A test is to be conducted with certain types of questions and each type of question weighs certain number of points.
A test would consist of two types of questions. These two types will be assigned variables that will denote the number of questions respectively as follows:
[tex]\begin{gathered} \text{True and False: x} \\ \text{MCQS : y} \end{gathered}[/tex]We are given that the entire test will consits of 20 questions. We can express the total number of questions on the test in terms of number of True and False questions ( x ) and number of MCQS ( y ) as follows:
[tex]\begin{gathered} \text{Total number of Questions = True and False + MCQS} \\ \textcolor{#FF7968}{20}\text{\textcolor{#FF7968}{ = x + y }}\textcolor{#FF7968}{\ldots Eq1} \end{gathered}[/tex]Further information is given to us in the questions regarding the number of points aloted to each type. The total weightage of each type of question on the test can be expressed as a product of ( number of each type * point weight of each type ).
The point weights for each type of questions are:
[tex]\begin{gathered} \text{True and False ( x ) : 3 points each} \\ \text{MCQs ( y ) : 11 points each} \end{gathered}[/tex]The total weights of each types of questions are:
[tex]\begin{gathered} \text{True and False ( points ) = 3}\cdot x \\ \text{MCQS ( points ) = 11}\cdot x \end{gathered}[/tex]We are given that the entire test is worth ( 100 points ). We express the total number of points of the test in terms of total weight of each type of question as follows:
[tex]\begin{gathered} test\text{ points = True and False ( points ) + MCQS ( points )} \\ \textcolor{#FF7968}{100}\text{\textcolor{#FF7968}{ = 3}}\textcolor{#FF7968}{\cdot x}\text{\textcolor{#FF7968}{ + 11}}\textcolor{#FF7968}{\cdot y\ldots}\text{\textcolor{#FF7968}{ Eq2}} \end{gathered}[/tex]We have two equations that express the total number of questions ( Eq 1 ) and total points ( Eq2 ) of the test in terms of number of True and False questions ( x ) and number of MCQs on the test ( y ).
[tex]\begin{gathered} \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ + y = 20 }}\textcolor{#FF7968}{\ldots Eq1} \\ \textcolor{#FF7968}{3x}\text{\textcolor{#FF7968}{ + 11y = 100 }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq2}} \end{gathered}[/tex]We will solve the above two equations simultaneously using Elimination method.
Step1: Multiply Eq1 with ( -3 )
[tex]\begin{gathered} -3\cdot\text{ ( x + y ) = -3}\cdot20 \\ \textcolor{#FF7968}{-3x}\text{\textcolor{#FF7968}{ - 3y = -60 }}\textcolor{#FF7968}{\ldots}\text{\textcolor{#FF7968}{ Eq3}} \end{gathered}[/tex]Step2: Add Eq 3 into Eq 2
[tex]\begin{gathered} -3x\text{ - 3y = -60 } \\ 3x\text{ + 11y = 100} \\ =========== \\ 8y\text{ = 40 } \\ \textcolor{#FF7968}{y}\text{\textcolor{#FF7968}{ = 5}} \\ =========== \end{gathered}[/tex]Step3: Back susbtitue the value of ( y ) into ( Eq1 )
[tex]\begin{gathered} x\text{ + ( 5 ) = 20 } \\ \textcolor{#FF7968}{x}\text{\textcolor{#FF7968}{ = 15 }} \end{gathered}[/tex]Therefore, the number of each type of questions that must be put on the test should be.
[tex]\begin{gathered} \text{\textcolor{#FF7968}{True and False ( x ) = 15}} \\ \text{\textcolor{#FF7968}{MCQs ( y ) = 5}} \end{gathered}[/tex]name the three congruent parts shown by the marks on each drawing
In this case the aswer is very simple. .
The congruent parts are the equal parts in the 2 triangles.
Therefore, the congruent parts would be:
1. side AB and side XY
2. ∠ A and ∠ X
3. side AC and side XZ
That is the solution. .
Let f(x) = 2x-1 and g(x) = x2 - 1. Find (f o g)(-7).
