We want to simplify the following expression:
[tex]\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}[/tex]This means that we want to "remove" the denominator".
STEP 1If we observe the denominator:
[tex](2-\sqrt[]{3})[/tex]If we multiply it by
2 + √3, then
[tex]\begin{gathered} (2-\sqrt[]{3})(2+\sqrt[]{3}) \\ =4-\sqrt[]{3}^2=4-3=1 \end{gathered}[/tex]STEP 2We know that if we multiply both sides of a fraction by the same number or expression, the fraction will remain the same, then we multiply both sides by 2 + √3:
[tex]\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}=\frac{(2+\sqrt[]{3})(2+\sqrt[]{3})}{(2-\sqrt[]{3})(2+\sqrt[]{3})}[/tex]For the denominator, as we analyzed before
[tex](2-\sqrt[]{3})(2+\sqrt[]{3})=1[/tex]For the denominator:
[tex](2+\sqrt[]{3})(2+\sqrt[]{3})=(2+\sqrt[]{3})^2[/tex]Then,
[tex]\frac{2+\sqrt[]{3}}{2-\sqrt[]{3}}=\frac{(2+\sqrt[]{3})(2+\sqrt[]{3})}{(2-\sqrt[]{3})(2+\sqrt[]{3})}=\frac{(2+\sqrt[]{3})^2}{1}=(2+\sqrt[]{3})^2[/tex]STEP 3Now, we can simplify the result:
[tex]\begin{gathered} (2+\sqrt[]{3})^2=(2+\sqrt[]{3})(2+\sqrt[]{3}) \\ =2^2+2\sqrt[]{3}+(\sqrt[]{3})^2+2\sqrt[]{3} \\ =4+4\sqrt[]{3}+3 \\ =7+4\sqrt[]{3} \end{gathered}[/tex]Answer: 7+4√3Given the points (3, -2) and (4, -1) find the slope
Slope is
[tex]\text{slope}=\frac{y2-y1}{x2-x1}[/tex]Then:
[tex]\text{slope}=\frac{-1-(-2)}{4-3}=\frac{-1+2}{1}=\frac{1}{1}=1[/tex]Answer: slope = 1
Which will hold more cake batter; the rectangular pan or two round pans? The volume of the rectangle pan is 234 in 3rd power. Find the volume of the two round pans and choose your decision. Round your answer to the nearest tenth if necessary.
Solution
Given a rectangular pan and two round pans
The volume, V, of the rectangular pan is 234 in³
A round pan is in the form of a cylinder
To find the volume, V, of the a round pan, the formula is
[tex]V=\pi r^2h[/tex]Where
[tex]\begin{gathered} d=8in \\ r=\frac{d}{2}=\frac{8}{2}=4in \\ r=4in \\ h=2in \end{gathered}[/tex]Substitute the variables into the formula above
[tex]\begin{gathered} V=\pi\left(4\right)^2\left(2\right)=100.53in^3\text{ \lparen two decimal places\rparen} \\ V=100.53in^3\text{ \lparen two decimal places\rparen} \end{gathered}[/tex]Since, there are two round pans, the volume of the two round pans will be
[tex]2\times100.53=201.06in^3[/tex]nce, the volume of the two round pans is 201.06 i³.
Since the volume of the two round pans (201.06in³) is less than the volume of the rectangular pan (234in³), the rectangular pan will hold more.
Hence, the answer is rectangular pan (option a)
Divide. −3.52−2.2 What is the quotient
The quotient of - 3.32 / - 2.2 is 8 ÷5.
What is the quotient?Given fraction:
-3.32 / -2.2
First step is to re-write the given fraction which is - 3.52 / - 2.2
3.52 / 2.2
Second step is to convert the decimal to fraction
352 ÷ 100 / 22 ÷10
Third step is to reduce the fraction
reducing 352/100
=(2^5 × 11)/(2^2 × 5^2)
= [(2^5 × 11) ÷ 2^2] / [(2^2 × 5^2) ÷ 2^2]
= (2^3 × 11)/5²
= 88/25
Reducing 22/10
Divide the numerator and denominator by the greatest common divisor
= 22 ÷ 2 / 10 ÷ 2
= 11 /5
Now let determine or find the quotient
88 ÷ 25 × 11 ÷ 5
= 8 ÷ 5
Therefore we can conclude that 8 ÷ 5 is the quotient.
