An integer is a number with a fractional part is true statement.
What is the equation of the line that passes through points (1,-19) and (-2,-7)?
This problem is about linear equations. We need to find the equation of the line w
A set of 12 data points is given above. Which of thefollowing is true of these data?
SOLUTION
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given data
[tex]\lbrace14.9,21.1,21.2,8.4,14.5,5.9,7.6,10.0,4.8,3.2,28.7,29.5\rbrace[/tex]STEP 2: Find the mean ofthe data
[tex]\begin{gathered} The\:arithemtic\:mean\:\left(average\right)\:is\:the\:sum\:of\:the\:values\:in\:the\:set\:divided\:by\:the\:number\:of\:elements\:in\:that\:set. \\ \mathrm{If\:our\:data\:set\:contains\:the\:values\:}a_1,\:\ldots \:,\:a_n\mathrm{\:\left(n\:elements\right)\:then\:the\:average}=\frac{1}{n}\sum _{i=1}^na_i\: \\ Sum=169.8 \\ n=12 \\ mean=\frac{169.8}{12} \\ mean=14.15 \end{gathered}[/tex]STEP 3: Find the median
[tex]\begin{gathered} \mathrm{The\:median\:is\:the\:value\:separating\:the\:higher\:half\:of\:the\:data\:set,\:from\:the\:lower\:half.} \\ \:the\:number\:of\:terms\:is\:odd,\:then\:the\:median\:is\:the\:middle\:element\:of\:the\:sorted\:set \\ If\:the\:number\:of\:terms\:\:is\:even,\:then\:the\:median\:is\:the\:arithmetic\:mean\:of\:the\:two\:middle\:elements\:of\:the\:sorted\:set \\ \\ \mathrm{Arrange\:the\:terms\:in\:ascending\:order} \\ 3.2,\:4.8,\:5.9,\:7.6,\:8.4,\:10,\:14.5,\:14.9,\:21.1,\:21.2,\:28.7,\:29.5 \\ median=12.25 \end{gathered}[/tex]Hence, it can be seen here that the mean is larger than median.
STEP 4: Find the Interquartile range
[tex]\begin{gathered} The\:interquartile\:range\:is\:the\:difference\:of\:the\:first\:and\:third\:quartiles \\ First\text{ Quartile}=6.75 \\ Third\text{ quartile}=21.15 \\ IQR=14.4 \end{gathered}[/tex]STEP 5: Find the standard deviation
[tex]\begin{gathered} \mathrm{The\:standard\:deviation,\:}\sigma \left(X\right)\mathrm{,\:is\:the\:square\:root\:of\:the\:variance:\quad }\sigma \left(X\right)=\sqrt{\frac{\sum _{i=1}^n\left(x_i-\bar{x}\right)^2}{n-1}} \\ Standard\text{ deviation}=9.11836 \end{gathered}[/tex]Hence, it can be seen from above that the interquartile range is larger than the standard deviation.
STEP 6: Find the range
[tex]\begin{gathered} \mathrm{The\:range\:of\:the\:data\:is\:the\:difference\:between\:the\:maximum\:and\:the\:minimum\:of\:the\:data\:set} \\ Minimum=3.2 \\ Maximum=29.5 \\ Range=26.3 \end{gathered}[/tex]STEP 7: Fnd the variance
[tex]\begin{gathered} \mathrm{The\:sample\:variance\:measures\:how\:much\:the\:data\:is\:spread\:out\:in\:the\:sample.} \\ \mathrm{For\:a\:data\:set\:}x_1,\:\ldots \:,\:x_n\mathrm{\:\left(n\:elements\right)\:with\:an\:average}\:\bar{x}\mathrm{,\:}Var\left(X\right)=\sum _{i=1}^n\frac{\left(x_i-\bar{x}\right)^2}{n-1} \\ Variance=83.14454 \end{gathered}[/tex]Hence, it can be seen that the range is not larger than the variance.
Therefore, the answer is I and II only.