Answer: (f o g)(-7) = 95
Step by step solution:
We have the two functions:
[tex]\begin{gathered} f(x)=2x-1 \\ g(x)=x^2-1 \end{gathered}[/tex]We need to find (f o g)(-7) or f(g(-7)), first we evaluate g(-7):
[tex](f\circ g)(-7)=f(g(-7))[/tex][tex]g(-7)=-7^2-1=49-1=48[/tex]Now we evaluate f(48):
[tex]f(48)=2\cdot48-1=96-1=95[/tex]can I please getsome help with this question here, I can't really figure out how to find side PQ
SOLUTION
The following diagram will help us solve the problem
(a) From the diagram, the height of the parallelogram is given as TR, and it is 40 mm
Now we can use the area which is given to us as 3,600 square-mm to find the base of the parallelogram, which is PQ
So,
[tex]\begin{gathered} \text{Area }of\text{ a parallelogram = base}\times height \\ So\text{ } \\ 3600=PQ\times TR \\ 3600=PQ\times40 \\ 3600=40PQ \\ \text{dividing by 40, we have } \\ \frac{3600}{40}=\frac{40PQ}{40} \\ PQ=90 \end{gathered}[/tex]Hence PQ is 90 mm
(b) Now, note that the side
[tex]PS=QR[/tex]So, we will find QR
Also, since we have PQ, we can find TQ, that is
[tex]\begin{gathered} PQ=PT+TQ \\ 90=60+TQ \\ TQ=90-60 \\ TQ=30mm \end{gathered}[/tex]Note that triangle QRT is a right-angle triangle, and QR is the hypotenuse or the longest side
From pythagoras
[tex]\text{hypotenuse}^2=opposite^2+adjacent^2[/tex]So,
[tex]\begin{gathered} QR^2=TR^2+TQ^2 \\ QR^2=40^2+30^2 \\ QR^2=1600+900 \\ QR^2=2,500 \\ QR=\sqrt[]{2,500} \\ QR=50mm \end{gathered}[/tex]Now, since
[tex]\begin{gathered} PS=QR \\ \text{then } \\ PS=50mm \end{gathered}[/tex]Hence PS is 50 mm
Write the decimal as a quotient of two integers in reduced form.
0.513
The given decimal can be written as a quotient of 513/1000.
What is quotient?
In maths, the result of dividing a number by any divisor is known as the quotient. It refers to how many times the dividend contains the divisor. The statement of division, which identifies the dividend, quotient, and divisor, is shown in the accompanying figure. The dividend 12 contains the divisor 2 six times. The quotient is always less than the dividend, whether it is larger or smaller than the divisor.
we can write the decimal given 0.513 as a answer of of 513 divided by 1000.
I.e.
[tex]0.513 = \frac{513}{1000}[/tex]
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What is the value of 3-(-2) how can I solve this questions
Explanation:
[tex]\text{Given: }3-\mleft(-2\mright)[/tex]To find the value od 3-(-2), we will multiply the sign at the outer with the inner
[tex]undefined[/tex]What is the average rate of change from g(1) to g(3)?Type the numerical value for your answer as a whole number, decimal or fractionMake sure answers are completely simplified
The average rate of change from g(1) to g(3)
[tex]\frac{g(x)_3-g(x)_1}{X_3-X_1}_{}[/tex]where
[tex]g(x)_3=-20,g(x)_1=-8,x_3=3,x_1=\text{ 1}[/tex][tex]\begin{gathered} =\frac{-20\text{ --8}}{3-1}\text{ = }\frac{-20\text{ +8}}{2} \\ =\frac{-12}{2} \\ -6 \end{gathered}[/tex]Hence the average rate of change is -6
Solve and graph on a number line. 2(x-1) 4 or 2 (x-1)>4
The given inequality is:
2 (x - 1
help meeeeeeeeee pleaseee !!!!!
Because x is continuous, we should use interval notation, the domain is:
D: [1, ∞)
How to find the domain?For a function y = f(x), we define the domain as the set of possible inputs of the function (possible values of x).
To identify the domain, we need to look at the horizontal axis. The minimum value is the one we can see in the left side, and the maximum is the one we could see on the right side.
There we can see that the domain starts at x = 1 and extends to the left, so the notation we can use for the domain is:
D: x ≥ 1
We know that the value x =1 belongs because there is a closed dot there.