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you want to buy one pair of pants, one t-shirt, and several pairs of socks. the pants cost $24.95, the t-shirt cost $22.50 and the socks are $5.95 per pair. how many pairs of socks can you buy if you have $80 to spend?*for this question you have to write and inequality solve it and answer the question in words*
ANSWER
5 pairs
EXPLANATION
We have that:
- one pair of pants cost $24.95
- one t-shirt costs $22.50
- one pair of socks cost $5.95
You want to buy one pair of pants, one t-shirt and several pairs of socks and you don't want to spend more than $80.
This means that everything you spend must be less than or equal to $80.
Let the number of pairs of socks you can buy be x.
This therefore means that:
[tex]\begin{gathered} 24.95+22.50+(5.95\cdot x)\text{ }\leq80 \\ \Rightarrow\text{ }24.95\text{ + 22.50 + 5.95x }\leq80 \\ \text{Find x:} \\ 47.45\text{ + 5.95x }\leq80 \\ \Rightarrow\text{ 5.95x }\leq80\text{ - 47.45} \\ 5.95x\text{ }\leq32.55 \\ x\text{ }\leq\frac{32.55}{5.95} \\ x\leq5.47 \end{gathered}[/tex]Because we know that the number of pairs of socks must be a whole number, we have to approximiate to whole number, which will be:
x = 5
Therefore, you can buy at most 5 pairs of socks.
Which of the following are maximum and minimum points of the function y = 2 cos x -1?
The graph of the function is shown below
From the options provided
Option A is correct because the maximum and minimum values are satisfied
shron spent 1 1/4 hours reading her book report and 2 2/5 hours doing her other homework. how much longer did sharon spent doing her homework than reading her book report
sharon spent 23/20 hour doing her homework than reading.
What is fraction?The fractional bar is a horizontal bar that divides the numerator and denominator of every fraction into these two halves.
The number of parts into which the whole has been divided is shown by the denominator. It is positioned in the fraction's lower portion, below the fractional bar.How many sections of the fraction are displayed or chosen is shown in the numerator. It is positioned above the fractional bar in the upper portion of the fraction.Given:
Sharon spent reading the book = [tex]1 \frac{1}{4}[/tex] = 5/4 hours
= 25/20 hours
Sharon spend doing homework = [tex]2 \frac{2}{5}[/tex] = 12/5 hours
= 48/20 hours
So, the difference between both activities
= 48/20- 25/20
= 23/20
Hence, sharon spent 23/20 hour doing her homework than reading.
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Finding a final amount in a word problem on exponential growth or decay
Answer:
19g
Explanation:
Given the starting amount as 155grams, the below amount will be left after the 1st half life;
[tex]\frac{155g}{2}=77.5g[/tex]After the 2nd half-life, the below amount will remain;
[tex]\frac{77.5g}{2}=38.75g[/tex]After the 3rd half-life, the below amount will remain;
[tex]\begin{gathered} \frac{38.75}{2}=19.375g\approx19g\text{ (rounding to the nearest gram)} \\ \end{gathered}[/tex]
Determine whether the statement is true or false.
2E{x|XEN and x is odd}
Is the statement true or false?
O True
O False
The given statement exists as false. An expression, rule, or law in mathematics establishes the link between an independent variable and a dependent variable.
What is meant by function?An expression, rule, or law in mathematics establishes the link between an independent variable and a dependent variable (the dependent variable).
The characteristic that every input is associated with exactly one output defines a function as a relationship between a set of inputs and a set of allowable outputs.
A relation between a collection of inputs and outputs is known as a function. A function exists, to put it simply, a relationship between inputs in which each input exists connected to precisely one output.
