The number of republicans and democrats will be 47 and 53 respectively.
Define equation.Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side. It shows that the expressions printed on the left and right sides have an equal relationship. In any mathematical equation, we have LHS = RHS (left hand side = right hand side). Equations can be solved to find the value of an unknown variable that represents an unknown quantity.
Given data -
Total number of members in U.S senate = 100
Let the number of republicans in U.S senate be x members
As republicans outnumber democrats by six,
Therefore, number of democrats = (x+6) members
According to the given data,
x + (x+6) = 100
2x + 6 = 100
2x = 94
x = 47
Number of members as a republicans will be 47.
Number of members as a democrats will be (100-47) = 53
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A university student is selecting courses for his next semester. He can choose from 8 science courses and 4 humanities courses. In how many ways can he choose 4 courses if more than 2 must be science courses
The number of ways which he can choose 4 courses if more than 2 must be science is; 224 ways.
Combination of outcomes;
He can choose from 4 humanities courses and 8 science courses.
If the condition requires that he chooses more than 2 science courses, it follows that;
He can only choose three science courses and only 1 humanities courses.
8C3 x 4C1 = 56x 4 = 224
On this note, the number of ways he can choose the required 4 courses is; 224 ways.
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18#Suppose that 303 out of a random sample of 375 letters mailed in the United States were delivered the day after they were mailed. Based on this, compute a 90% confidence interval for the proportion of all letters mailed in the United States that were delivered the day after they were mailed. Then find the lower limit and upper limit of the 90% confidence interval. Carry your intermediate computations to at least three decimal places. Round your answers to two decimal places. (If necessary, consult a list of formulas.)Lower limit:Upper limit:
ANSWER:
Lower limit: 0.77
Upper limit: 0.84
STEP-BY-STEP EXPLANATION:
Given:
x = 303
n = 375
We calculate the value of the proportion in the following way:
[tex]\begin{gathered} p=\frac{x}{n}=\frac{303}{375} \\ \\ p=0.808 \end{gathered}[/tex]For a 90% confidence interval we have the following:
[tex]\begin{gathered} \alpha=100\%-90\%=10\%=0.1 \\ \\ \alpha\text{/2}=0.1=0.05 \\ \\ \text{ For the normal table this corresponds to:} \\ \\ Z_{\alpha\text{/2}}=1.645 \end{gathered}[/tex]We calculate the limits of the 90% confidence interval using the following formula:
[tex]\begin{gathered} \text{ Lower limit: }p-Z_{\alpha\text{/2}}\cdot\sqrt{\frac{p\cdot(1-p)}{n}}=\:0.808-1.645\cdot\sqrt{\frac{0.808\cdot\left(1-0.808\right)}{375}}\:=0.77 \\ \\ \:\text{Upper limit: }p-Z_{\alpha\text{/2}}\cdot\sqrt{\frac{p\cdot\left(1-p\right)}{n}}\:=0.808+1.645\cdot\sqrt{\frac{0.808\cdot\left(1-0.808\right)}{375}}=0.84 \end{gathered}[/tex]sin38° = ? (Write the Trigonometic ratio as a fraction)
Solution
The trigonometric ratio of sin 38 =
[tex]\begin{gathered} \sin \text{ 38 =}\frac{opposite}{hypothenus} \\ \text{opposite = a} \\ hypothenuse\text{ = c} \\ Sin\text{ 38 =}\frac{a}{c} \\ \end{gathered}[/tex][tex]Sin38^o\text{ = 0.6157= }\frac{6157}{10000}[/tex]Factor by grouping c^2-8c +16 -4d^2
INFORMATION:
We have the following expression
[tex]c^2-8c+16-4d^2[/tex]And we must factor it by grouping
STEP BY STEP EXPLANATION:
To factor it by grouping, we must:
1. group the first 3 terms of the expression
[tex](c^2-8c+16)-4d^2[/tex]2. factor the expression in the parenthesis
[tex](c-4)^2-4d^2[/tex]3. rewrite 4d^2 as unique exponential expression
[tex](c-4)^2-(2d)^2[/tex]4. factor by square difference
[tex]((c-4)+2d)((c-4)-2d)[/tex]5. simplify
[tex]=(c+2d-4)(c-2d-4)[/tex]ANSWER:
the factoring for c^2-8c +16 -4d^2 by grouping is
[tex](c+2d-4)(c-2d-4)[/tex]Rewrite (4x4 + 8x2 + 3)/(4x2) in the form q(x) + r(x)/b(x) where q(x) = quotient, r(x) = remainder, and b(x) = divisor.