Josh's grocery bill is $56.00 and the sales tax in his state is 7% how much extra does he have to pay? I assume 125?
Given:
Josh's grocery bill is $56.00
The sales tax in his state is 7%
[tex]\begin{gathered} \text{Extra money he have to pay=56}\times\frac{7}{100} \\ \text{Extra money he have to pay=}\frac{392}{100} \\ \text{Extra money he have to pay= \$3.92} \end{gathered}[/tex]On total Josh has to pay $59.92
Josh has to pay extra money as tax is $3.92
Draw a graph, and label and scale both axes. Plot the points (-2, 3) and (1, -5), clearly labeling them.
Answer:
Explanation:
First, we draw the graph, label the x and y-axis. Then we plot the points given.
To plot the point (-2, 3), we first move to units to the left from the origin and the 3 units up as sketched below
Similarly for (1, -5), we have
We moved one unit to the right and 5 units down.
To find the slope, we take the vertical distance and the horizontal distance and then the slope will be
slope vertical distance / horizontal distance
Therefore the slope is
[tex]Slope=-\frac{8}{3}[/tex]
a large community college has professors and lectures. total number of facility members is 228. the school reported that they had five professors for every 14 lectures. how many of each type of faculty member does the community college employee?
a large community college has professors and lectures. total number of facility members is 228. the school reported that they had five professors for every 14 lectures. how many of each type of faculty member does the community college employee?
Let
x -----> number of professors
y ----> number of lectures
we have that
x+y=228
x=228-y -------> equation A
x/y=5/14
x=(5/14)y ------> equation B
equate equation A and equation B
228-y=(5/14)y
solve for y
(5/14)y+y=228
(19/14)y=228
y=228*14/19
y=168
Find the value of x
x=228-168=60
therefore
number of professors is 60number of lectures is 168need help with this problem, find the length of the darkened arc. C is the center of the circle
Notice that the central angle measures 138 degrees, We have a property of the circle that says that the measure of a central angle is equal to the arc between its sides.
Therefore, the arc measures 138 degrees
need help assap look at file attached
Answer:
length is 27, width is 9
Step-by-step explanation:
72/4= 18
2x27+ 2x9 = 54 + 18 = 72
Solve equation for x. 5x^2 - 4x =6
Answer:
Step-by-step explanation:
use the quadratic formula
5x^2-4x-6
4+-[tex]\sqrt{16+120}[/tex] all over 10
4+-[tex]2\sqrt{34}[/tex]/10
2+-[tex]\sqrt{34}[/tex]/5
D
Select from the drop-down menus to correctly complete each statement.
The opposite of −358 is on the
Choose...
side of zero on a number line as −358. The opposite of 429is on the
Choose...
side of zero on a number line as 429.
The opposite of −3 5/8 is on the opposite side of zero on a number line as −3 5/8 . The opposite of 4 2/9 is on the opposite side of zero on a number line as 4 2/9 .
What is a number line?A number line is a type of graph with a graduated straight line which contains both positive and negative numbers that are typically placed at equal intervals along its length.
What are opposites?In Mathematics, opposites simply refers to numbers that are located on opposite sides of zero (0) on any number line. Additionally, opposites generally have the same distance from zero (0) on any given number line.
In conclusion, -3 5/8 is a number that is located on the opposite side of zero (0) on a number line while 4 2/9 is a number that is also located on the opposite side of zero (0) on a number line.
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Complete Question:
Select from the drop-down menus to correctly complete each statement. The opposite of −3 5/8 is on the ______ side of zero on a number line as −3 5/8 . The opposite of 4 2/9 is on the ______ side of zero on a number line as 4 2/9 .
What are the values of n in the following equation? Select all that apply. 4n + 2 = 34
The given equation is
[tex]4x+2=34[/tex]First, we subtract 2 from each side
[tex]\begin{gathered} 4x+2-2=34-2 \\ 4x=32 \end{gathered}[/tex]Then, we divide the equation by 4
[tex]\begin{gathered} \frac{4x}{4}=\frac{32}{4} \\ x=8 \end{gathered}[/tex]Hence, the answer is x = 8.what is the value of 6 3/4 (-11.5)
We are given the following expression
[tex]6\frac{3}{4}(-11.5)[/tex]As you can see, a mixed number is being multiplied with a negative decimal number.