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Which expressions are equivalent to z + (z + 6)? Choose all answers that apply: A (2 + 2) + (2 + 6) 00 (z + 6 + 6 © 2(z + 3)
ANSWER:
[tex]2\cdot(z+3)[/tex]STEP-BY-STEP EXPLANATION:
We have the following expression
[tex]z+(z+6)[/tex]We operate and we have
[tex]z+z+6=2z+6=2\cdot(z+3)[/tex]which equation represents a line having a slope of 5/2 and a y intercept of (0,-4)
First you must know the standard equation of a line and this is expressed as:
[tex]y\text{ = mx+c}[/tex]where:
m is the slope of the line
c is the intercept
Given
Slope m = 5/2
Next is to get the intercept c:
To do that, you will substitute m = 5/2 and the coordinate (0, -4) into the equation above as shown:
[tex]\begin{gathered} -4\text{ = 5/2(0)+c} \\ -4\text{ = 0+c} \\ c\text{ = -4} \end{gathered}[/tex]Next is to get the required equation by substituting m = 5/2 and c = -4 into the equation above as shown:
[tex]\begin{gathered} y\text{ = mx + c} \\ y\text{ = }\frac{5}{2}x\text{ +(-4)} \\ y\text{ = }\frac{5}{2}x\text{ - 4} \end{gathered}[/tex]Hence the required equation is espressed as:
[tex]y\text{ = }\frac{5}{2}x-4[/tex]The formula k=5/9(f-32)+273.15 converts temperature of the object in a laboratory is cooled to 1.5 kelvin. What is the temperature of the object in degrees fahrenheit?
The temperature of the object is -456.97 degrees
What is the max/min of the quadratic equation in factored form: f(x) = 0.5(x +3)(x-7)
F(x) = 1/2(x+3)(X-7)
Step 1 ; expand the function
F(x)= 1/2(x²-7x+3x-21)
F(x) = 1/2(x² - 4x-21)
F(x) = 1/2x² - 2x-21/2
Step 2 : Take the second derivative of F(x)
This means you are to differentiate F(X) twice
[tex]\begin{gathered} F(x)=\frac{1}{2}x^2-2x-\frac{21}{2} \\ \text{First derivative is} \\ F^!(x)\text{=x-2} \\ F^{!!}(x)=1 \\ \text{the second derivative =1} \end{gathered}[/tex]The second derivative is greater than 0, so it is a minimum point
Put x=1 in F(x) to find the value
[tex]\begin{gathered} f(x)=\frac{1}{2}(1)^2_{}-\text{ 2(1)-}\frac{21}{2} \\ f(x)=\frac{1}{2}-2-\frac{21}{2} \\ f(x)=-2-\frac{20}{2} \\ f(x)\text{ =-12} \end{gathered}[/tex]The minimum of the quadratic equation is -12
Point B is on line segment AC. Given BC = 10 and AB = 5, determine the lengthAC.Answer: AC= Anyone know how to solve these???
3
1) Let's sketch that, to better understand this:
2) Considering the Segment Addition Postulate, we can write that:
DF = DE + EF Plug into that the given values
9 = 6 + EF
9-6 = 6-6 + EF
3 = EF
EF =3
3) Hence, the line segment EF is 3 units long
Is the following sequence arithmetic, geometric, or neither?1, 5, 25, 125, 625
This is a geometric sequence
This is because we can find the common ratio and not common difference
difference between managerial and financial accounting?
The main difference between managerial accounting and financial accounting is who the statement is being prepared for.
What differentiates managerial and financial accounting?There are several things that differentiate managerial accounting from financial accounting but the main one is the target of the financial statements.
Financial accounting is used for financial statements for those outside the company such as investors, customers, and the government. As such, it has to abide by certain standards.
Managerial accounting on the other hand, is for managers and decision makers in the company. It is less restrictive as the main goal is to provide information for managers and not those outside.
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- A chemist mixes 2,362 milliliters of a solution. The solution must be divided equally among 8 beakers. How much solution should be poured into each beaker?
Answer:
295.25mm
Explanation:
If the chemist mixes 2362mm of a solution and needs to divide it equally into 8 breakers, to determine how much solution should be poured into each breaker, we have to divide 2362mm divide 8;
[tex]\frac{2362}{8}=295.25\operatorname{mm}[/tex]the volume v of a fixed amount of a gas variety directly as the temperature T and inversely as the pressure P. suppose that V =42cm3 when T=84 kelvin and P=8kg/cm2 find the temperature when V =74cm3 and P=10 kg/cm2
ok
I'll use the law of gases to solve this problem
V1 = 42 cm^3
T1 = 84 °K
P = 8 Kg/cm^2
T2 = x
V2 = 74 cm^3
P2 = 10 kg/cm^2
Equation
P1V1/T1 = P2V2/T2
Solve for T2
T2 = P2V2T1 / P1V1
Substitution
T2 = (10*74*84) / (8*42)
Simplification
T2 = 62160 / 336
Result
T2 = 185°K
23 – 4u < 11 what is the answer
23 - 4u < 11
23 is adding on the left, then it will subtract on the right
-4u < 11 - 23
-4u < -12
-4 is multiplying on the left, then it will divide on the right. Remember that dividing by a negative number changes the sign.