ANSWER
(x² + 2) + 3/4x²
EXPLANATION
Since the divisor is a single-term polynomial, to write the answer in the requested form, we can distribute the divisor into each of the terms of the dividend,
[tex]\frac{4x^4+8x^2+3}{4x^2}=\frac{4x^4}{4x^2}+\frac{8x^2}{4x^2}+\frac{3}{4x^2}[/tex]And simplify,
[tex]\frac{4x^4}{4x^2}+\frac{8x^2}{4x^2}+\frac{3}{4x^2}=(x^2^{}+2)^{}+\frac{3}{4x^2}[/tex]Hence, the answer is (x² + 2) + 3/4x².
Verify algebratically if each function is odd, even, or neither. For question #5 only
Answer:
[tex]\text{ odd}[/tex]Explanation:
Here, we want to check if the given function is even or odd
To do that, we find g(x) and g(-x)
If g(x) equals g(-x), the the function is even. Otherwise, the function is odd
We find the functions as follows:
[tex]\begin{gathered} g(x)=7x^3\text{ - x} \\ g(-x)=7(-x)^3-(-x) \\ g(-x)=-7x^3\text{ + x} \end{gathered}[/tex]Finally:
[tex]\begin{gathered} \text{ since g(x) }\ne\text{ g(-x) } \\ \text{Function g(x) is odd} \end{gathered}[/tex]the lines are perpendicular if the slope of one line is 4/7 what is the slope of the other line
if two lines are perpendicular, it is true that:
[tex]\begin{gathered} m1\cdot m2=-1 \\ Let\colon \\ m1=\frac{4}{7} \\ m2=other_{\text{ }}line \\ \frac{4}{7}\cdot m2=-1 \\ solve_{\text{ }}for_{\text{ }}m2 \\ m2=-1\cdot\frac{7}{4} \\ m2=-\frac{7}{4} \end{gathered}[/tex]Which function has the following characteristics? • A vertical asymptote at x = 3 • A horizontal asymptote at y = 2 Domain: {** +3] 2x - 8 X - 3 y=x2-9 2 9 x² - 4 4 OB. V C. 2x2 - 18 x² - 4 4 2x2 - 8 O D. ** - 9
SOLUTION
To get this, note that the vertical asymptote can be gotten by setting the denominator to be equal to 0.
If we do this, we will notice that the vertical asymptote of option A and option D is x = 3
That is
for option A
[tex]\begin{gathered} y=\frac{2x-8}{x-3} \\ x-3=0 \\ x=3 \end{gathered}[/tex]For option D
[tex]\begin{gathered} y=\frac{2x^2-8}{x^2-9} \\ x^2-9=0 \\ x^2=9 \\ x=3 \end{gathered}[/tex]So, the answer is either option A or D.
But to get the correct answer, let us look at the graphs for both functions
Graph of A
[tex]y=\frac{2x-8}{x-3}[/tex]From the graph, you can see that the domain is defined at x = 3. Notice that the green line cut across x = 3.
Now let's check Graph of option D
[tex]y=\frac{2x^2-8}{x^2-9}[/tex]From the graph, you can see that the domain is defined at x = -3 and x = 3. Notice that the purple and green line cut across x = -3 and x = 3. So, the domain here is
[tex]x=\pm3[/tex]Hence
Two mechanics worked on a car. The first mechanic charged $105 per hour, and the second mechanic charged $120 per hour. The mechanics worked for a combined total of 20 hours, and together they charged a total of $2175. How long did each mechanic work?
