Plot the point given by the following polar coordinates on the graph below. Each circular grid line is 0.5 units apart.230(2.5. -,
Answers
Solution:
Given:
[tex](2.5,-\frac{2\pi}{3})[/tex]Related Questions
Each coordinate grid shows the graph of a system of two equations. Which graph represents a system of equations with no solution? Select all that apply.
Answers
System of Linear Equations with No Solutions
A system has no solutions if two equations are parallel.
Therefore, The answer would be option:
Figure A is a scale image of Figure B.27Figure AFigure B4535What is the value of x?
Answers
Answer:
x = 21
Explanation:
Figure A is a scaled version of figure B. This means that the ratio between any two sides must be the same for both figures.
It follows then
[tex]\frac{27}{45}=\frac{x}{35}[/tex]which just means that the ratio f sides 27 with 45 must be the same as the ratio between side x and 35. Why? Because these two sides are the same across the two figures and therefore their size with respect to each other must not change.
Now to find the value of x, we simply need to solve for x.
We do this by multipying both sides by 35:
[tex]undefined[/tex]A storage box has a volume of 56 cubic inches. The base of the box is 4 inches by 4 inches. Lin’s teacher uses the box to store her set of cubes with an edge length of 1/2 inch.If the box is completely full how many cubes are in the set?
Answers
To answer this question, we have to find the volume of one of the cubes from Lin's set. To do it use the formula to find the volume of a cube, which is the edgelength raised to 3:
[tex]\begin{gathered} V=ed^{3^{}} \\ V=(\frac{1}{2})^3 \\ V=\frac{1}{8} \end{gathered}[/tex]In this case, each cube has a volume of 1/8 cubic inches. To find the number of cubes in the set, perform the division of the volume of the box by the volume of one cube:
[tex]n=\frac{V_{box}}{V_{cube}}=\frac{56}{\frac{1}{8}}=8\cdot56=448[/tex]The set has 448 cubes.
8+7t=22 in verbal sentence
Answers
Eight plus Seven times t equals twenty-two
Explanation
Step 1
Let
a number= t
seven times a number= 7t
the sum of eigth and seven times a number=8+7t
the sum of eigth and seven times a number equals twenty-two=8+7t=22
or,in other words
Eight plus Seven times t equals twenty-two
I hope this helps you
The number of microbes in a tissue sample is given by the functionN (t) = 34.8 + In(1 + 1.2t)where N(t) is the number of microbes (in thousands) in the sample after thours.a.) How many microbes are present initially?b.) How fast are the microbes increasing after 10 hours?
Answers
Explanation
[tex]N(t)=34.8+\ln (1+1.2t)[/tex]we have a function where the number of microbes ( N) depends on the time(t)
hence
Step 1
a.) How many microbes are present initially?
to know this, we need replace time I= t = zero, because it was "initially"
so
when t=0
replace.
[tex]\begin{gathered} N(t)=34.8+\ln (1+1.2t) \\ N(0)=34.8+\ln (1+1.2\cdot0) \\ N(0)=34.8+\ln (1) \\ N(0)=34.8+0 \\ N(0)=34.8 \end{gathered}[/tex]so, initially there were 34.8 microbes
Step 2
b)How fast are the microbes increasing after 10 hours?
to know this, let t=10
so
[tex]\begin{gathered} N(t)=34.8+\ln (1+1.2t) \\ N(10)=34.8+\ln (1+1.2\cdot10) \\ N(10)=34.8+\ln (1+12) \\ N(10)=34.8+\ln (13) \\ N(10)=34.8+2.56 \\ N(10)=37.36 \end{gathered}[/tex]therefore , after 10 hours the number of microbes is 37.36
I hope this helps you
Which lines are parallel?
M: y + 1 = -3 (x-1)
K: y =3(x+2)
P: y + 4 = 3x
Answers
The most appropriate choice for equation of line in slope intercept form will be given by-
line K is parallel to line P
line M is not parallel to both line K and line P.
What is equation of line in slope intercept form?
The most general form of equation of line in slope intercept form is given by y = mx +c
Where m is the slope of the line and c is the y intercept of the line.
Slope of a line is the tangent of the angle that the line makes with the positive direction of x axis.