u > (-12)/(-4)
u > 3
The two angles shown are supplementary.162°Which equation can be solved to find the value of x, and what is the value of x?A.162° + Xo = 90°; x = 72B.162° + x° = 180°; x = 18C.162° + x = 360°; x= 198D.162° + x = 180°; x = 242
Supplementary angles are two angles whose sum is exactly 180, therefore:
162 + x = 180
Solving for x:
subtract 162 from both sides:
x = 180 - 162
x = 18
A half-marathon has 53 runners. A first-, second-, and third-place trophy will be awarded. Howmany different ways can the trophies be awarded?
Let's use the combination formula:
[tex]\begin{gathered} C(n,k)=nCk=\frac{n!}{k!(n-k)!} \\ n=53 \\ k=3 \\ C(53,3)=53C3=\frac{53!}{3!(50)!}=23426 \end{gathered}[/tex]on a trip of 2,300 miles, a missionary went 9 times as far by plane as by car. How for did the missionary travel by plane
Let the trip by car be c and the trip by plane be p.
The missionary travelled 9 times as far by plane as he did by car. This means if his trip by car is modelled by c, then the trip by plane would be 9c.
Hence, knowing that the entire trip of 2300 miles is by plane and by car;
[tex]\begin{gathered} c+p=2300 \\ \text{When p=9c, then} \\ c+9c=2300 \\ 10c=2300 \\ \text{Divide both sides by 10} \\ c=230 \\ \text{Therefore, his trip by plane would be derived as;} \\ c+p=2300 \\ 230+p=2300 \\ \text{Subtract 230from both sides} \\ p=2070 \end{gathered}[/tex]csc 0 (sin2 0 + cos2 0 tan 0)=sin 0 + cos 0= 1
Okay, here we have this:
Considering the provided expression, we are going to prove the identity, so we obtain the following:
[tex]\frac{csc\theta(sin^2\theta+cos^2\theta tan\theta)}{sin\theta+cos\theta}=1[/tex][tex]\frac{\frac{1}{sin\vartheta}(sin^2\theta+cos^2\theta\frac{sin\theta}{cos\theta})}{sin\theta+cos\theta}=1[/tex][tex]\frac{\frac{1}{sin\vartheta}(sin^2\theta+cos\text{ }\theta sin\theta)}{sin\theta+cos\theta}=1[/tex][tex]\frac{(\frac{sin^2\theta}{sin\theta}+\frac{cos\text{ }\theta sin\theta}{sin\theta})}{sin\theta+cos\theta}=1[/tex][tex]\frac{(sin\text{ }\theta+cos\text{ }\theta)}{sin\theta+cos\theta}=1[/tex][tex]\frac{1}{1}=1[/tex][tex]1=1[/tex]A committee must be formed with 5 teachers and 4 students. If there are 6 teachers to choose from, and 15 students, how many different ways could the committee be made?