Solution
The first mechanic charged $105 per hour.
The second mechanic charged $120 per hour.
The mechanics worked for a combined total of 20 hours
Let the first mechanic work for x hours
Then
[tex]\begin{gathered} 105x+(20-x)120\text{ =2175} \\ 105x+2400-120x=2175 \\ \text{collect like terms} \\ 105x-120x=2175-2400 \\ -15x=-225 \\ \\ \text{Divide both sides by -15} \\ \frac{\text{-15x}}{\text{-15}}=-\frac{225}{\text{-15}} \\ \\ x=15 \end{gathered}[/tex]The first mechanic work for x hours which 15hours
The second mechanic work for (20-x ) hours which is 20-15=5hours
.
You have the option of borrowing money from one source that charges simple interest or from another source that charges the same APR but compounds the interest monthly. Which would you choose, and why?
I would choose the source that charges simple interest. This is because simple interest is based only on the principal (The amount borrowed), but compound interest is based on the principal and also the interest that has been generated from it.
The admission fee at an amusement park is $1.50 for children and $4 for adults. On a certain day, 277 people entered the park, and the admission fees collected totaled 828.00 dollars. How many children and how many adults were admitted?
Given:
Let x be the number of children.
Let y be the number of adults.
In total, there were 277 people.
So,
[tex]x+y=277\ldots\ldots\ldots(1)[/tex]According to the question, the fee of $1.50 for children and $4 for adults and the total fees collected is $828.
So,
[tex]1.5x+4y=828\ldots\ldots\ldots(2)[/tex]Multiply by 4 in equation (1),
[tex]4x+4y=1108\ldots\ldots\ldots(3)[/tex]Subtracting the equation (2) from (3), we get
[tex]\begin{gathered} 2.5x=280_{} \\ x=112 \end{gathered}[/tex]Substitute x=112 in equation (1), we get
[tex]\begin{gathered} 112+y=277 \\ y=165 \end{gathered}[/tex]Thus,
• The number of children is x = 112.
,• The number of adults is y = 165.
24. Simplify x^2 + 7x + 12 x + 3
We reduce like terms:
[tex]\begin{gathered} x^2+7x+12x+3 \\ x^2+19x+3 \end{gathered}[/tex]therefore, the answer is x^2+19x+3
if I make 9.75 hour and work 30 hours a week. how much I make in a week? how much I make in a month? how much in a year?
Since you make $9.75 per hour and you work 30 hours a week that means that you make:
[tex]9.75\cdot30=292.5[/tex]Therefore you make $292.5 in a week.
A month has 4 1/3 weeks, then per month you earn:
[tex]292.5\cdot4\frac{1}{3}=1267.5[/tex]Therefore you earn $1267.5 in a month.
Finally since each year has 12 month you earn:
[tex]1267.5\cdot12=15210[/tex]Therefore you earn $15210 in a year.
Grocery store A is selling bananas for $9.75 for 1/2 pound .Grocery store B is selling 5 pounds of Bananas for $3.75 which store us offering the best unit rate
Grocery Store B ($1.33 per pound of bananas)
1) With these data we can write the following, and ,
Grocery Store A:
$ pounds
9.75 1/2
x--------------- 1
1/2x=9.75 x 2
x =19.5
Cross multiplying it:
Grocery Store B
$ pounds
5 3.75
y 1
3.75y=5
y=5/3.75
y=1.33
2) The best unit rate is at Grocery Store B ($1.33 per pound of bananas)
kelly is knitting a scarf for her brother. it took her 1/3 hour to knit 3/8 foot of the scarf. How fast is Kelly's knitting speed, in feet per hour?A[tex]4 \frac{1}{2} [/tex]B[tex]3[/tex]C[tex]2\frac{1}{2} [/tex]D[tex]1 \frac{1}{8} [/tex]
We need to divide the number of foot of scarf knitted by the number of time, in hours, taken.
