The linear model for the data is expressed as: R = 20p - 160.
How to Write a Linear Model?Using two pairs of values from the table values, say, (32, 480) and (33, 500), find the unit rate (m).
Unit rate (m) = (500 - 480)/(33 - 32) = 20/1
Unit rate (m) = 20.
Substitute (p, R) = (32, 480) and m = 20 into R = mp + b to find b
480 = 20(32) + b
480 = 640 + b
480 - 640 = b
b = -160
To write the linear model, substitute m = 20 and b = -160 into R = mp + b:
R = 20p - 160
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2. A map is drawn so that every 1 inch on the map represents 50 actual miles. The table below can be used to record the actual distances between towns on the map. Actual Distance (mi.) Map Distance (in.) Actual Distances and Map Distances 50 1 Complete the table. What is the actual distance between two towns that are 4 inches apart on the map? 2. A map is drawn so that every 1 inch on the map represents 50 actual miles . The table below can be used to record the actual distances between towns on the map . Actual Distance ( mi . ) Map Distance ( in . ) Actual Distances and Map Distances 50 1 Complete the table . What is the actual distance between two towns that are 4 inches apart on the map ?
Answer:
200miles
Step-by-step explanation
4 x 50 = 200
As for the table, just multiply each inch by 50.
For example; 7in = 350 miles because 7x50 = 350
URGENT PLEASE ANSWER THESE
Answer:
1A) 2 gallons of 20% solution and 3 gallons 15% solution needed
1B) 4 gallons of 20% solution and 1 gallons 15% solution needed
Step-by-step explanation:
1A) adding 20% salt and 15% water making 5 gallons of 17%
20% salt + 15% salt = 5 gallons of 17% salt
0.20x + 0.15(5 - x) = 0.17(5)
0.20x + 0.75 - 0.15x = 0.85
0.20x + 0.75 - 0.75 - 0.15x = 0.85 - 0.75
0.20x - 0.15x = 0.10
0.05x = 0.10
0.05x/ 0.05 = 0.10/0.05
x = 2 gallons (amount of 20% solution needed)
5 - x = 5 - 2 = 3 gallons (amount of 15% solution needed)
1B)
0.20x + 0.15(5 - x) = 0.19(5)
0.20x + 0.75 - 0.15x = 0.95
0.20x + 0.75 - 0.75 - 0.15x = 0.95 - 0.75
0.20x - 0.15x = 0.20
0.05x = 0.20
0.05x / 0.05 = 0.20/0.05
x = 4 gallons (amount of 20% solution needed)
5 - x = 5 - 4 = 1 gallons (amount of 15% solution needed)
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Consider the following scenario describing the residents of Charlestown: The list pairs resident's ages with their zip codes.
Does this scenario represent a function?
Answer:
Step-by-step explanation:
how to solve part ii and iii
(i) Given that
[tex]\tan^{-1}(x) + \tan^{-1}(y) + \tan^{-1}(xy) = \dfrac{7\pi}{12}[/tex]
when [tex]x=1[/tex] this reduces to
[tex]\tan^{-1}(1) + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]\dfrac\pi4 + 2 \tan^{-1}(y) = \dfrac{7\pi}{12}[/tex]
[tex]2 \tan^{-1}(y) = \dfrac\pi3[/tex]
[tex]\tan^{-1}(y) = \dfrac\pi6[/tex]
[tex]\tan\left(\tan^{-1}(y)\right) = \tan\left(\dfrac\pi6\right)[/tex]
[tex]\implies \boxed{y = \dfrac1{\sqrt3}}[/tex]
(ii) Differentiate [tex]\tan^{-1}(xy)[/tex] implicitly with respect to [tex]x[/tex]. By the chain and product rules,
[tex]\dfrac d{dx} \tan^{-1}(xy) = \dfrac1{1+(xy)^2} \times \dfrac d{dx}xy = \boxed{\dfrac{y + x\frac{dy}{dx}}{1 + x^2y^2}}[/tex]
(iii) Differentiating both sides of the given equation leads to
[tex]\dfrac1{1+x^2} + \dfrac1{1+y^2} \dfrac{dy}{dx} + \dfrac{y + x\frac{dy}{dx}}{1+x^2y^2} = 0[/tex]
where we use the result from (ii) for the derivative of [tex]\tan^{-1}(xy)[/tex].