The correct option is A, because the domain is continuous (as we can see in the graph), we should use interval notation. In this case the domain can be written as:
D: [1, ∞)
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Help me with my schoolwork what is the slope of line /
The two points given on the line are
[tex]\begin{gathered} (x_1,y_1)\Rightarrow(-2,9) \\ (x_2,y_2)\Rightarrow(6,1) \end{gathered}[/tex]The slope of line that passes through (x1,y1) and (x2,y2) is gotten using the formula below
[tex]\begin{gathered} m=\frac{\text{change in y}}{\text{change in x}} \\ m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}[/tex]By substituting the values, we will have
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{1-9}{6-(-2)} \\ m=-\frac{8}{6+2} \\ m=-\frac{8}{8} \\ m=-1 \end{gathered}[/tex]Therefore,
The slope of the line = -1
An airplane is taking off at angle of 9 degrees and traveling at a speed of 200 feet per second in relation to the ground. If the clouds begin at an altitude of 4,000 feet, how many seconds will it take for the airplane to be in the clouds?
ANSWER
[tex]\begin{equation*} 127.85\text{ }seconds \end{equation*}[/tex]EXPLANATION
First, let us make a sketch of the problem:
To find the time it will take the airplane to be in the clouds, we first have to find the distance flown by the airplane in attaining that height, x.
To do this, apply trigonometric ratios SOHCAHTOA for right triangles:
[tex]\sin9=\frac{4000}{x}[/tex]Solve for x:
[tex]\begin{gathered} x=\frac{4000}{\sin9} \\ x=25,569.81\text{ }ft \end{gathered}[/tex]Now, that we have the distance, we can solve for the time by applying the relationship between speed and distance:
[tex]\begin{gathered} speed=\frac{distance}{time} \\ \Rightarrow time=\frac{distance}{speed} \end{gathered}[/tex]Substitute the given values into the formula above and solve for time:
[tex]\begin{gathered} time=\frac{25569.81}{200} \\ time=127.85\text{ }seconds \end{gathered}[/tex]That is the number of seconds that it will take.
21. Juanita is packing a box that is 18 inches long and 9 inches high. The total volume of the box.1,944 cubic inches. Use the formula V = lwh to find the width of the box. Show your work
The width of the box is 12 inches
Explanations:
The formula for calculating the volume of a rectangular box is expressed as:
[tex]V=\text{lwh}[/tex]where:
• l is the ,length ,of the box
,• w is the ,width, of the box
,• h is the ,height ,of the box
Given the following parameters
• length = 18 inches
,• heigh = 9 inches
,• volume = 1,944 cubic inches
Substitute the given parameters into the formula to calculate the width of the box as shown:
[tex]\begin{gathered} 1944=18\times w\times9 \\ 1944=162w \end{gathered}[/tex]Divide both sides by 162 to have:
[tex]\begin{gathered} 162w=1944 \\ \frac{\cancel{162}w}{\cancel{162}}=\frac{1944}{162} \\ w=12\text{inches} \end{gathered}[/tex]Hence the width of the box is 12 inches
The function f(x) = 40(0.9)^x represents the deer population in a forest x years after it was first studied. What was the deer population when it was first studied?a. 44b.40c. 36d.49
We are given the function that models a deer population:
[tex]f(x)=40(0.9)^x[/tex]Where x is the years since the study started. If we want to know the initial population, we want to find the population at x = 0 years.
Thus:
[tex]f(0)=40(0.9)^0=40\cdot1=40[/tex]The correct answer is option b. 40
The linear regressionequation andcorrelation coefficientfrom the above datawas calculated to be:Predicted y = 16.2+2.45(x) with r = 0.98What is the coefficientof determination?Answer Choices:A. Coefficient of determination = 0.98B. Coefficient of determination = 0.96C. Coefficient of determination = 0.99D. Coefficient of determination cannot be determined with only the given information.
Given:
[tex]\text{ coefficient of correlation \lparen r\rparen = 0.98}[/tex]To find:
Coefficient of determination
Explanation:
The coefficient of determination is also known as the R squared value, which is the output of the regression analysis method.
If the value of R square is zero, the dependent variable cannot be predicted from the independent variable.
So, here the required coefficient of determination is:
[tex]r^2=(0.98)^2=0.9604\approx0.96[/tex]Final answer:
Hence, the required coefficient of determination is (B) 0.96.