If [tex]\theta[/tex] is the angle that the line makes with the positive direction of x axis, then slope (m) is given by
m = [tex]tan\theta[/tex]
The distance from the origin to the point where the line cuts the x axis is the x intercept of the line
The distance from the origin to the point where the line cuts the y axis is the y intercept of the line
For M
y + 1 = -3(x - 1)
y + 1 = -3x + 3
y = -3x + 3-1
y = -3x + 2
Slope of M = -3
For K
y = 3(x+2)
y = 3x + 6
Slope of K = 3
For P
y + 4 = 3x
y = 3x - 4
slope of P = 3
Since Slope of K = Slope of P,
line K is parallel to line P
Since slope of M is different from slope of both K and P,
line M is not parallel to both line K and line P.
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The angle of depression from the top of a sheer cliff to point A on the ground is 35º. If point A is280 feet from the base of the cliff, how tall is the cliff? Round the answer to the nearest tenth of afoot.
Answers
In this case, to calculate the tall of the cliff, consider the distance from the base of the cliff to the point A, as a hypotenuse of a right triangle.
The tall of the clift is given by:
h = 280 sin(35)
h = 280(0.573)
h = 160.60
Hence, the tal of the clift is 160.60 feet
Yea I think and her dad is doing great so
Answers
Given the following function:
[tex]tan\text{ }\theta=\frac{10}{y}[/tex]Both θ and y are functions of the time (t)
We will find the derivatives of θ and y with respect of the time (t) as follows:
[tex]sec^2θ*\frac{dθ}{dt}=-\frac{10}{y^2}*\frac{dy}{dt}[/tex]Now, we will find dy/dt when θ = π/6 and dθ/dt = π/12
First, we need to find the value of y when θ = π/6
[tex]\begin{gathered} tan(\frac{\pi}{6})=\frac{10}{y} \\ \frac{1}{\sqrt{3}}=\frac{10}{y} \\ \\ y=10\sqrt{3} \end{gathered}[/tex]so, we will substitute the values to find dy/dt as follows:
[tex]\begin{gathered} sec^2(\frac{\pi}{6})*\frac{\pi}{12}=-\frac{10}{(10\sqrt{3})^2}*\frac{dy}{dt} \\ \\ so,\frac{dy}{dt}=-\frac{(10\sqrt{3})^2}{10}*sec^2(\frac{\pi}{6})*\frac{\pi}{12}=-10.4719755 \end{gathered}[/tex]Rounding to 2 decimal places
So, the answer will be:
[tex]\frac{dy}{dt}=-10.47\text{ feet/hour}[/tex]PLEASE ITS URGENT I NEED HELP!!! I BEG YOU GUYS PLEEAASEEE THANKS..
Answers
Explanation
remember some properties of the exponents
[tex]\begin{gathered} a^m\cdot a^n=a^{m+n} \\ (a^m)^n=a^{m\cdot n} \\ a^{-m}=\frac{1}{a^m} \end{gathered}[/tex]then, to solve this solve each option and compare
Step 1
[tex]6^{-5}\cdot6^2[/tex]solve
[tex]\begin{gathered} 6^{-5}\cdot6^2=6^{-5+2}=6^{-3} \\ \end{gathered}[/tex]so, this is not an answer
Step 2
[tex](\frac{1}{6^2})^5[/tex]solve
[tex]\begin{gathered} (\frac{1}{6^2})^5=(6^{-2})^5=6^{(-2\cdot5)}=6^{-10} \\ \end{gathered}[/tex]so, this is an answer
Step 3
[tex]\begin{gathered} (6^{-5})^2 \\ \text{solve} \\ (6^{-5})^2=6^{-5\cdot2}=6^{-10} \end{gathered}[/tex]so, this is an answer
Step 4
[tex]\begin{gathered} \frac{6^{-3}}{6^7} \\ \text{solve} \\ \frac{6^{-3}}{6^7}=\frac{1}{6^3\cdot6^7}=\frac{1}{6^{3+7}}=\frac{1}{6^{10}}=6^{-10} \end{gathered}[/tex]so, this is an answer
Step 5
[tex]\begin{gathered} \frac{6^5\cdot6^{-3}}{6^{-8}} \\ \text{solve} \\ \frac{6^5\cdot6^{-3}}{6^{-8}}=\frac{6^{5-3}}{6^{-8}}=\frac{6^2}{6^{-8}}=6^2\cdot\frac{1}{6^{-8}}=6^2\cdot6^8=6^{10} \end{gathered}[/tex]so, this is not an answer
I hope this helps you
Introduction to Probability
Answers
The probability of birthday being in July is 1/12.
What do you mean by probability?