First, convert the mixed number to a simple fraction then multiply with the decimal number
[tex]6\frac{3}{4}=\frac{6\cdot4+3}{4}=\frac{24+3}{4}=\frac{27}{4}[/tex]Now multiply it with the negative decimal number
[tex]\frac{27}{4}(-11.5)=-\frac{310.5}{4}=-77.625[/tex]So the resultant decimal number may be written back into the mixed form as
[tex]-77.625=-77\frac{5}{8}[/tex]Therefore, the result of the given expression is -77 5/8
X + 2y = 3x = 5Enter your answer as a point using parenthesis and a comma. Do not use any spaces in youranswer.If there are no solutions, type "no solutions." If there are infinitely many solutions, type"infinitely many."Answer:
thats the explanation
answer is : (5, -1)
polygon wxyz has vertices W( 1, 5 ), X( 6, 5), Y( 6, 10), and Z(1, 10)
If w' x' y' z' is a dilation of wxyz with scale factor 5, give the coordinates of w' x' y' z'
The coordinates of W'X'Y'Z' are W'(5, 25), X'(30, 25), Y'(30, 60) and Z'(5, 10) respectively.
Given that, polygon WXYZ has vertices W( 1, 5 ), X( 6, 5), Y( 6, 10), and Z(1, 10).
What is a dilation?Dilation is the process of resizing or transforming an object. It is a transformation that makes the objects smaller or larger with the help of the given scale factor. The new figure obtained after dilation is called the image and the original image is called the pre-image.
We know that, scale factor = Dimension of the new shape ÷ Dimension of the original shape
Dimension of the original shape W'= 5(1, 5) = (5, 25)
X' =5(6, 5) = (30, 25)
Y' =5(6, 10) = (30, 60)
Z' =5(1, 10) = (5, 10)
Therefore, the coordinates of W'X'Y'Z' are W'(5, 25), X'(30, 25), Y'(30, 60) and Z'(5, 10) respectively.
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Solve the system of equations using the elimination method. Note that the method of elimination may be referred to as the addition method. (If there is no solution, enter NO SOLUTION.)0.2x + 0.7y = 2.20.9x − 0.2y = 3.2(x, y) =
To solve the system of equations
[tex]\begin{gathered} 0.2x+0.7y=2.2 \\ 0.9x-0.2y=3.2 \end{gathered}[/tex]we need to make the coefficients of one of the variables opposite, that is, they need to have the same value with different sign; let's do this with the y variable, so let's multiply the second equation by 0.7 and the first equation by 0.2; then we have:
[tex]\begin{gathered} 0.04x+0.14y=0.44 \\ 0.63x-0.14y=2.24 \end{gathered}[/tex]Now we add the equations and solve the resulting equation for x:
[tex]\begin{gathered} 0.04x+0.14y+0.63x-0.14y=0.44+2.24 \\ 1.64x=2.68 \\ x=\frac{2.68}{0.67} \\ x=4 \end{gathered}[/tex]Now that we have the value of x we plug it in one of the original equations and solve for y:
[tex]\begin{gathered} 0.2(4)+0.7y=2.2 \\ 0.8+0.7y=2.2 \\ 0.7y=2.2-0.8 \\ 0.7y=1.4 \\ y=\frac{1.4}{0.7} \\ y=2 \end{gathered}[/tex]Therefore, the solution of the system of equation is (4,2)
at the casino Robert has currently lost 72$ he plays another game during which he loses 35$ find the total amount of money Robert has lost
we know that
Robert has currently lost 72$
he plays another game during which he loses 35$
to find out the total money Robert has lost, adds the amount 1 plus the amount 2
so
72+35=$107
Triangle MNO has its vertices at the following coordinates:M(2, 2) N(-1,3) O(1,5)Give the coordinates of the image triangle M'N'O' after a 90° counterclockwise rotation about the origin.
