=pi/6 and x=7pi/6
The negative answer also gives us two solutions:
[tex]\tan x=-\frac{\sqrt[]{3}}{3}[/tex]x=5pi/6, 11pi/6
Summarizing the solutions are:
{
Related Questions
The base of the pyramid is a square.158Perimeter of the base =Area of the base =Slant height =Lateral area =square unitsSurface area =square unitsBlank 1:Blank 2:Blank 3:Blank 4:Blank 5:
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The pyramid has a square base.
i) Perimeter of the base implies the perimeter of the square.
Perimeter of a square is given as:
[tex]\begin{gathered} P=4L \\ L=8 \\ P=4\times8 \\ =32 \end{gathered}[/tex]ii) Area of the base implies the area of the square.
Area of a square is given as:
[tex]\begin{gathered} A=L^2 \\ L=8 \\ A=8^2 \\ A=64 \end{gathered}[/tex]iii) The slant height can be obtained by using the pythagoras theorem.
From the diagram, the hypotenuse side is the unknown slant height, the other two(2) sides are of length 15 and 8.
Thus, we have:
[tex]\begin{gathered} H^2=O^2+A^2\text{ (Pythagoras theorem)} \\ H^2=15^2+8^2 \\ H^2=225+64 \\ H^2=289 \\ H=\sqrt[]{289} \\ H=17 \end{gathered}[/tex]Hence, the slant height is 17
iv) The lateral area of a square pyramid is the sum of the areas of all its 4 triangular side faces.
The area of a triangle is given as:
[tex]\begin{gathered} A=\frac{1}{2}\times Base\times Height \\ \text{Base}=8;\text{ Height=15} \\ A=\frac{1}{2}\times8\times15 \\ A=\frac{120}{2} \\ A=60 \\ \text{Hence, the lateral area is 4}\times60\text{ ( since there are 4 triangular faces)} \\ \text{Lateral area= 240} \end{gathered}[/tex]v) The surface area is the sum of the lateral area and the base area.
The lateral area has been obtained to be 240.
The base area has been obtained to be 64.
Thus, the surface area = 240 + 64
Hence, the surface area is 304
principal hernadez bought 640 t shirts for the student in his school. the t shirts came in packs of 20. how many packs did he buy
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We know that
• Hernandez bought 640 t-shirts.
,• They came in packs of 20.
To know how many packs he bought, we just need to divide
[tex]\frac{640}{20}=32[/tex]Therefore, principal Hernandez bought 32 packs of t-shirts.4^4x - 1 = 8^2xSolve for x.
Answers
x = 1
Explanation:
[tex]4^{4x-1}=8^{2x}[/tex]Make the base have the same number:
8 = 2³
4 = 2²
The base becomes 2
[tex]2^{2(4x-1)}=2^{3(}^{2x)}[/tex]Then we simplify:
[tex]\begin{gathered} \text{The base cancels out:} \\ 2(4x-1)\text{ = 3(2x)} \\ 8x\text{ - 2 = 6x} \\ \text{collect like terms: } \\ 8x\text{ - 6x = 2} \\ 2x\text{ = 2} \\ x\text{ = 2/2} \\ x\text{ = 1} \end{gathered}[/tex]Therefore, x = 1
check:
[tex]\begin{gathered} 4^{4(1)-1}=8^{2(1)} \\ 4^{4-1}=8^2 \\ 4^3\text{ }=8^2 \\ 4\times4\times4=\text{ 64} \\ 8\times8\text{ = 64} \\ 64\text{ = 64 (x=1 is correct)} \end{gathered}[/tex]Business that offers repayment plan for purchases are required by law to disclose the interest rate but that doesn’t mean they go out of their way to let you know what it is you have to read all the paperwork find the interest rate for the following purchaseTo finance a new laptop Emily is offered a five year payment plan with low monthly payments of $31 and50 Cent the cost of the laptop is $884.43 including tax round to the decimal place if necessary
Answers
Answer:
The number of years for the repayment is
[tex]t=5years[/tex]The amount to be repaid back monthly is
[tex]\text{ \$31.50}[/tex]The cost of the laptop given is
[tex]\text{ P=\$884.43}[/tex]Step 1:
Calculate the total amount of money to be repaid after 5 years
[tex]\begin{gathered} Amount\text{ repayable=monthly payments}\times number\text{ of years}\times12 \\ Amount\text{ repayable=31.50}\times5\times12 \\ Amount\text{ repayable=1890} \end{gathered}[/tex]Step 2:
Calculate the interest on the laptop
[tex]\begin{gathered} interest=Amount\text{ repayable-cost of the laptop} \\ interest=1890-884.43 \\ interest=1005.57 \end{gathered}[/tex]Step 3:
To calculate the interest rate, we will use the formula below
[tex]\begin{gathered} I=\frac{PRT}{100} \\ \frac{100I}{PT}=\frac{PRT}{PT} \\ R=\frac{100I}{PT} \\ T=5,P=884.43,I=1005.57 \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} R=\frac{100I}{PT} \\ R=\frac{100\times1005.57}{884.43\times5} \\ R=\frac{100557}{4422.15} \\ R=22.7\% \end{gathered}[/tex]Hence,
The interest rate will be
[tex]\Rightarrow22.7\%[/tex]For the dinner special at a restaurant, the customer must choose an appetizer, a salad, an entree, a side dish, and a dessert.Here are the items to choose from.ChoicePossible items Appetizer Buffalo wings, Cheese sticks, Potato skins, Nachos Salad Caesar, Southwest, Garden Entree Lamb, Roast turkey, Salmon, Pot roast, Grilled chicken Side dish Mixed vegetables, Baked potato, Rice, French fries, Corn Dessert Chocolate cake, Brownies, Cheesecake How many dinner specials are possible?