The House of Pizza say that their pizzas are 14 inches wide, but when you measured it, the pizza was 12 inches. What is your percent error? Make sure to include your percent sign! (Round to 2 decimals)
The percent error of the house of the pizza would be 2.
The difference between the estimated and actual values in comparison to the actual value is expressed as a percentage. In other words, the relative error multiplied by 100 equals the percent error.
How to calculate the percent error?Percent errors indicate the magnitude of our errors when measuring something in an analysis process. Lower percentage errors indicate that we are getting close to the accepted or original value.
Suppose the actual value and the estimated values after the measurement are obtained. Then we have:
Error = Actual value - Estimated value
To determine the percent error, we will measure how much percent of the actual value, the error is, in the estimated value.
We have been given that House of Pizza says that their pizzas are 14 inches wide, but when measured, the pizza was 12 inches.
WE know that Error = Actual value - Estimated value
Then Error = 14 - 12 = 2
Therefore, the percent error would be 2.
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Which of these would not produce a representative sample that determines the favoritesport of the students at the local high school?ask every tenth student from a list of names in the student directoryask every tenth student who arrives at school on Wednesdayask ten students wearing football jerseys each day for a weekask five students from each classroom chosen by picking numbersMy Progress >
Answer: ask ten students wearing football jerseys each day for a week
This sample wouldn't b representative because, the use of a footblla
“Use the properties to rewrite this expression with the fewest terms possible:3+7(x - y) + 2x - 5y”
Expanding 7(x - y) in the above expression gives
[tex]-5y^{}+2x+7x-7y+3[/tex]adding the like terms (2x+ 7x) and (-5y-7y) gives
[tex](-5y-7y)+(2x+7x)+3[/tex][tex]\rightarrow\textcolor{#FF7968}{-12y+8x+3.}[/tex]The last expression is the simplest form we can convert our expression into.
The following distribution represents the number of credit cards that customers of a bank have. Find the mean number of credit cards.Number of cards X01234Probability P(X)0.140.40.210.160.09
To solve this problem we have a formula at hand: the mean (m) number of credits cards is
[tex]m=\sum ^{}_XX\cdot P(X)[/tex]Then,
[tex]m=0\cdot0.14+1\cdot0.4+2\cdot0.21+3\cdot0.16+4\cdot0.09=1.66[/tex]The profit of a cell-phone manufacturer is found by the function y= -2x2 + 108x + 75 , where x is the cost of the cell phone. At what price should the manufacturer sell the phone tomaximize its profits? What will the maximum profit be?
Hello!
First, let's rewrite the function:
[tex]y=-2x^2+108x+75[/tex]Now, let's find each coefficient of it:
• a = -2
,• b = 108
,• c = 75
As we have a < 0, the concavity of the parabola will face downwards.
So, it will have a maximum point.
To find this maximum point, we must obtain the coordinates of the vertex, using the formulas below:
[tex]\begin{gathered} X_V=-\frac{b}{2\cdot a} \\ \\ Y_V=-\frac{\Delta}{4\cdot a} \end{gathered}[/tex]First, let's calculate the coordinate X by replacing the values of the coefficients:[tex]\begin{gathered} X_V=-\frac{b}{2\cdot a} \\ \\ X_V=-\frac{108}{2\cdot(-2)}=-\frac{108}{-4}=\frac{108}{4}=\frac{54}{2}=27 \end{gathered}[/tex]So, the coordinate x = 27.
Now, let's find the y coordinate:[tex]\begin{gathered} Y_V=-\frac{\Delta}{4\cdot a} \\ \\ Y_V=-\frac{b^2-4\cdot a\cdot c}{4\cdot a} \\ \\ Y_V=-\frac{108^2-4\cdot(-2)\cdot75}{4\cdot(-2)} \\ \\ Y_V=-\frac{11664+600}{-8}=\frac{12264}{8}=1533 \end{gathered}[/tex]The coordinate y = 1533.
Answer:
The maximum profit will be 1533 (value of y) when x = 27.
Raphael has an odd-shaped field shown in Figure 13-2. He wants to put a four-strand barbed wire fence around it for his cattle.A. What is the perimeter of the field?b. How many 80-rod rolls of barbed wire does he need topurchase?c. How many acres will be fenced?