The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a numeric value and 1, where, broadly speaking, 0 denotes the event's impossibility and 1 denotes certainty. The likelihood that an event will occur increases with its probability. A straightforward illustration is flipping a fair (impartial) coin. The chance of either "heads" or "tails" is half because there are only two possible outcomes (heads or tails), and because the coin is fair, both outcomes (heads and tails) are equally likely.
Probability of birthday being in July = 1/12 (12 months in a year)
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Scientists are conducting an experiment with a gas in a sealed container. The mass of the gas is measured and the scientists realize that the gas is leaking over time in a linear way. Eight minutes since the experiment started the gas had a mass of 302.4 grams. Seventeen minutes since the experiment started the gas had a mass of 226.8 gramsLet x be the number of minutes that have passed since the experiment started and let y be the mass of the gas in grams at that moment. Use a linear equation to model the weight of the gas over time.a) This lines slope-intercept equation is [ ] b) 39 minutes after the experiment started, there would be [ ] grams of gas left. c) if a linear model continues to be accurate, [ ] minutes since the experiment started all gas in the container will be gone.
Answers
Here, we want to model an experiment linearlly
From the question, we have it that;
The coordinates are written as;
(number of minutes, mass of gas)
So, what we have to do know is to set up the two given points
These are the points;
(8,302.4) and (17,226.8)
Now, using these two points, we can model the equation
We start by getting the slope of the line that passes through these two points
To do this, we shall use the slope equation
We have this as;
[tex]\begin{gathered} \text{slope m = }\frac{y_2-y_1}{x_2-x_1} \\ \\ (x_1,y_1)\text{ = (8,302.4)} \\ (x_2,y_2)\text{ = (17,226.8)} \\ \text{substituting these values;} \\ m\text{ = }\frac{226.8-302.4}{17-8}\text{ = }\frac{-75.6}{9}\text{ = -8.4} \end{gathered}[/tex]The general equation representing a linear model is ;
[tex]\begin{gathered} y\text{ = mx + b} \\ m\text{ is slope} \\ b\text{ is y-intercept} \\ y\text{ = -8.4x + b} \end{gathered}[/tex]To get the y-intercept so as to write the complete equation, we use any of the two points and substitute its coordinates
Let us substitute the coordinates of the first point
[tex]\begin{gathered} 302.4\text{ = -8.4(8) + b} \\ 302.4\text{ = -67.2 + b} \\ b\text{ = 67.2 + 302.4} \\ b\text{ = 369.6 } \end{gathered}[/tex]a) Thus, we have the complete linear model as;
[tex]y\text{ = -8.4x + 369.6}[/tex]b) To get this, we simply substitute the value of x given into the linear model
[tex]\begin{gathered} y\text{ = -8.4(39) + 369.6} \\ y\text{ = -327.6 + 369.6} \\ y\text{ = 42} \end{gathered}[/tex]39 minutes after the experiment started, there would be 42 grams
c) If all the gas is gone, then the value of y will br zero at this point
To get the corresponding x-value which is the time, we have it that;
[tex]\begin{gathered} 0\text{ = -8.4x +369.6} \\ 8.4x=\text{ 369.6} \\ x\text{ = }\frac{369.6}{8.4} \\ x\text{ = 44} \end{gathered}[/tex]In 44 minutes, all the gas in the container will be gone
I need to double check 15 I got answer B
Answers
We will have that the area of one sector of the circle will be:
[tex]A=(\frac{45}{360})\pi(20in)^2\Rightarrow A=\frac{25\pi}{2}in^2[/tex]So, the solution is option B.
tom has a rectangular prism - shaped suitcase that measures 9 inches by 9 inches by 24 inches. he needs a second suitcase that has the same volume but smaller surface than his current suitcase. which suitcase size would fit Toms needs
Answers
ANSWER:
18 inches by 9 inches by 12 inches
EXPLANATION:
The volume of Tom's rectangular prism-shaped suitcase which measures 9 inches by 9 inches by 24 inches is;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =9*9*24 \\ \\ =1944\text{ }square\text{ }inches \end{gathered}[/tex]So the volume of Tom's suitcase is 1944 cubic inches
The surface area will be;
[tex]\begin{gathered} SA=2(lw+wh+hl) \\ \\ =2(9*9+9*24+24*9) \\ \\ =2(81+216+216) \\ \\ =2(513) \\ \\ =1026\text{ }square\text{ }inches \end{gathered}[/tex]So the volume of the suitcase is 1026 square inches
*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 18 inches by 6 inches;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =18*18*6 \\ \\ =1944\text{ cubic inches} \end{gathered}[/tex][tex]\begin{gathered} Surface\text{ }Area=2(18*18+18*6+6*18) \\ \\ =2(324+108+108) \\ \\ =2(540) \\ \\ =1080\text{ square inches} \end{gathered}[/tex]We can see that the suitcase that measures 18 inches by 18 inches by 6 inches has the same volume as the first one but a higher surface area which doesn't fit Tom's needs
*Let's go ahead and determine the volume of a suitcase that measures 12 inches by 10 inches by 9 inches;
[tex]\begin{gathered} Volume=12*10*9 \\ \\ =1080\text{ cubic inches} \end{gathered}[/tex]We can see that the suitcase that measures 12 inches by 10 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.