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Solution
Step 1:
There are 4 Appetizers, 3 Salad, 5 Entrees, 5 Side dishes, and 3 Desserts.
Step 2:
Possible dinner special
[tex]\begin{gathered} =\text{ 4 }\times\text{ 3 }\times\text{ 5 }\times\text{ 5 }\times\text{ 3} \\ \\ =\text{ 900 } \end{gathered}[/tex]Final answer
900 specials possible dinner.
900
What type of number is 27t?Choose all answers that apply:Whole numberBIntegerRationalDIrrational
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2π is an irrational number as π is irrational. An irrational number is any real number that cannot be expressed as the quotient of two integers.
The answer is the option D.
is 3/3 x 3/4 less than, greater than, or equal to 3/4
Answers
Given data:
The given expression is 3/3 x 3/4.
The given expression can be written as,
[tex]\begin{gathered} \frac{3}{3}\times\frac{3}{4}=1\times\frac{3}{4} \\ =\frac{3}{4} \end{gathered}[/tex]Thus, the expression 3/3 x 3/4 is equal to 3/4.
solve the system of equations by the addition method 5x + 2y = 64x - 3y = 14
Answers
Given the system
[tex]\begin{gathered} 5x+2y=6 \\ 4x-3y=14 \end{gathered}[/tex]To solve it you have apply the addition method, this means that you have to add both equations.
You have to keep in mind that is only valid to add the terms that have the same variables.
The solution is 9x-y=20
Consider the following types of data that were obtained from a random sample of 49 credit card accounts. Identify all the averages (mean, median, or mode) that can be used to summarize the data.(Select all that apply.)(a) Outstanding balance on each account.A. modeB. medianC. mean(b) Name of credit card (e.g., MasterCard, Visa, American Express, etc.).A. modeB. medianC. mean(c) Dollar amount due on next payment.A. modeB. medianC. mean
Answers
Answer:
(a)A,B.C
(b)A
(c)A, B, C
Explanation:
Part A
The outstanding balance on each account can be summarized using the mean, median, and mode since it is quantitative data.
Options A, B, and C are correct here.
Part B
The name of credit cards can only be summarized by frequency. That is, how many belong to each company (i.e. MasterCard, Visa, American Express, etc.)
The only correct option here is Mode. (Option A).
Part C
The dollar amount due on the next payment is a numerical data, we can find the mean, median, and mode for any numerical data.
Options A, B, and C are correct here.
12. A block of ice is in theshape of a cube with sidelengths of 1.8 inches. The icehas a density of 876 kg percubic inch. Find the mass ofthe block of ice to the nearesttenth of a kg.
Answers
SOLUTION:
Step 1:
In this question, we are given the following:
A block of ice is in the shape of a cube with side lengths of 1.8 inches.
The ice has a density of 876 kg per cubic inch.
Find the mass of the block of ice to the nearest tenth of a kg.