The counter clockwise rotation of any point X(x,y) about origin results in change of coordinates as,
[tex]X(x,y)\rightarrow X^{\prime}(-y,x)[/tex]Determine the coordinates of the vertices of the triangle M'N'O'.
[tex]M(2,2)\rightarrow M^{\prime}(-2,2)[/tex][tex]N(-1,3)\rightarrow(-3,-1)[/tex][tex]O(1,5)\rightarrow(-5,1)[/tex]So coordinates of triangle M'N'O' are;
M'(-2,2)
N'(-3,-1)
O'(-5,1)
the table below shows an inspectors measurement of the lengths of four bridges
Given
Table which ahows an inspectors measurement of the lengths of four bridges
Find
Order from shortest to longest lengths of bridges.
Explanation
First we convert the lengths of Bridges in improper form , then in decimal form.
[tex]\begin{gathered} M=1\frac{49}{60}=\frac{109}{60}=1.817 \\ \\ N=1\frac{79}{100}=\frac{179}{100}=1.79 \\ \\ O=1\frac{52}{75}=\frac{127}{75}=1.693 \\ \\ P=1\frac{97}{120}=\frac{217}{120}=1.80 \end{gathered}[/tex]so , O has shortest length and M has longest length.
Final Answer
Therefore ,
the order from shortest to longest is
Bridge O , Bridge N , Bridge P and Bridge M , so the correct option is 1
For each ordered pair, determine whether it is a solution to 4x - 5y = -13.Is it a solution?x$(x, y)YesNo(-7, -3)(3, -4)OO(-2, 1)oO(6, 7)0
The equation is 4x - 5y = -13.
Substitute -7 for x and -3 for y in the equation to check whether ordered pair is solution of the equation.
[tex]\begin{gathered} 4\cdot(-7)-5\cdot(-3)=-13 \\ -28+15=-13 \\ -13=-13 \end{gathered}[/tex]The ordered pair satisfy the equation so point (-7,-3) is solution of equation.
Substitute 3 for x and -4 for y in the equation to check whether ordered pair is solution of the equation.
[tex]\begin{gathered} 4\cdot3-5\cdot(-4)=-13 \\ 12+20=-13 \\ 32\ne-13 \end{gathered}[/tex]The ordered pair not satisfy the equation. So point (3,-4) is not a solution of the equation.
Substitute -2 for x and 1 for y in the equation to check whether ordered pair is solution of the equation.
[tex]\begin{gathered} 4\cdot(-2)-5\cdot1=-13 \\ -8-5=-13 \\ -13=-13 \end{gathered}[/tex]The ordered pair satisfy the equation. So point (-2,1) is solution of equation.
Substitute 6 for x and 7 for y in the equation to check whether ordered pair is solution of the equation.
[tex]\begin{gathered} 4\cdot6-5\cdot7=-13 \\ 24-35=-13 \\ -11\ne-13 \end{gathered}[/tex]The orderedpair not satisfy the equation. So point (6,7) is not a solution of the equation.
Henry had a batting average of 0.341 last season (out of 1000 at-bats, he had 341 hits). Given that thisbatting average will stay the same this year, answer the following questions,What is the probabilitythat his first hit willoccur within his first 5at-bats? Answer choice. 0.654. 0.765. 0.821. 0.876
The probability of a successful batting is 0.341; we need to find the probability of at least 1 hit within the first 5 at-bats; thus,
[tex]P(Hit)=1-P(NoHit)[/tex]Therefore, we need to calculate the probability of not hitting the ball within the first 5 at-bats.