Solve for [tex]\frac{dy}{dx}[/tex] :
[tex]\dfrac1{1+x^2} + \left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} + \dfrac y{1+x^2y^2} = 0[/tex]
[tex]\left(\dfrac1{1+y^2} + \dfrac x{1+x^2y^2}\right) \dfrac{dy}{dx} = -\left(\dfrac1{1+x^2} + \dfrac y{1+x^2y^2}\right)[/tex]
[tex]\dfrac{1+x^2y^2 + x(1+y^2)}{(1+y^2)(1+x^2y^2)} \dfrac{dy}{dx} = - \dfrac{1+x^2y^2 + y(1+x^2)}{(1+x^2)(1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = - \dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2) (1 + x^2y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2) (1+x^2y^2)}[/tex]
[tex]\implies \dfrac{dy}{dx} = -\dfrac{(1 + x^2y^2 + y + x^2y) (1 + y^2)}{(1 + x^2y^2 + x + xy^2) (1+x^2)}[/tex]
From part (i), we have [tex]x=1[/tex] and [tex]y=\frac1{\sqrt3}[/tex], and substituting these leads to
[tex]\dfrac{dy}{dx} = -\dfrac{\left(1 + \frac13 + \frac1{\sqrt3} + \frac1{\sqrt3}\right) \left(1 + \frac13\right)}{\left(1 + \frac13 + 1 + \frac13\right) \left(1 + 1\right)}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac{\left(\frac43 + \frac2{\sqrt3}\right) \times \frac43}{\frac83 \times 2}[/tex]
[tex]\dfrac{dy}{dx} = -\dfrac13 - \dfrac1{2\sqrt3}[/tex]
as required.
Researchers at a food company are interested in how a new ketchup made from green tomatoes (and green in color) will compare to their traditional red ketchup. They randomly assign participants to either the green or red ketchup condition. Participants indicate the tastiness of the sauce on a 20-point scale. Tastiness scores tended to be skewed (not normal). What statistical test should be used to analyze that data
The statistical test which can be used to analyze the data is chi square test.
Given two categories of tomatoes , one is red and other one is green.
When we want to compare more than two categorical variables, we can use the chi-square test. The main objective of chi square test is to compare the distribution of responses or the proportions of participants in each response category. This test does not require mean, standard deviation or anything. In the given problem we have not given sample mean, population mean, population standard deviation, etc. So we cannot use z test, t test, f test because they require sample mean, population mean , standard deviation in their calculation. The formula to be use in chi square test is as under:
[tex]X^{2}=[/tex]∑[tex](f_{0} -f_{e}) ^{2} /f_{e}[/tex]
where [tex]f_{0}[/tex]= observed frequency of each of response category.
[tex]f_{1}[/tex]=expected frequency in each of the response categories.
We can find the critical value in a table of probabilities for the chi square distribution with degree of freedom df=K-1.
Hence we can use chi square test to analyze the data.
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Kirsten is knitting and hates to donate to a local charity. Each hat takes 3 hours to make. If Kirsten knitted 3 hats and a third of another hat. how many hours did she spend knitting?
Please help? I’ve been stuck on this question
Answer:
y-intercept = -3
another point on the line = (6,-1)
Step-by-step explanation:
1. Underline important parts of the question (slope of 1/3, point (3, -2))
2. mark the point that is given (3, -2)
3. slope form is rise over run which basically means that the number one is how much the line rises, and the number three is how much the line goes to the right (since the slope is positive then it runs to the right, if it was negative it would go to the left.) Basically 1/3 = rise over run
4. put your pencil on the point (3, -2) and go up one point and over three points. continue doing so from each new point (two or three points is good enough)
5. take a ruler and draw a line that connects all three and keep it going until it crosses the y- axis (which is the vertical line)
6. the point that crosses through the y -axis is the y-intercept.