Step 2:
The details of the solution are as follows:
Recall that:
[tex]\begin{gathered} Density\text{ = }\frac{Mass}{Volume} \\ where\text{ Volume of the cube = length x length x length} \\ Volume\text{ of the cube = 1. 8 x 1. 8 x 1. 8 = 5.832 inches}^3 \end{gathered}[/tex][tex]Density\text{ = 876 kg per cubic inch}[/tex][tex]\begin{gathered} Therfore,\text{ mass of of the block of ice = Density x volume} \\ Mass\text{ of the bolock of ice = 876 x 5.832 = 5108. 832 kg}\approx\text{ 5108.8 kg} \\ (\text{ to the nearest tenth of a kg \rparen} \end{gathered}[/tex]CONCLUSION:
The mass of the block of ice to the nearest tenth of a kg =
[tex]5108.8\text{ kg }[/tex]
Find the lowest common multiple of each group. show the factors you used.1) 5,22 and 121
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Solution:
Given:
[tex]5,22,121[/tex]By prime factorization method:
[tex]\begin{gathered} 5=5 \\ 22=2\times11 \\ 121=11\times11 \\ \\ Hence,\text{ the LCM is:} \\ 2\times5\times11\times11 \\ LCM=1210 \end{gathered}[/tex]Therefore, the LCM is 1,210
Using a deck of 52 standard playing cards, find the probability for P(Clubs or Spades).
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Solution
There are 13 clubs and 13 spade
=> probability for P(Clubs or Spades) = 13/52 + 13/52
[tex]\frac{13}{52}+\frac{13}{52}\text{ =}\frac{26}{52}=\frac{1}{2}[/tex]simplify the expression x² - 3xy - 5xy - 7y² + 4x² + 8y²
Answers
x² - 3xy - 5xy - 7y² + 4x² + 8y²
First, let's re-arrange
x² + 4x² - 3xy - 5xy - 7y² + 8y²
5x² - 8xy + y²
Please make this simple and easy, and help me quickly, thank you!
Answers
Given:
Total distance = 60 km
Upstream time = 5 hours
Downstream time = 3 hours
Find-:
The rate of boat in still water.
Explanation-:
Let
The speed of the boat in still water is"x"
The speed of the water is "y"
For upstream ( Against the current )
Speed of boat is:
[tex]=x-y[/tex][tex]\begin{gathered} \text{ Distance }=60\text{ km} \\ \\ \text{ Time }=5\text{ hours} \end{gathered}[/tex]Use the formula:
[tex]\text{ Speed}=\frac{\text{ Distance}}{\text{ Time}}[/tex]For upstream speed is:
[tex]\begin{gathered} \text{ Speed}=\frac{\text{ Distance}}{\text{ Time}} \\ \\ x-y=\frac{60}{5} \\ \\ x-y=12..............(1) \end{gathered}[/tex]For downstream,
[tex]\text{ Speed}=x+y[/tex][tex]\begin{gathered} \text{ Distance}=60\text{ km} \\ \\ \text{ Time}=3\text{ hours} \end{gathered}[/tex]Use the formula of distance
[tex]\begin{gathered} \text{ Speed}=\frac{\text{ Distance}}{\text{ Time}} \\ \\ x+y=\frac{60}{3} \\ \\ x+y=20 \\ \\ y=20-x................(2) \end{gathered}[/tex]From eq(2) put the value of "y" in eq(1) then the value is:
[tex]\begin{gathered} x-y=12...........(1) \\ \\ y=20-x...............(2) \end{gathered}[/tex][tex]\begin{gathered} x-y=12 \\ \\ x-(20-x)=12 \\ \\ x-20+x=12 \\ \\ 2x-20=12 \\ \\ 2x=12+20 \\ \\ 2x=32 \\ \\ x=\frac{32}{2} \\ \\ x=16 \end{gathered}[/tex]The rate of the boat in still water is 16 km/hour
1 1/3 divided by 40 using the algorithm
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the given expression is,
1 1/3 = 4/3
[tex]\frac{\frac{4}{3}}{40}=\frac{4}{120}[/tex]Please see the picture for the question and my answer is wrong
Answers
Given sentence:
Five more than half the input is the output
y is the output, x is the input
To interpret the sentence, we should break it into parts:
-half the input: we half the input
- 5 more than we add 5 to the new input
- the sum of these is the output
The equation that represents the given sentence:
[tex]y\text{ = 5 + }\frac{1}{2}x[/tex]Answer:
[tex]y\text{ = 5 + }\frac{1}{2}x[/tex]Special Right Triangles(Radical Answer) Problem I would appreciate the help:0
Answers
In order to determine the value of x, use the
Name Danielle Klein Datealillar S4: Linear Equations, Functions, and Inequalities T6: Finding Solution Sets to Systems of Equations Using Substitution and Graphing Independent Practice 1. Last Monday, two law students met up at Café Literatura after school to read the pages they were assigned in the Legal Methods class. Alejandro can read 1 page per minute, and he has read 28 pages so far. Carly, who has a reading speed of 2 pages per minute, has read 12 pages so far. Part A: Define the variables and write two equations to represent the number of pages that each student read. DE 4 Variables: X-Minutes they real they head Alejandro:X-XF28 x= Number of payes Carly:apGraph both equations , find when Alejandro has read more pages than Carly, and when they have read the same amount of pages.