Let's go ahead and determine the volume of a suitcase that measures 16 inches by 5 inches by 9 inches;
[tex]\begin{gathered} Volume=16*5*9 \\ \\ =720\text{ cubic inches} \end{gathered}[/tex]We can see that the suitcase that measures 16 inches by 5 inches by 9 inches has a different volume from the first one which doesn't fit Tom's needs.
*Let's go ahead and determine the volume and surface area of a suitcase that measures 18 inches by 9 inches by 12 inches;
[tex]\begin{gathered} Volume=l*w*h \\ \\ =18*9*12 \\ \\ =1944\text{ cubic inches} \end{gathered}[/tex][tex]\begin{gathered} Surface\text{ }Area=2(18*9+9*12+12*18) \\ \\ =2(162+108+216) \\ \\ =2(486) \\ \\ =972\text{ square inches} \end{gathered}[/tex]We can see that the suitcase that measures 18 inches by 9 inches by 12 inches has the same volume as the first one and s smaller surface area which fits Tom's needs
Eliana drove her car 81 km and used 9 liters of fuel. She wants to know how many kilometres she can drive on 22 liters of fuel. She assumes her car will continue consuming fuel at the same rate. How far can Eliana drive on 22 liters of fuel? What if Eliana plans to drive from Dubai to Abu Dhabi via Sheikh Zayed Bin Sultan which is 139.4 km? How many liters of fuel does she need?
Answers
Eliana can drive 198 km with 22 liters of fuel.
If Eliana plans to drive from Dubai to Abu Dhabi via Sheikh Zayed Bin Sultan which is 139.4 km then she would need 15.5 liters of fuel
In this question, we have been given Eliana drove her car 81 km and used 9 liters of fuel.
81 km=9 liters
9 km= 1 liter
She wants to know the distance she can drive on 22 liters of fuel. She assumes her car will continue consuming fuel at the same rate.
By unitary method,
22 liters = 22 × 9 km
= 198 km
Also, given that if Eliana plans to drive from Dubai to Abu Dhabi via Sheikh Zayed Bin Sultan which is 139.4 km.
We need to find the amount of fuel she would need.
Let 139.4 km = x liters
By unitary method,
x = 139.4 / 9
x = 15.5 liters
Therefore, Eliana can drive 198 km with 22 liters of fuel.
If Eliana plans to drive from Dubai to Abu Dhabi via Sheikh Zayed Bin Sultan which is 139.4 km then she would need 15.5 liters of fuel
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As an incoming college freshman, Tina received a 10-year $15,100 Federal Direct
Unsubsidized Loan with an interest rate of 4.29%. She knows that she can begin making
loan payments 6 months after graduation but interest will accrue from the moment the
funds are credited to his account. How much interest will accrue while she is still in
school and over the 6-month grace period for this freshman year loan?
O $2,813.28
O $2,915.06
O $3,001.32
O $3,102.38
Answers
The interest that will accrue while the college freshman is still in school and over the 6-month grace period for this freshman year loan is, approximately, B. $2,915.06.
How is the interest determined?The interest for federal college loans is based on the simple interest formula instead of compounding.
By compounding, we mean that interest is computed on the principal and accumulated interest.
Federal Direct Unsubsidized Loan = $15,100
Number of years for college = 4 years
The number of years before repayment = 4.5 years (4 years + 6 months)
Simple interest for 4.5 years = $2,915.06 ($15,100 x 4.29% x 4.5 years)
On the other hand, one can compute the compounded interest using an online finance calculator.
Compounded Interest:N (# of periods) = 4.5 years
I/Y (Interest per year) = 4.29%
PV (Present Value) = $15,100
PMT (periodic payment) during the 4.5 years = $0
Results:
FV = $18,241.85
Total Interest = $3,141.85
Thus, the interest is Option B.