The binomial distribution states that
[tex]\begin{gathered} P(X=k)=(nBinomialk)p^k(1-p)^{n-k} \\ n\rightarrow\text{ total number of trials} \\ k\rightarrow\text{ number of successful trials} \\ p\rightarrow\text{ probability of a successful trial} \end{gathered}[/tex]Thus, in our case,
[tex]P(k=0)=(5Binomial0)(0.341)^0(0.659)^5=1*1*0.124287...[/tex]Then,
[tex]P(Hit)=1-0.124287...\approx0.876[/tex]Therefore, the answer is 0.876Angel Corporation produces calculators selling for $25.99. Its unit cost is $18.95. Assuming a fixed cost of $80,960, what is the breakeven point in units?
The breakeven point is 11,500, meaning that if the sell that number of units, the profit will be zero.
How to get the breakeven point?
Here we know that the unit price is $25.99, so if they sell x units, the revenue is
R(x) = $25.99*x
And the cost per unit is $18.95, plus a fixed cost of $80,960
Then the cost of x units is:
C(x) = $80,960 + $18.95*x
The breakeven point is the value of x such that the cost is equal to the revenue, so we need to solve:
$25.99*x = $80,960 + $18.95*x
$25.99*x - $18.95*x = $80,960
$7.04*x = $80,960
x = $80,960/%7.04 = 11,500
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What is the reference angle for 289°? A. 71° B. 19° C. 11° D. 89°
Given:
Angle θ=289°.
For angles from 270° to 360°, the reference angle can be calculated by subtracting the given angle from 360° .
The reference angle of θ can be calculated as:
[tex]\begin{gathered} 360\degree-\theta=360\degree-289\degree \\ =71\degree \end{gathered}[/tex]Therefore, reference angle of 289° is 71°.
find 2x:3y if x:y = 2:5
4 : 15
Explanation:[tex]\begin{gathered} \text{x : y = 2: 5} \\ \frac{x}{y}\text{ = }\frac{2}{5} \\ \\ 2x\text{ : 3y = ?} \end{gathered}[/tex][tex]\begin{gathered} 2x\colon\text{ 3y = }\frac{2x}{3y} \\ 2x\colon3y\text{ = }\frac{2}{3}\times\frac{x}{y} \end{gathered}[/tex][tex]\begin{gathered} \text{substitute for x/y in 2x:3y} \\ \frac{2}{3}\times\frac{x}{y}\text{ =}\frac{2}{3}\times\frac{2}{5} \\ =\text{ }\frac{4}{15} \\ \\ \text{Hence, 2x:3y = }\frac{4}{15} \\ or \\ 4\colon15 \end{gathered}[/tex]EFG IS dilated with scale factor of 4 to create triangle E’F’G’ the measure of angle F’ is 53 degrees what is the measurement of angle F
The measurement of ∠ F = 53 °.
Given,
Triangle EFG is dilated with a scale factor of 4 to create Δ E ' F ' G ' .
The measure of ∠ F ' is 53 °.
To find the measurement of angle F.
We know that a dilation creates similar figures i.e. it preserves the measure of angles.
Therefore, if Triangle EFG is dilated to form Δ E ' F ' G ', then the measure of ∠ F' = measure of ∠ F [Corresponding angles remains same]
⇒ The measure of ∠ F' = measure of ∠ F = 53 °
Hence, The measurement of ∠ F = 53 °.
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A boat heading out to sea starts out at Point A, at a horizontal distance of 1357 feet
from a lighthouse/the shore. From that point, the boat's crew measures the angle of
elevation to the lighthouse's beacon-light from that point to be 9°. At some later time
the crew measures the angle of elevation from point B to be 3°. Find the distance
from point A to point B. Round your answer to the nearest tenth of a foot if
necessary.
The distance from point A to point B is 2744.1 feet.
Define Trigonometric ratio
The ratios of the sides of a right triangle are called trigonometric ratios. Three common trigonometric ratios are the sine (sin), cosine (cos), and tangent (tan).