7. now you can go and look at your points that you have made and choose one of them to answer the question with
(I hope this cleared it up a bit I'm not very good at explaining things this is the best I can do)
Simplify the expression to a polynomial in
standard form:
(3x-1)(3x^2+3x+8)
Answer:
9x^3+6x^2+21x-8
Step-by-step explanation:
(3x-1) (3x ^ 2 + 3x + 8) =3x(3x ^ 2 + 3x + 8) +(-1)(3x ^ 2 + 3x + 8) =9x^3+9x^2+24x-3x^2-3x-8 = 9x^3+6x^2+21x-8
Answer:
[tex]\huge\boxed{\sf 9x^3+6x^2+21x-8}[/tex]
Step-by-step explanation:
Given expression:= [tex](3x-1)(3x^2+3x+8)[/tex]
[tex]Expand[/tex]
[tex]= 3x(3x^2+3x+8)-1(3x^2+3x+8)\\\\Multiply\\\\=9x^3+9x^2+24x-3x^2-3x-8\\\\Combine \ like \ terms\\\\= 9x^3+9x^2-3x^2+24x-3x-8\\\\= 9x^3+6x^2+21x-8\\\\\rule[225]{225}{2}[/tex]
change the unit of length 6ft 2in= ___ft?
Answer:
Step-by-step explanation:
Comment
You don't have to do anything to the 6 feet. It already is in feet. It's the 2 inches you have to worry about.
2 inches = 2/12 of a foot.
2/12 = 1/6 = 0.167 feet
Answer
6 feet 2 inches = 6.167 feet
Step-by-step explanation:
True or False? If ab<0, then a<0, b>0 or a>0, b<0
Answer:
True
Step-by-step explanation:
If ab < 0, then ab = negative #.
In order for ab to be a negative #, one of them has to be negative while the other one needs to be positive.
Example:
a = -2, b = 1
ab < 0
(-2)(1) < 0
-2 < 0, TRUE
a < 0
-2 < 0, TRUE
b > 0
1 > 0, TRUE
If I switch a = -2 to 1 and b = 1 to -2, a > 0 and b < 0 is true too.
Which graph represents the function f(x)=8 1/3 x?
The graph that represents the exponential function [tex]y = 8\left(\frac{1}{3}\right)^x[/tex] is given at the end of the answer.
What is an exponential function?An exponential function is modeled by:
[tex]y = ab^x[/tex]
In which:
a is the initial value. that is, y when x = 0.b is the rate of change.In this problem, the function is:
[tex]y = 8\left(\frac{1}{3}\right)^x[/tex]
It has an initial value of 8 and rate of change of 1/3, as shown in the graph given at the end of the answer.
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When drawing the stretchout arc for the pattern of a cone fitting, the radius should be equal to _____.
When drawing the stretch-out arc for the pattern of a cone fitting, the radius should be equal to the true length of the side.
The 3 tactics typically used in developing patterns are parallel-line, radial-line, and triangular improvement.
To make such objects, a drafter has to first draw them as a pattern or pattern improvement. A pattern improvement additionally known as a stretch out or simply an improvement is a format of an object made on a single flat aircraft.
In radial traces, we develop styles for shapes that have a taper, all element lines (bends) should radiate returned to a not unusual factor, a radius point. We want matters for this procedure to paintings: A radius factor is in the center (right cone). A radius point is inside an inexpensive distance.
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Given the following information: calculate the perimeter of triangle rst. assume rzx is similar to rst. rz=21, ry= 24, rx= 39, zy=9 and sw=18
If triangle RST and RZX are similar then the perimeter of triangle RZX is 150 units.