[tex]\frac{\frac{3}{8}\text{ ft }}{\frac{1}{3}\text{ hour}}=\frac{3}{8}\cdot3\frac{\text{ ft}}{\text{ hour}}=\frac{9}{8}\frac{\text{ ft}}{\text{ hour}}=\frac{8+1}{8}\frac{\text{ ft}}{\text{ hour}}=1\frac{1}{8}\frac{\text{ ft}}{\text{ hour}}[/tex]Using a standard 52-card deck, Michelle will draw 6 cards with replacement. If Event A = drawing all hearts and Event B =drawing no face cards, which of the following best describes events A and B?
The described events can be classified as independent.
Mainly because the probability of one event won't change the probability of the other event.
Hence, the answer is independent.
If {an) is an arithmetic sequence where a1=-23 and the common difference is 6, find a79
Given:
The first term
[tex]a_1=-23[/tex]The common difference, d=6
To find
[tex]a_{79}[/tex]Using the nth term formula,
[tex]\begin{gathered} a_n=a+(n-1)d \\ a_{79}=-23+(79-1)6 \\ =-23+(78)6 \\ =-23+468 \\ =445 \end{gathered}[/tex]Hence, the answer is,
[tex]a_n=445[/tex]What is the midpoint of the line segment graphed below?10-(5,9)-10-10-(2,-1)10
Step 1
The midpoint formula is given as;
[tex]\begin{gathered} \frac{x_1+x_2}{2},\frac{y_1+y_2}{2} \\ =\frac{5+2}{2},\frac{9-1}{2} \\ =3.5,4 \end{gathered}[/tex]Answer;
[tex](\frac{7}{2},4)[/tex]Which of the following are the xintercepts on the graph of the function shown below? f(x)=(x+2)(x-7)
Which of the following are the xintercepts on the graph of the function shown below? f(x)=(x+2)(x-7)
we have the function
f(x)=(x+2)(x-7)
This is a vertical parabola written in factored form
The zeros or x-intercepts of the function are
x=-2 and x=7
Remember that the x-intercepts are the values of x when the value of the function is equal to zero
therefore
the answer is
x=-2 and x=7What are the magnitude and direction of w = ❬–10, –12❭? Round your answer to the thousandths place.
The direction of a vector is the orientation of the vector, that is, the angle it makes with the x-axis.
The magnitude of a vector is its length.
The formulas to find the magnitude and direction of a vector are:
[tex]\begin{gathered} u=❬x,y❭\Rightarrow\text{ Vector} \\ \mleft\Vert u|\mright|=\sqrt[]{x^2+y^2}\Rightarrow\text{ Magnitude} \\ \theta=\tan ^{-1}(\frac{y}{x})\Rightarrow\text{ Direction} \end{gathered}[/tex]In this case, we have:
• Magnitude
[tex]\begin{gathered} w=❬-10,-12❭ \\ \Vert w||=\sqrt[]{(-10)^2+(-12)^2} \\ \Vert w||=\sqrt[]{100+144} \\ \Vert w||=\sqrt[]{244} \\ \Vert w||\approx15.620\Rightarrow\text{ The symbol }\approx\text{ is read 'approximately'} \end{gathered}[/tex]• Direction
[tex]\begin{gathered} w=❬-10,-12❭ \\ \theta=\tan ^{-1}(\frac{-12}{-10}) \\ \theta=\tan ^{-1}(\frac{12}{10}) \\ \theta\approx50.194\text{\degree} \\ \text{ Add 180\degree} \\ \theta\approx50.194\text{\degree}+180\text{\degree} \\ \theta\approx230.194\text{\degree} \end{gathered}[/tex]Therefore, the magnitude and direction of the vector are:
[tex]\begin{gathered} \Vert w||\approx15.620 \\ \theta\approx230.194\text{\degree} \end{gathered}[/tex]Consider the line . 7x-8y=-1Find the equation of the line that is parallel to this line and passes through the point . (-3,-6)Find the equation of the line that is perpendicular to this line and passes through the point . (-3.-6)