Given, horizontal distance = 1357 feet
Let h = height of the lighthouse
tan(A) = perpendicular / base
where, perpendicular = h
base = 1357 feet
Angle is 9°
so now, put these value in trigonometric ratio
tan(9) = h / 1357
where, tan(9) = 0.1584
h = 1357 * 0.1584
h = 214.9 feet
Let d = distance from point B to the Lighthouse base
Angle is 3°
h = 214.9 feet
so, tan(3) = 214.9 / d
where, tan(3) = 0.0524
0.0524 = 214.9 / d
d = 214.9 / 0.0524
d = 4101.1 feet
Distance between point A to B = 4101.1 - 1357
= 2744.1 feet
Therefore, the distance from point A to point B is 2744.1 feet
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Subtracting algorithm
Using subtracting algorithm, we have 69,000 - 42,547 = 26,453 and 152,681 - 34,820 = 117,861.
According to the question,
We have the following expressions:
_ _ , 0 0 0
-4 2, _ _ _
_________
2 6 , 4 5 3
Now, we will first solve this.
The spaces has to be filed with the digits from 0 to 9 and in both the questions, no digit has to be repeated.
Now, we will solve it from the right hand side.
(Note that we will solve each row that is we will solve vertically.)
0 is given, then we have blank, and in result we have 3.
It is clear that no number can be subtracted from 0. So, we have to regroup it from other digits. So, it will be 10.
Now, 10-3 = 7
So, this blank will be 7.
Now, come to next one.
We have 0 and a blank and then 5 in the result.
Now, it should be 9 because 1 was taken away from it.
So, 9-5 = 4
This blank has to be filled with 4.
Now, we will solve next one.
We have 0, a blank and then 4 in the result.
Now, this 0 will make 9.
9-4 = 5
This blank has to be filled with 5.
Now, we have a blank, 2 and 6 in the result.
1 has been carried from this digit.
So, we have:
6+2+1 = 9
This blank has to be filled with 9.
Next, we have a blank, 4 and 2 in the result.
4+2 = 6
This has to be filled with 6.
Now, we will solve the second question like this.
_ 5 2, 6 8 1
- _ 4, _ _ _
__________
1 1 7, 8 6 1
Now, moving from right to left vertically:
We have 1, a blank and 1 in the result.
1-1 = 0
Blank = 0
We have 8, a blank and 6 in the result.
8-6 = 2
Blank = 2
We have 6, a blank and 8 in the result.
Now, 8 can not be subtracted from 6. So, it has to be 16.
16-8 = 8
Blank = 8
Now, next one is complete.
So, we have next one.
We have 5, a blank and 1 in the result.
We will subtract 1 from 4 because 1 was carried away from 5.
4-1 = 3
Blank = 3
Now, we have a blank and 1 in the result.
Blank = 1
Hence, the answers would be 69,000 - 42,547 = 26,453 and 152,681 - 34,820 = 117,861.
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1. A table is 2 feet wide. It is 6 times as long as it is wide. Table= A-Label the diagram with the dimensions of the table.B-find the perimeter of the table
Width(w) = 2 feet
Length (L)= 6w = 6(2) = 12 feet
a. Table = 2 x 12
b: Perimeter of a rectangle:
P = 2w+ 2L = 2(2)+2(12) = 4+24 = 28 feet
A store is having a sale to celebrate President’s Day. Every item in the store is advertised as one- fourth off the original price. If an item is marked with a sale price of , what was its original price?
If the discount is one fourth off, it means the discount is 1/4 = 25% of the original price, so the final price will be 75% or 3/4 of the original price.
In order to find the original price, we just need to divide the final price by 3/4, this way we "remove" the discount.
For example, if the sale price is $75, the original price would be:
[tex]\text{original price}=\frac{75}{\frac{3}{4}}=75\cdot\frac{4}{3}=25\cdot4=100[/tex]So for a sale price of $75, the original price would be $100.