Given triangle RZX and RST are similar and RZ=28, RY=40,ZY=16 and SW=24.
In the similar triangles the ratio of sides are in proportion to the sides of other triangle.
The ratio of ZY to SW is 16:24 meaning it is 2:3. So all the sides of RST are 1.5 times as long as RZX.
RZ=28 so RS=28*1.5=42
RX=40 so RT=40*1.5=60
ZX=ZY+XY and since they are the of the same length , ZX=32 (ZY given)
Then ST=48
Perimeter of a triangle is the sum of sides of the triangle.
Perimeter of RST=RS+ST+RT
=42+60+48
=150
Hence the perimeter of triangle RST is 150 units.
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Question is incorrect as it should be as under:
Given the following information, calculate the perimeter of triangle RST. Assume triangle RZX is similar to RST. RZ=28, RY=40, ZY=16 and SW=24.
Figure should also be given.
You design a logo for your soccer team. The logo is 3 inches by 5 inches. You decide to dilate the logo to 1.5 inches by 2.5 inches. What is the scale factor of this dilation?
Answer:
1/2
Step-by-step explanation:
The dimensions are cut in half ....scale factor 1/2
find x in the 45 45 90 tri!
Step-by-step explanation:
simple answer mate..
under root 2 is right because angle are same.. option d
Answer:
√2
Step-by-step explanation:
Angle ( θ ) = 45
Hypotenuse side = 2
Opposite side = x
Formula : -
sin θ = Opposite side / Hypotenuse side
Note :
The value of sin 45 = 1/√2
sin 45 = x/2
1/√2 = x/2
Cross multiply,
x*√2 = 2*1
x√2 = 2
2 can be simplified as √2 * √2
x√2 = √2*√2
Divide √2 on both sides,
x = √2
complete the square to rewrite y=x^2-6x+14 in vertex form. then state whether the vertex is a maximum or minimium and give its coordinates
The vertex form of the Quadratic equation is y = [tex](x-3)^{2}[/tex] +5. The vertex is (3, 5), which is a maximum point.
what is a quadratic equation?A quadratic equation is an algebraic equation with the power of its variable 2.
it can be written in standard form which is called vertex form or in the general form.
Analysis:
y = [tex]x^{2}[/tex] -6x +14
y = [tex]x^{2}[/tex] -6x +9 +5
Simplifying this perfect square expression y = [tex]x^{2}[/tex] -6x +9
y = [tex]x^{2}[/tex] -3x -3x +9
y = x(x-3) -3(x-3) + 5
y = (x-3)(x-3) +5
y = [tex](x-3)^{2}[/tex] + 5
In conclusion, the Vertex form of the equation is y = [tex](x-3)^{2}[/tex] + 5
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Can someone help me with question 3 please. Please make sure you actually answer the question and not type something that isn't regarding the question.
Answer:
Option C
Step-by-step explanation:
We can see that these two triangles are similar triangles. And based on the sides we also know triangle ZXY and triangle FDE have a ratio of 3:2. We can find side x by the equation:
3:2=9:x
3x=18
x=6 or option C
Answer:
C.6
Step-by-step explanation:
12x=8
x=8/12=2/3
2/3*9=18/3=6
PLEASE HELP ME
6 x 6 x ? = 261
Answer:
7.25
Step-by-step explanation:
Simplify
36x = 261
Divide both sides by 36
x = 261/36
Simplify
x = 7.25
Hope this helps!
Answer:
7.25 or 7[tex]\frac{1\\}{4}[/tex]
Step-by-step explanation:
261 ÷ 6 ÷ 6 = 7.25
helllppppppppppp please
Answer:
No, it will not pass specifications.
Step-by-step explanation:
It says that the max slope of the wheelchair ramp should be 1/12.
To calculate slope, you do rise over run or y values over x values. In this case, the rise over run is 5/50. 5/50 simplifies to 1/10.