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Let t be the time and P be the number of pages that each students has read. In both cases, the equation that relates P and t is a linear equation. The slope-intercept form of the equation of a line is:
[tex]y=mx+b[/tex]Where m represents the rate of change of y with respect to x and b represents the initial value when x=0.
In this case, where P represents the number of pages and t represents the time, the relation can be written as:
[tex]P=mt+b[/tex]Adjust the paramenters m and b for each student.
Since Alejandro can read 1 page per minute, then the rate of change of the number of pages with respect to time is 1. Since he has read 28 pages so far, then the initial value is 28. The number of pages that Alejandro reads, is:
[tex]P=t+28[/tex]Since Carly can read 2 pages per minute, the rate of change is 2. Since she has read 12 pages so far, the initial value is 12. The equation for Carly, is:
[tex]P=2t+12[/tex]To graph each equation, evaluate it on two different values of t to find the corresponding values of P.
For Alejandro, let's use t=0 and t=1:
[tex]\begin{gathered} t=0\Rightarrow P=0+28\Rightarrow P=28 \\ t=1\Rightarrow P=1+28\Rightarrow P=29 \end{gathered}[/tex]Plot the points (0,28) and (1,29) in a coordinate plane:
Then, draw a line through them:
Do the same for Carly's equation. We can see that two points on the line would be (0,12) and (1,14):
To find when has Alejandro read more pages than Carly, write an inequality. After t minutes, Alejandro has read t+28 pages, and Carly has read 2t+12 pages. We want t+28 to be greater than 2t+12, then:
[tex]t+28>2t+12[/tex]Substract t from both sides:
[tex]\begin{gathered} t+28-t>2t+12-t \\ \Rightarrow28>t+12 \end{gathered}[/tex]Substract 12 from both sides:
[tex]\begin{gathered} 28-12>t+12-12 \\ \Rightarrow16>t \end{gathered}[/tex]Therefore, whenever t is less than 16 minutes, Alejandro has read more pages than Carly.
Notice that if we replace the ">" sign for a "=" sign, we would find that they have read the same amount of pages when t=16 minutes.
How many solutions does the graph have?
Answers
Answer:
One solution
Step-by-step explanation:
The lines only intersect one time, this indicates that there are only one solution
i need help please.Which of the following expressions are equivalent to 7x + 14 − 3x + 12?1. 21x + 8x2. 7x − 3x + 14 + 123. 4x + 14 + 124. 4x − 25. 7x + 26 − 3x
Answers
The expression is given as ;
7x + 14 -3x + 12 -------collect like terms
7x - 3x + 14 + 12 --------equation 2
perform operation {addition and subtraction}
4x + 26
However at equation 2 above you can write it as ;
7x + 14 + 12-3x --------add the numbers
7x + 26 - 3x ----------equation 5
Additionally at equation 2 above you can subtract the terms with x's as;
7 x-3x + 14 +12
4x + 14 + 12 ------------equation 3
Answer :
2, 3, 5
what might a postitive number mean in thid content ,what about a negatuve number
Answers
1.
A positive number might mean that the number of bottles in the machine is bigger than the number of bottles sold.
A negative number might mean that the number of bottles sold is bigger than the number of bottles in the machine.
2.
A cero might mean that the number of bottles sold is equal to the number of bottles in the machine.
How can you find a common denominator for one third and one fifth
Answers
Given to find the common denominator for one-third and one-fifth.
Here, the denominators of both fractions are 3 and 5. Since 3 and 5 do not have any factors common. So, the LCD will be the product of 3 and 5, which is equal to 15.
Thus, LCD of 1/3 and 1/5 is 15.
AN Question 7 (5 points) A recent Nielson rating poll contact a random sample of Americans to determine the amount of time their family watched television on a Tuesday night. Exactly 250 people were involved in the poll with 37 people watching no television. 51 people watching 30 minutes of television. 17 people watching 45 minutes of television. 20 people watching 60 minutes of television. 19 people watching 75 minutes of television, 11 people watching 90 minutes of television. 50 people watching 120 minutes of television, and 45 people watching 240 minutes of television. Determine the mean from the given Nielson rating poli. Do not round your answer.