Answer: Total perimeter = 9, 962.01 feet
The figure is a composite structure
It contains a rectangle and triangle
The perimeter of a rectangle is given as
Perimeter = 2( length + width)
length of the rectangle = 1500ft
Width of the rectangle = 1390 ft
Perimeter = 2( 1500 + 1390)
Perimeter = 2(2890)
Perimeter = 5780 ft
To calculate the perimeter of a triangle
[tex]\begin{gathered} \text{Perimeter = a + b + }\sqrt[]{a^2+b^2} \\ a\text{ = 1050ft and b = 1390 ft} \\ \text{Perimeter = 1050 + 1390 + }\sqrt[]{1050^2+1390^2} \\ \text{Perimeter = 2440 + }\sqrt[]{1,102,\text{ 500 + 1, 932, 100}} \\ \text{Perimeter = 2400 + }\sqrt[]{3,034,600} \\ \text{Perimeter = 2440 + 1,742,01} \\ \text{Perimeter = }4182.01\text{ f}eet \end{gathered}[/tex]The total perimeter of the field = Perimeter of the rectangle + perimeter of the right triangle
Total perimeter = 5780 + 4182.01
Total perimeter = 9, 962.01 feet
Given the conversion factor which cube has the larger surface area?
Given the surface area of a cube as
[tex]\begin{gathered} SA=6l^2 \\ \text{where l is the length} \end{gathered}[/tex]Given Cubes A and B
[tex]\begin{gathered} \text{Cube A} \\ l=19.5ft \end{gathered}[/tex][tex]\begin{gathered} \text{Cube B } \\ l=6m\text{ } \\ \text{ in ft}\Rightarrow\text{ 1m =3.28ft} \\ l=6\times3.28ft=19.68ft \end{gathered}[/tex]Find the surface area of the cubes and compare them to know which one is larger
[tex]\begin{gathered} \text{Cube A} \\ SA=6\times19.5^2=6\times380.25=2281.5ft^2 \end{gathered}[/tex][tex]\begin{gathered} \text{Cube B} \\ SA=6\times19.68^2=6\times387.3024=2323.8144ft^2 \end{gathered}[/tex]Hence, from the surface area gotten above, Cube B has a larger surface area than Cube A
I thought of a number. from ²/₇ parts of that number I subtracted 0,4 and got ⅗. The number is: A: ²⁄₇ B: ⅖ C: 3,5D: 4,5
Note : The use of comma as number separator represent point in this solution
Step 1: Let the number be x, thus, 2/7 parts of the number means
[tex]\frac{2}{7}x[/tex]Step 2: Subtract 0,4 from 2/7 parts of x
[tex]\frac{2}{7}x-0,4\Rightarrow\frac{2}{7}x-\frac{4}{10}[/tex]Step 3: Equate the expression above to 3/5
[tex]\frac{2}{7}x-\frac{4}{10}=\frac{3}{5}[/tex]Step 4: Simplify the equation above
[tex]\begin{gathered} \frac{2}{7}x-\frac{4}{10}=\frac{3}{5} \\ \frac{20x-28}{70}=\frac{3}{5}(\text{cross multiply)} \\ 5(20x-28)=70(3) \\ 100x-140=210 \\ 100x=210+140 \\ 100x=350 \\ \frac{100x}{100}=\frac{350}{100}(\text{Divide both side by 100)} \\ x=3,5 \end{gathered}[/tex]Hence, the number is 3,5
Option C is correct
hannah paid 15.79 for a dress that was originally marked 24.99 what js the percent of discount
The percentage of discount is 37%
Here, we want to calculate the percentage of discount
The first thing we need to do here is to calculate the discount amount
Mathematically, we have this as;
[tex]24.99-15.79\text{ = 9.2}[/tex]Now, we find the percentage of 24.99 is this discount
We have this as;
[tex]\frac{9.2}{24.99}\text{ }\times100\text{ \% = 36.8\%}[/tex]The percentage of discount is approximately 37%
One ton (2,000 pounds) is equivalent to 907 kilograms. A baby elephant weighs about 91 kilograms atbirth. Approximately how many pounds (lbs.) is this?A 200 lbs.B 400 lbs.C 600 lbs.D 1,000 lbs.