1/10 > 1/12, meaning that it is more than the maximum slope.
Joseph noticed that a fork in the road created 3 angles. If two angles are as shown (both 160) , what is the measure of angle x?
The measure of angle x that is formed around a point is: 145°.
What is the Angle Around a Point?The measure of the angle around a point equals 360 degrees.
The the angles formed by the fork in the road equals 360 degrees.
Therefore, we would have the following equation:
x + 165 + 50 = 360
x + 215 = 360
Subtract both sides by 215
x = 360 - 215
x = 145°
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−9x+2>18 OR 13x+15≤−4 solve for x please
Inequalities help us to compare two unequal expressions. The value of x must be less than -19/13.
What are inequalities?Inequalities help us to compare two unequal expressions. Also, it helps us to compare the non-equal expressions so that an equation can be formed. It is mostly denoted by the symbol <, >, ≤, and ≥.
The given inequalities can be simplified as,
1.)
−9x+2>18
−9x>18-2
−9x>16
x < -16/9
Thus, the value of x must be less than -16/9.
2.)
13x+15≤−4
13x ≤ -19
x ≤ -19/13
Thus, the value of x must be less than -19/13.
For the given condition −9x+2>18 OR 13x+15≤−4, the value of x must follow any one of the inequality. Hence, the value of x must be less than -19/13.
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The following confidence interval is obtained for a population proportion, p: (0.688, 0.724). Use these confidence interval limits to find the point estimate, p
The point estimation based on the given interval limits is 0.706.
What is a point estimate?The use of sample data to compute a single value that will serve as a "best guess" or "best estimate" of an unknown population characteristic is referred to as point estimation.
The point estimation based on the given interval limits is,
Point Estimate = (0.688+0.724)/2 = 0.706
Hence, The point estimation based on the given interval limits is 0.706.
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Identify the dependent variable in the following research description: As a researcher, I examine the effect of different clothing materials on sleep quality In this experiment, participants are randomly assigned to groups. One group of participants wears organic cotton shirts before they go to bed. Another group wears organic linen shirts before they go to bed. Sleep quality is measured. a. participants.
b. sleep quality.
c. organic cotton shirts and organic linen shirts. d. different clothing materials.
The dependent variable in the given research description is sleep quality.
Given variables :1) participants, 2) sleep quality, 3) organic cotton shirts and organic linen shirts, 4) different clothing materials.
We have to identify the dependent variable among all the variables.
Dependent variable is a variable whose quantity depends on other variables.
Independent variable is a variable whose value does not depend on other variables.
In the given research there are four variables:
1) participants,
2) sleep quality,
3) organic cotton shirts and organic linen shirts,
4) different clothing materials.
Among all these sleep quality is a dependent variable because it depends on the quality of clothes whether they or organic shirts or organic linen shirts.
It may be possible that some people get quality sleep in organic cotton shirts and some may get in organic linen shirts.
Hence the dependent variable is sleep quality.
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what is the decimal of 1/2
Answer:
0.5
one divide by two is zero point five
[tex]\frak{Hi!}[/tex]
[tex]\orange\hspace{300pt}\above2[/tex]
We need to find the decimal form of [tex]\boldsymbol{\sf{\displaystyle\frac{1}{2}}}[/tex].
But first we need to convert it into a fraction whose
denominator is 10, or 100, or 1,000, etc.
The closest number that's divisible by 2 is 10.
So that's the new denominator of the fraction.
Now we should also multiply the top times the number
that we multiplied the bottom by. See how this works?
[tex]\boldsymbol{\sf{\displaystyle\frac{1\times5}{2\times5}}}[/tex]. This yields
[tex]\boldsymbol{\sf{\displaystyle\frac{5}{10}}}[/tex]. And 5/10 in decimal-form yields
[tex]\boldsymbol{\sf{0.5}}[/tex].