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cabrinha run 3/10 mile each day for 6 days how many miles did she run in off
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3/10 mile per day for 6 days.
To find how many miles did she run multiply the miles per day by 6days:
[tex]\frac{3\text{mile}}{10\text{day}}\cdot6\text{days}=\frac{18}{10}\text{mile}=\frac{9}{5}\text{mile}[/tex]Then, in 6 days she run 9/5 milewhich describes the solution of the inequality y>-15? a) solid vertical line through (0,-15) with shading to the left of the line. b) dashed vertical line through (0,-15) with shading to the left of line. c) solid horizontal line through (0,-15) with shaing below line. d) dashed horizontal line through (0,-15) with shaing above line.
Answers
The solution to the inequality y > - 15 is all values of y greater than -15. This means the number -15 itself is not included; therefore, the line is a dashed line that passes through (0, -15). Furthermore, the > sign implies that the shaded region is found above the dashed line. Hence, the solution to our inequality is a dashed horizontal line through (0, -15), with shading above the line.
On the graph below, what is the length of side AB? B ...
Answers
The distance between two points in the plane is:
[tex]d(P,Q)=\sqrt[]{(x_2-x_1)^2+(y_2}-y_1)^2[/tex]The points A and B have coordinates A(5,3) and B(5,6). Then the distance between them is:
[tex]\begin{gathered} d(A,B)=\sqrt[]{(5-5)^2+(6-3)^2} \\ =\sqrt[]{(3)^2} \\ =\sqrt[]{9} \\ =3 \end{gathered}[/tex]Therefore, the length of the side AB is 3 units.
is this equation no solution, one solution, or infinitely may solutions
Answers
Given:
[tex]\begin{gathered} x+4y=8\ldots\ldots\ldots\ldots(1) \\ y=-\frac{1}{4}x+2\ldots\ldots\ldots\ldots(2) \end{gathered}[/tex]To solve for x and y:
Substitute the equation (2) in (1) we get,
[tex]\begin{gathered} x+4(-\frac{1}{4}x+2)=8 \\ x-x+8=8 \\ 8=8 \end{gathered}[/tex]Therefore, the given system has infinitely many solutions.
find the values of x and y that maximize the objective function c = 3x + 4y for the graph
Answers
Answer: The correct answer is x=0 and y=4 or (0,4) per the graph
Step-by-step explanation:
To find the maximum value, we must test each point using the equation:
Check for (0,4):
C=3x+4y
C=3(0)+4(4)
C=16
Check for (2,2):
C=3(2)+4(2)
C=6+8
C=14
Check for (4,0):
C=3(4)+4(0)
C=12
Answer:
Step-by-step explanation:
Earth's Moon is 384,400 km from Earth. What is the correct way to write this distance in scientific notation? O A. 3.844 x 105 km OB. 38.44 x 10-4 km O C. 38.44 x 104 km O D. 3.844 x 10-5 km SUBMIT
Answers
To do this, move the decimal in such a way that there is a non-zero digit to the left of the decimal point. The number of decimal places you shift will be the exponent by 10. If the decimal is shifted to the right the exponent will be negative. If the decimal is shifted to the left, the exponent will be positive.
So, in this case, you have
Therefore, the correct way to write this distance in scientific notation is
[tex]3.844\times10^5[/tex]And the correct answer is
[tex]undefined[/tex]Suppose that the probability that you will win a contest is 0.0002, what is theprobability that you will not win the contest? Leave your answer as a decimal and donot round or estimate your answer.
Answers
Answer:
0.9998
Explanation:
The probability that you will not win the contest can be calculated as 1 less the probability that you will win a contest, so
1 - 0.0002 = 0.9998
Therefore, the answer is 0.9998
4. Driving on the highway, you can safely drive 65 miles per hour. How far can you drive in ‘h’ hours? What is the domain of the function which defines this situation?A) 65B) the number of hours you driveC) the distance you driveD)the amount of gas you use
Answers
Answer:
[tex]\text{ The number of hours you drive.}[/tex]Step-by-step explanation:
For distance, we can apply the following equation:
[tex]\begin{gathered} d=s*h \\ where, \\ s=\text{ speed} \\ h=\text{ hours} \end{gathered}[/tex]Since we know that we can safely drive 65 miles per hour, the domain will be defined by the number of hours you drive:
[tex]d=65h[/tex]Decide whether or not the number 46/45 Could represent the probability
Answers
Solution
- A probability, by definition, is a fraction that is less than or equal to 1.
- The number 46/45 is a number greater than 1.