Since 2000 pounds = 907 kilograms, use the conversion factor:
[tex]\frac{2000\text{ pounds}}{907\operatorname{kg}}[/tex]To find out what 91 kg are equal to, measured in pounds:
[tex]91\operatorname{kg}=\frac{2000\text{ pounds}}{907\operatorname{kg}}=\frac{91\cdot2000}{907}\text{ pounds =200.66 pounds}[/tex]Therefore, a baby elephant weighs about 200 lbs.
Consider the function f(x)= square root 5x-10 for the domain [2, +infinity). find f^-1(x), where f^-1 is the inverse of f. also state the domain of f^-1 in interval notation.edit: PLEASE DOUBLE CHECK ANSWERS.
let f(x) = y
To find the inverse of f(x), we would interchange x and y:
[tex]\begin{gathered} y\text{ = }\sqrt[]{5x\text{ - 10}} \\ \text{Interchanging:} \\ x\text{ = }\sqrt[]{5y\text{ - 10}} \end{gathered}[/tex]Then we would make the subject of formula:
[tex]\begin{gathered} \text{square both sides:} \\ x^2\text{ = (}\sqrt[]{5y-10)^2} \\ x^2\text{ = 5y - 10} \end{gathered}[/tex][tex]\begin{gathered} \text{Add 5 to both sides:} \\ x^2+10\text{ = 5y} \\ y\text{ = }\frac{x^2+10}{5} \\ \text{The result above is }f^{\mleft\{-1\mright\}}\mleft(x\mright) \end{gathered}[/tex][tex]\begin{gathered} f^{\mleft\{-1\mright\}}\mleft(x\mright)\text{ = }\frac{x^2+10}{5} \\ The\text{ domain of the inverse is all real numbers} \\ \text{That is from negative infinity to positive infinity} \end{gathered}[/tex]In interval notation:
[tex]\begin{gathered} \text{Domain = (-}\infty,\text{ }\infty) \\ f^{\{-1\}}(x)\text{ = }\frac{x^2+10}{5}\text{for domain (-}\infty,\text{ }\infty) \end{gathered}[/tex]Find the sum of the arithmetic series given a1 =2, an =35 an n = 12
Given:
[tex]a_1=2,a_n=35,n=12[/tex]Required:
Find the sum of the arithmetic series.
Explanation:
The sum of the arithmetic series when the first and the last term is given by the formula.
[tex]S_n=\frac{n}{2}(a_1+a_n)[/tex]Substitute the given values in the formula.
[tex]\begin{gathered} S_n=\frac{12}{2}(2+35) \\ =6(37) \\ =222 \end{gathered}[/tex]Final Answer:
Option D is the correct answer.
The data shows the total number of employee medical leave days taken for on-the-job accidents in the first six months of the year: 12, 6, 15, 9, 28, 12. Use the data for the exercise. Find the standard deviation.
ANSWER:
The standard deviation is 7
STEP-BY-STEP EXPLANATION:
The standard deviation formula is as follows
[tex]\sigma=\sqrt[]{\frac{\sum^N_i(x_i-\mu)^2_{}}{N}}[/tex]The first thing is to calculate the average of the sample like this:
[tex]\begin{gathered} \mu=\frac{12+6+15+9+28+12}{6} \\ \mu=\frac{82}{6}=13.67 \end{gathered}[/tex]Replacing and calculate the standard deviation:
[tex]\begin{gathered} \sigma=\sqrt[]{\frac{(12_{}-13.67)^2_{}+(6_{}-13.67)^2_{}+(15_{}-13.67)^2_{}+(9_{}-13.67)^2_{}+(28-13.67)^2_{}+(12_{}-13.67)^2_{}}{6}} \\ \sigma=\sqrt[]{\frac{293.33}{6}} \\ \sigma=6.99\cong7 \end{gathered}[/tex]Pour subtracted from the product of 10 and a number is at most-20,
we have
four subtracted from the product of 10 and a number is at most-20
Let
n ----> the number
so
[tex]10n-4\leq-20[/tex]solve for n
[tex]\begin{gathered} 10n\leq-20+4 \\ 10n\leq-16 \\ n\leq-1.6 \end{gathered}[/tex]the solution for n is the interval (-infinite, -1.6]
All real numbers less than or equal to negative 1.6
Use the Binomial Theorem to expand the expression.(x +6)^3
ok
[tex]\begin{gathered} (x+6)^3=^{}x^3+3(x)^2(6)+3(x)(6)^2+6^3 \\ \text{ = x}^3+18x^2\text{ + 3(36)x + 216} \\ \text{ = x}^3+18x^2\text{ + 108x + 216} \end{gathered}[/tex][tex]\begin{gathered} (a+b)^3\text{ } \\ first\text{ term = a} \\ \text{second term = b} \\ \text{theorem } \\ (a+b)^3=a^3+3a^2b+3ab^2+b^3 \end{gathered}[/tex]that is the rule
just identify a and b in your problem
a = x
b = 6
Substitute in the theorem, and simplify
Determine the common ratio for each of the following geometric series and determine which one(s) have an infinite sum.