[tex]\orange\hspace{300pt}\above3[/tex]
[tex]\LARGE\boldsymbol{\sf{calligraphy}}[/tex]
Which exponential equation is equivalent to this logarithmic equation? log x = 4
Answer:
[tex]x=10^{4}[/tex]
Step-by-step explanation:
Switching from Logarithmic to Exponential form:An exponential base is the inverse of a logarithm with the same base. Given the equation log x = 4, note that the base of the logarithm isn't written in (it has no subscript). By default, the base of the logarithm function is base-10. So, to rewrite the equation using an exponential, we need to undo the logarithm with an exponential of the same base.
[tex]\log(x)=4[/tex]
[tex]10^{\log{(x)}}=10^{4}[/tex]
[tex]x=10^{4}[/tex]
Analogies with Addition/SubtractionThink about the equation x + 7 = 10.
I could tell you this is the equation in "addition form". What if I asked you to write an equation in "subtraction form"? While neither "addition form" nor "subtraction form" have been defined explicitly, one could undo the addition, and get an equation with subtraction in it, which one could argue is a "subtraction form" of the equation.
[tex]x+7=10[/tex]
Subtracting 7 from both sides to undo the addition...
[tex](x+7)-7=(10)-7[/tex]
Simplifying the left side since the "+7" and "-7" cancel...
[tex]x=10-7[/tex]
This is arguably in a "subtraction form" (there's a subtraction in it), whereas the original equation had addition in it.
While we could simplify this particular equation's right-hand side, we might not always be able to (what if the 10 had been a "y"... then the "y" and the "7" aren't like terms, and they would have to remain separate)
Similarly, in the logarithm problem, one could simplify 10^4 (it's 10,000), but one doesn't have to, and one won't always be able to.
For the logarithm problem, x=10^4 is the exponential form, as requested.
Which represents where f(x) = g(x)?
The equation represents the intersections. The correct option is A:
f(2) = g(2) = 0f(0) = g(0) = 4.Which represents where f(x) = g(x)?
If we have two functions:
y = f(x) and y = g(x), the equation:
f(x) = g(x) gives the value of x such that the two functions have the same output. So, that equation gives the intersection points between the two graphs.
By looking at the graph, we can see that we have two intersections, one at:
(2, 0) and other at (0, 4).
This means that:
f(2) = g(2) = 0f(0) = g(0) = 4.So the correct option is the first one.
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HELP!!
[tex]\sf If \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 47, \: and \: x > 0, \: then \: what \: is \: value \: of \: x + \dfrac{1}{x} [/tex]
DON'T SPAM.
Answer:
[tex]x+\cfrac{1}{x} =7[/tex]================
Given:[tex]x^2+\cfrac{1}{x^2} =47[/tex]Add 2 to both sides of equation:
[tex]x^2+\cfrac{1}{x^2}+2 =47+2[/tex]Then follow the steps:
[tex]x^2+\cfrac{1}{x^2}+2*x*\cfrac{1}{x} =49[/tex][tex](x+\cfrac{1}{x})^2=7^2[/tex]Take square root of both sides to get:
[tex]x+\cfrac{1}{x} =\pm\ 7[/tex]We are taking the positive value as we are told x > 0, hence the answer is 7.