In general, for a discount of x%, the original price (given the sale price) can be calculated as:
[tex]\text{original price}=\frac{\text{sale price}}{1-\frac{x}{100}}[/tex]PLEASE HELP: Which of the following are identities? Check all that apply. A. (sin x + cos x)^2 = 1 + sin 2x B. sin 3x - sinx/ cos3x + cosx = tan xC. sin 6x = 2 sin3x cos3x D. sin 3x/sin x cos x = 4 cos x - sec x
All the options are correct
Explanations:A quick and smart way is to substitute a value for x in each of the options and verify if the right hand side equals the left hand side
Let x = 30
A) (sin x + cos x)² = 1 + sin 2x
(sin 30 + cos 30)² = 1.866
1 + sin 2(30) = 1.866
Therefore (sin x + cos x)² = 1 + sin 2x
B)
[tex]\begin{gathered} \frac{\sin3x-\sin x}{\cos3x+\cos x}=\tan x \\ \frac{\sin3(30)-\sin30}{\cos3(30)+\cos30}=0.577 \\ \tan \text{ 30 = 0.577} \end{gathered}[/tex]Therefore:
[tex]\frac{\sin3x-\sin x}{\cos3x+\cos x}=\tan x[/tex]C) sin 6x = 2 sin3x cos3x
sin 6(30) = 0
2 sin3(30) cos3(30) = 0
Therefore sin 6x = 2 sin3x cos3x
This can also be justified by sin2A = 2sinAcosA
D.
[tex]\frac{\sin3x}{\sin x\cos x}=\text{ 4}\cos x-\sec x[/tex][tex]\begin{gathered} \frac{\sin 3(30)}{\sin 30\cos 30}=\text{ 2.31} \\ 4\cos 30-\sec 30=\text{ }2.31 \end{gathered}[/tex]Options A to D are correct
1. A jar contains 5 red marbles numbered 1 to 5 and 6 blue marbles numbered 1 to 6. A marble is drawn at random from the jar. Find the probability that the marble is blue or odd-numbered.
We will use the following formula:
[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B).[/tex]First, we compute the probability that we get a blue marble:
[tex]P(\text{Blue)}=\frac{6}{5+6}=\frac{6}{11}\text{.}[/tex]Now, we compute the probability of getting an odd-numbered marble:
[tex]P(\text{odd-num)}=\frac{6}{11}\text{.}[/tex]Finally, the probability that we draw a blue and odd-numbered marble is:
[tex]P(\text{blue and odd)=}\frac{3}{11}.[/tex]Answer: The probability that the marble is blue or odd-numbered is:
[tex]\begin{gathered} P(\text{blue or odd)=P(blue)+P(odd-num)-P(blue and odd)=}\frac{6}{11}+\frac{6}{11}-\frac{3}{11}=\frac{9}{11}. \\ P(\text{blue or odd)}=\frac{9}{11}\text{.} \end{gathered}[/tex]For questions 5&6 find F -1(x), the inverse of F(x)
To find the inverse function, we can follow the next steps:
First Function1. Replace x with y as follows:
[tex]y=3x+7\Rightarrow x=3y+7[/tex]2. Solve the resulting equation for y. Subtract 7 from both sides of the equation:
[tex]x-7=3y+7-7\Rightarrow x-7=3y[/tex]3. Divide both sides of the equation by 3:
[tex]\frac{(x-7)}{3}=\frac{3}{3}y\Rightarrow y=\frac{(x-7)}{3}=\frac{1}{3}(x-7)=\frac{x}{3}-\frac{7}{3}[/tex]Second FunctionWe need to repeat the process to obtain the inverse of this function:
1. Replace x with y:
[tex]y=8x\Rightarrow x=8y[/tex]2. Solve for y. Divide both sides by 8:
[tex]\frac{x}{8}=\frac{8}{8}y\Rightarrow y=\frac{x}{8}[/tex]In summary, we have that the inverse functions are:
For function
[tex]y=3x+7[/tex]The inverse function is:
[tex]y=F^{-1}^{}(x)=\frac{(x-7)}{3}[/tex]And, for the function
[tex]y=8x[/tex]The inverse function is:
[tex]y=f^{-1}(x)=\frac{1}{8}x[/tex]