I. 4+5+25/4+…
II. -7+7/4-7/9+…
III. 1/2-1+2…
IV. 4- ++...
A. III only
B. II, IV only
C. I, Ill only
D. I, II, IV only
The correct answer is Option A ( III Only). I . -16 sum cannot be negative, II. Not a G.P, III. Sum = 1/4, and IV. Not a G.P.
Solution:Given geometric series,
I. 4 +5 +25 /4 ….
The common ratio(r) is (5/1)/(4/1) = 5/4.
S∞ = a / ( 1 - r)
= 4 / ( 1 - 5/4)
= 4 / -1/4
S∞ = -16.
Since sum cannot be negative.
II . -7 + 7/3 - 7/9+ ....
Here common ratio = -7 / (7/3) = -1/3
but - 7/9 / 7 /3 = 7/9
Here there is no common ratio so this not a G.P.
iii. 1/2 -1 + 2.....
Common ratio = -1 / (1/2) = -2
S∞ = a / ( 1 - r)
= 1/2 / (1 -(-2))
S∞ = 1/4.
iv 4 - 8/5 +16/5.....
Here there is no common ratio.
So this is not a G.P.
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match the system of equations with the solution set.hint: solve algebraically using substitution method.A. no solutionB. infinite solutionsC. (-8/3, 5)D. (2, 1)
We will solve all the systems by substitution method .
System 1.
By substituting the second equation into the first one, we get
[tex]x-3(\frac{1}{3}x-2)=6[/tex]which gives
[tex]\begin{gathered} x-x+6=6 \\ 6=6 \end{gathered}[/tex]this means that the given equations are the same. Then, the answer is B: infinite solutions.
System 2.
By substituting the first equation into the second one, we have
[tex]6x+3(-2x+3)=-5[/tex]which gives
[tex]\begin{gathered} 6x-6x+9=-5 \\ 9=-5 \end{gathered}[/tex]but this result is an absurd. This means that the equations represent parallel lines. Then, the answer is option A: no solution.
System 3.
By substituting the first equation into the second one, we obtain
[tex]-\frac{3}{2}x+1=-\frac{3}{4}x+3[/tex]by moving -3/4x to the left hand side and +1 to the right hand side, we get
[tex]-\frac{3}{2}x+\frac{3}{4}x=3-1[/tex]By combining similar terms, we have
[tex]-\frac{3}{4}x=2[/tex]this leads to
[tex]x=-\frac{4\times2}{3}[/tex]then, x is given by
[tex]x=-\frac{8}{3}[/tex]Now, we can substitute this result into the first equation and get
[tex]y=-\frac{3}{2}(-\frac{8}{3})+1[/tex]which leads to
[tex]\begin{gathered} y=4+1 \\ y=5 \end{gathered}[/tex]then, the answer is option C: (-8/3, 5)
System 4.
By substituting the second equation into the first one, we get
[tex]-5x+(2x-3)=-9[/tex]By combing similar terms, we have
[tex]\begin{gathered} -3x-3=-9 \\ -3x=-9+3 \\ -3x=-6 \\ x=\frac{-6}{-3} \\ x=2 \end{gathered}[/tex]By substituting this result into the second equation, we have
[tex]\begin{gathered} y=2(2)-3 \\ y=4-3 \\ y=1 \end{gathered}[/tex]then, the answer is option D