[tex] \qquad \qquad \bf \huge\star \: \: \large{ \underline{Answer} } \huge \: \: \star[/tex]
x + 1/x = 7[tex]\textsf{ \underline{\underline{Steps to solve the problem} }:}[/tex]
[tex] \qquad❖ \: \sf \: {x}^{2} + \cfrac{1}{ {x}^{2} } = 47[/tex]
[tex] \qquad❖ \: \sf \: {x}^{2} + \cfrac{1}{ {x}^{2} } + 2 - 2 = 47[/tex]
( adding and subtracting 2, doesn't change the value )
[tex] \qquad❖ \: \sf \: {x}^{2} + \cfrac{1}{ {x}^{2} } + 2 = 47 + 2[/tex]
[tex] \qquad❖ \: \sf \: {x}^{2} + \cfrac{1}{ {x}^{2} } + \bigg(2 \sdot x \sdot \cfrac{1}{x} \bigg) = 49[/tex]
( x cancel out, so no change in value )
[tex] \qquad❖ \: \sf \: \bigg(x + \cfrac{1}{x} \bigg) {}^{2} = 49[/tex]
( use identity a² + 2ab + b² = (a + b)² )
[tex] \qquad❖ \: \sf \: \bigg(x + \cfrac{1}{x} \bigg) {}^{} = \sqrt{49} [/tex]
[tex] \qquad❖ \: \sf \: x + \cfrac{1}{x} {}^{} = \pm7[/tex]
since we have to take positive value, i.e greater than 0
[tex] \qquad❖ \: \sf \: x + \cfrac{1}{x} {}^{} = 7[/tex]
[tex] \qquad \large \sf {Conclusion} : [/tex]
Therefore, the required value is 7
The equation y = ax describes the graph of a line. If the value of a is negative,
the line:
Answer:
The line has a downward slope
Step-by-step explanation:
If y = ax, it means that a represents the slope of the line. So, if A is negative, the line has a downward slope. In other words, for every increase in the value of x, there is a decrease in the value of y (increase in the negative direction). You can imagine the line y = -x, where the value of a is -1. When x is 1, y is -1; when x is 2, y is -2; when x is 5, y is -5. You can imagine that the graph looks like a diagonal line, much like y = x, except the function values are positive in the second quadrant and negative in the fourth (it slants downward to the right side). I hope that answers your question.
Determine the number of possible solutions for a triangle with b = 37, a = 32, and b = 27.
There are two(2) possible solutions for the given parameters of a triangle. Using the law of cosines, the parameters of the triangle are calculated.
What is the law of cosines?The law of cosines is used to relate the angles and side lengths of a triangle.
They are given as follows:
a² = b² + c² -2bc cos(A)
b² = a² + c² - 2ac cos(B)
c² = a² + b² - 2ab cos(C)
Calculation:The given triangle has B = 37°, a = 32 and b = 27
So, using the law b² = a² + c² - 2ac cos(B)
On substituting,
27² = 32² + c² - 2(32)c cos(37°)
⇒ c² - 50.56 c + 1024 = 729
⇒ c² - 50.56 c + 295 = 0
Since the obtained equation is quadratic, it has 2 solutions.
Using quadratic formula the solutions are calculated.
c = [50.56 ± [tex]\sqrt{50.56^2-4(1)(295)}[/tex]]/2(1)
On simplifying,
c = 43.8 and c = 6.7
Therefore, there are 2 possible solutions for the given triangle.
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A tennis tournament has 2n contestants. We want to pair them up for the first round of singles matches. Show that the number of different ways in which first round matches can be conducted is
There are
[tex]\dbinom{2n}2 = \dfrac{(2n)!}{2! (2n-2)!}[/tex]
ways of pairing up any 2 members from the pool of [tex]2n[/tex] contestants. Note that
[tex](2n)! = 1\times2\times3\times4\times\cdots\times(2n-2)\times(2n-1)\times(2n) = (2n-2)! \times(2n-1) \times(2n)[/tex]
so that
[tex]\dbinom{2n}2 = \dfrac{(2n)\times(2n-1)\times(2n-2)!}{2! (2n-2)!} = \boxed{n(2n-1)}[/tex]
The number of different ways in which first-round matches can be conducted is n (2n - 1).
We have,
Number of contestant = 2n
Number of contestants in each match = 2
Now,
The number of different ways in which first-round matches can be conducted.
A combination formula is used.
= [tex]^{2n}C_2[/tex]
= 2n! / 2! (2n - 2)!
= 2n (2n - 1)(2n - 2)! / [2 x (2n - 2)!]
= 2n (2n - 1) / 2
= n (2n - 1)
Thus,
The number of different ways in which first-round matches can be conducted is n (2n - 1).
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