Answers
Mean = sum of the terms/number of terms
Replacing with data:
Mean = (37*0 + 51*30 + 17*45 + 20*60 + 19*75 + 11*90 + 50*120 + 45*240)/250
= 90.84 minutes
For the function f(x) = x2 - 4. construct and simplify the difference quotient f(x+h)-f(x) h
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Find the length of AB.A140/6 inAB = [ ? ]inBRound your answer to the nearest hundredth.
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The length of AB is given by:
AB = θ*r
Basically AB = L
Where:
L = Arc length of the minor sector.
Now:
r = 6 in
θ = 140° = 7/9 π
Therefore:
AB = (7/9 π) * 6 = 14/3 π ≈ 14.66 in
May I please get help with this. I can’t seem to figure out the right answers for them
Answers
Given
Solution
Using similar triangles
[tex]\begin{gathered} \frac{35}{57}=\frac{22.8}{x} \\ \\ \text{cross multiply} \\ 35x=1299.6 \\ \text{Divide both sides by 35} \\ \frac{35x}{35}=\frac{1299.6}{35} \\ \\ x=37.1314\ldots \end{gathered}[/tex]The final answer
[tex]37m[/tex]How do I solve these?1) Evaluate the expression, 2b+(c-3a) when a= -1/2, b= 3/4, c= 1/4.2) A store manager uses a markup rate of 24% on all appliances. Find the cost of a coffee maker that sells for $77.50. Use formula S=C+r•C, where S is the selling price, C is the cost, and r is the markup rate.
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To evaluate the expression
[tex]2b+(c-3a)[/tex]Plug in the corresponding values that are given and operate, as following:
[tex]\begin{gathered} 2b+(c-3a) \\ \\ \rightarrow2(\frac{3}{4})+(\frac{1}{4}-3(-\frac{1}{2})) \\ \\ \rightarrow\frac{3}{2}+(\frac{1}{4}+\frac{3}{2}) \\ \\ \rightarrow\frac{3}{2}+\frac{1}{4}+\frac{3}{2} \end{gathered}[/tex]At this point, we have to have all our fractions with the same denominator to add them up. To do so, let's find the LCM (Least Common Multiple) between our different denominators (In this case, 2 and 4)
The LCM is 4. Now, we need to find a way for all our fractions to have 4 as a denominator. Notice that we can do so by multiplying both 3/2 by 2 (both in the numerator and denominator), as following:
[tex]\begin{gathered} \frac{3}{2}+\frac{1}{4}+\frac{3}{2}\rightarrow\frac{2\cdot3}{2\cdot2}+\frac{1}{4}+\frac{2\cdot3}{2\cdot2} \\ \\ \rightarrow\frac{6}{4}+\frac{1}{4}+\frac{6}{4} \end{gathered}[/tex]Now, we can add them up:
[tex]\frac{6}{4}+\frac{1}{4}+\frac{6}{4}\rightarrow\frac{6+1+6}{4}\rightarrow\frac{13}{4}[/tex]This way, the answer is 13/4
for every 5 tacos,julianna uses 2 cups of shredded cheese. complete the table to show the relationships between the number of tacos and the number of cups of cheese
Answers
For 5 taccos 2 cups cheese is needed.
So 10 taccos is 2*(5 taccos)=2*2 cups cheese =4cups
[tex]\begin{gathered} As\text{ 5 taccos=2cup cheese} \\ 1\text{ tacco=}\frac{2}{5}\text{cup cheese} \\ As\text{5 taccos=2cup cheese} \\ 1\text{ cup cheese=}\frac{5}{2}\text{taccos} \\ So\text{ 10 tacco=10}\ast\frac{2}{5}\text{cup cheese=2}\cdot2=4\text{cups} \\ 10\text{ cup cheese =10}\ast\frac{5}{2}\text{tacco}=5\ast5=25tac\cos \end{gathered}[/tex][tex]35\text{ taccos}=35\ast\frac{2}{5}=7\cdot2=14\text{ cup cheese}[/tex][tex]24\text{ cup cheese=}24\ast\frac{5}{2}=12\ast5=60tac\cos [/tex]So to convert taccos to cups of cheese multiply by 2/5 and from cups of cheese to taccos multiply by 5/2.
A local diner purchased a jukebox for $9400 the business makes a down payment of $1000 and agrees to 36 monthly payments of $275 each estimate the annual percentage rate to the nearest 10th of a percent
Answers
Given:-
Jukebox for $9400.
Payment done is $1000.
Agrees to make down payment for 36 months each month of $275.
To find
Rewrite the function for the following transformation: the graph is shifted to the left 5 units.