There are 6 teachers and 15 students to choose from
To form a committee of 5 teachers and 4 students
The combination rule will be applied
From 6 teachers, The number of ways 5 teachers can be selected is
[tex]^6C_5[/tex]From 15 students, the number of ways 4 students can be selected is
[tex]^{15}C_{4^{}}[/tex]Therefore, the total number of ways a committee of 5 teachers and 4 students can be formed from 6 teachers and 15 students is
[tex]^6C_5\times^{15}C_{4^{}}[/tex]Simplifying this gives
[tex]^6C_5\times^{15}C_{4^{}}=\frac{6!}{(6-5)!\times5!}\times\frac{15!}{(15-4)!\times4!}[/tex]This further gives
[tex]\begin{gathered} ^6C_5\times^{15}C_{4^{}}=\frac{6!}{1!\times5!}\times\frac{15!}{11!\times4!} \\ ^6C_5\times^{15}C_{4^{}}=\frac{6\times5!}{1\times5!}\times\frac{15\times14\times13\times12\times11!}{11!\times4\times3\times2\times1} \end{gathered}[/tex]Cancel out common factors
[tex]\begin{gathered} ^6C_5\times^{15}C_{4^{}}=6\times\frac{15\times14\times13\times12}{4\times3\times2\times1} \\ ^6C_5\times^{15}C_{4^{}}=6\times\frac{32760}{24} \\ ^6C_5\times^{15}C_{4^{}}=6\times1365 \\ ^6C_5\times^{15}C_{4^{}}=8190 \end{gathered}[/tex]Therefore, the number of ways the committee can be formed is 8190 ways
Graph the solution of the linear inequality and answer the questions on the bottom
The inequality is given
[tex]-2y\leq6x+18[/tex]To draw the graph of the inequality which is less than equal to
The problem is below, we know the man weighs 60, the cat weighs 10 but we’re having a hard time explaining how
Given data:
The weight of man and daughter = 90
The weight of man and cat is = 70
The weight of cat and daughter = 40 .
The man weights 60 kg.
then daughter weight = 90-60 = 30.
Thus the daughter weight is 30kg.
therefore the cat weight iwith daughter, 40-30 = 10 .
also with father, 70-60 = 10 .
Thus, the father weight is 60 kg.
The daughter weight is 30 kg and
The cat weight is 10 kg.
find the values of x y and z.The answers are in degrees.
Answer
x = 35°
y = 145°
z = 25°
Explanation
We are told to solve for x, y and z.
Considering the first triangle with angles 55°, x° and the right angle (90°).
The sum of angles in a triangle is 180°.
So,
x° + 55° + 90° = 180° (Sum of angles in a triangle is 180°)
x° + 145° = 180°
x = 180° - 145° = 35°
Then, we can solve for y. Angles x and y are on the same straight line, and the sum of angles on a straight line is 180°
x° + y° = 180°
35° + y° = 180°
y° = 180° - 35°
y° = 145°
We can then solve for z°. The big triangle has angles (55° + 10°), z° and the right angle (90°).
The sum of angles in a triangle is 180°.
So,
55° + 10° + z° + 90° = 180°
z° + 155° = 180°
z = 180° - 155°
z° = 25°
Hope this Helps!!!
A person chooses a number in a set containing the first 5 cubic numbers. Find the set representing the event E of choosing a number that can be evenly divided by 2. Give your answer as a set, e.g. {1,2,3}, using the cubed number (not the base number) and do not include E= in your answer.
The first 5 cubic numbers are:
[tex]\lbrace1,8,27,64,125\rbrace[/tex]To find the set that represents event E we have to choose the numbers from the set above that are evenly divided by 2; this means that we have to choose the numbers that are multiple of 2. We notice that this numbers are 8 and 64, therefore the set that represents event E is:
[tex]\lbrace8,64\rbrace[/tex]What is the standard form of the complex number that point A represents?
Answer
-3 + 4i
Explanation
The standard form for a complex number is given by:
[tex]\begin{gathered} Z=a+bi \\ \text{Where:} \\ a\text{ is the real part,} \\ b\text{ is the imaginary part} \end{gathered}[/tex]From the graph, the coordinates of A corresponding to the real axis and imaginary axis is traced in blue color in the graph below:
Hence, the standard form of the complex number that a represents is: -3 + 4i
The ages of 12 selected participants in a Students’ Congress are:
16, 24, 23, 20, 12, 27, 19, 25, 21, 23, 14, 13
What is the mean age of the participants?
The mean age of the 12 participants in a student congress is 19.75.
given 16,24,23,20,12,27,19,25,21,23,14,13
Mean generally means average. The formula of Mean is as follows
Add the given numbers 16+24+23+20+21+27+19+25+21+23+14+13 = 237
Divide the total number by the number of participants that is
237/12 = 19.75
The mean age of the participants = 19.75
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Solve the inequality
-1/3x - 12 > 21 or -6x + 10 < -2
x < ? or x > ?
solve for both
Answer:
x < 2 or x > -11
Step-by-step explanation:
b. 1. add 12 to both sides to get -1/3x > 33
2. multiply by -3/1 to both sides to get x > -11
a. 1) subtract 10 to both sides
2) divide by -6 to both sides