- Thus, the number 46/45 CANNOT be a probability
Find the volume of the figure. Round to the nearest hundredths place if necessary.
Answers
The volume of a Pyramid
Given a pyramid of base area A and height H, the volume is calculated as:
[tex]V=\frac{A\cdot H}{3}[/tex]The base of this pyramid is a right triangle, with a hypotenuse of c=19.3 mm and one leg of a=16.8 mm. The other leg can be calculated by using the Pythagora's Theorem:
[tex]c^2=a^2+b^2[/tex]Solving for b:
[tex]b^{}=\sqrt[]{c^2-a^2}=\sqrt[]{19.3^2-16.8^2}=9.5\operatorname{mm}[/tex]The area of the base is the semi-product of the legs:
[tex]A=\frac{16.8\cdot9.5}{2}=79.8\operatorname{mm}^2[/tex]Now the volume of the pyramid:
[tex]V=\frac{79.8\operatorname{mm}\cdot12\operatorname{mm}}{3}=319.2\operatorname{mm}^3[/tex]The volume of the figure is 319.2 cubic millimeters
A child has an empty box that measures 4 inches by 6 inches by 3 inches. View the figure.What is the length of the longest pencil that will fit into the box, given that the length of the pencil must be a whole number of inches? Do not round until your final answer.
Answers
Solution
For this case we can do the following:
We can find the value of s on this way:
[tex]s=\sqrt[]{6^2+4^2}=\sqrt[]{52}=7.21[/tex]And solving for r we got:
[tex]r=\sqrt[]{6^2+3^2}=\sqrt[]{45}=6.71[/tex]Then the answer for this case would be:
[tex]\sqrt[]{52}=7.21[/tex]The first three terms of a sequence are given. Round to the nearest thousandth (ifnecessary).15, 18, 108/5. find the 8th term
Answers
SOLUTION
The following is a geometric series
We will use the formula
[tex]T_n=ar^{n-1}[/tex]Where Tn is the nth term of the series,
n is the number of terms = 8,
a is the first term = 15
And r is the common ratio = 1.2 (to find r, divide the second term, 18 by the first term which is 15
Now let's solve
[tex]\begin{gathered} T_n=ar^{n-1} \\ T_8=15\times1.2^{8-1} \\ T_8=15\times1.2^7 \\ T_8=15\times3.583 \\ T_8=53.748 \end{gathered}[/tex]So the 8th term = 53.748
rewrite using a single exponent. 9 4, 9 4
Answers
We need to represent the product of two exponents as one single exponent. To do that we need to calculate their product, when the bases are equal we can conserve the base and add the exponents. We will do this below:
[tex]9^49^4=9^{4+4}=9^8[/tex]If the 10 letters are {aa,aa,aa,aa,bb,bb,cc,cc RR,RR} are available and all 10 of them are to be selected without replacement,what is the number of different permutations?
Answers
In order to calculate the number of permutations, first we start with the factorial of the number of letters.
There are 10 letters, so we start with the factorial of 10.
Then, we need to check the number of repetitions. Each repetition will be a factorial in the denominator:
[tex]x=\frac{10!}{a!\cdot b!\operatorname{\cdot}...}[/tex]We have four repetitions of aa, two repetitions of bb, two repetitions of cc and two repetitions of RR, therefore the final expression for the number of permutations is:
[tex]x=\frac{10!}{4!2!2!2!}[/tex]Calculating this expression, we have:
[tex]x=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5\operatorname{\cdot}4!}{4!\operatorname{\cdot}2\operatorname{\cdot}2\operatorname{\cdot}2}=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5}{8}=18900[/tex]Therefore there are 18900 permutations.
In the triangle below, if B = 69°, A = 32°, c = 5.7, use the Law of Sines to find a. Round your answer to the nearest hundredth.
Answers
We know that the interior angles have to add to 180°, then we have that:
[tex]\begin{gathered} C=180-69-32 \\ C=79 \end{gathered}[/tex]Hence angle C=79°.
Now that we know the angle C we can use the law of sines to find a; the law of sines states that:
[tex]\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}[/tex]From this we have the equation:
[tex]\frac{\sin A}{a}=\frac{\sin C}{c}[/tex]Plugging the values given and solving for a we have:
[tex]\begin{gathered} \frac{\sin32}{a}=\frac{\sin 79}{5.7} \\ a=(5.7)\frac{\sin 32}{\sin 79} \\ a=3.08 \end{gathered}[/tex]Therefore a=3.08