we have the line
7x-8y=-1
Find out the slope of the given line
isolate the variable y
8y=7x+1
y=(7/8)x+1/8
the slope is m=7/8
Part a
Find the equation of the line that is parallel to this line and passes through the point . (-3,-6)
Remember that
If two lines are parallel, then their slopes are equal
so
The slope of the parallel line is m=7/8 too
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=7/8
point (-3,-6)
substitute and solve for b
-6=(7/8)(-3)+b
b=-6+(21/8)
b=-27/8
therefore
The equation is
y=(7/8)x-27/8Part b
Find the equation of the line that is perpendicular to this line and passes through the point . (-3.-6)
Remember that
If two lines are perpendicular, then their slopes are negative reciprocal
so
The slope of the perpendicular line is m=-8/7
Find out the equation of the line in slope-intercept form
y=mx+b
we have
m=-8/7
point (-3,-6)
substitute and solve for b
-6=-(8/7)(-3)+b
b=-6-(24/7)
b=-66/7
therefore
the equation is
y=-(8/7)x-66/7Use the pair of functions f and g to find the following values if they exist[tex] f(x) = \sqrt{x + 2} [/tex][tex]g(x) = 3x - 2[/tex]a. (f+g)(2)b.(f/g)(0)c.(f-g)(-1)
Solve by completing the square. x2 - 8x + 5 = 0
Answer:
[tex]\begin{gathered} x_1=4+\sqrt[]{11} \\ x_2=4-\sqrt[]{11} \end{gathered}[/tex]Step-by-step explanation:
Solve the following quadratic completing the square:
[tex]x^2-8x+5=0[/tex]Keep x terms on the left and move the constant to the right side:
[tex]x^2-8x=-5[/tex]Then, take half of the x-term and square it.
[tex](-8\cdot\frac{1}{2})^2=16[/tex]Now, add this result to both sides of the equation:
[tex]x^2-8x+16=-5+16[/tex]Rewrite the perfect square on the left.
[tex]\begin{gathered} (x-4)^2=-5+16 \\ (x-4)^2=11 \end{gathered}[/tex]Take the square root of both sides:
[tex]\begin{gathered} \sqrt[]{(x-4)^2}=\pm\sqrt[]{11} \\ x-4=\pm\sqrt[]{11} \\ x=\pm\sqrt[]{11}+4 \end{gathered}[/tex]Hence, the two solutions of the equation are:
[tex]\begin{gathered} x_1=4+\sqrt[]{11} \\ x_2=4-\sqrt[]{11} \end{gathered}[/tex]2. FR has a midpoint M. Use the given information to find the missing endpoint. F(-2,3) and M(3,0)
ANSWER:
R(8, -3)
STEP-BY-STEP EXPLANATION:
We have that the midpoint formula is the following:
[tex]M(m_1,m_2)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})[/tex]In this case, we know the midpoint M, that is, m1 and m2 and the startpoint F, that is, we know, x1 andy1, we replace to calculate the values of R, the endpoint:
[tex]\begin{gathered} 3=\frac{-2+x_2}{2} \\ 6=-2+x_2 \\ x_2=6+2=8 \\ \\ 0=\frac{3+y_2}{2} \\ 0=3+y_2 \\ y_2=-3 \\ \\ \text{Therefore, the missing endpoint is: (8,-3)} \end{gathered}[/tex]I need quick answers please, is due soon. i need assistance finding 5 points. 2 to the left of vertex, i need the vertex, and 2 to the right of the vertex. the graph only goes up to 14. thank you!
We have to find 5 points of the parabola:
[tex]y=x^2+8x+11[/tex]and then graph it.
We can find the vertex by completing the square:
[tex]\begin{gathered} y=x^2+8x+11 \\ y=x^2+2\cdot4x+16-16+11 \\ y=(x+4)^2-5 \end{gathered}[/tex]As we now have the vertex form of the parabola, we can see that the vertex is at (x,y) = (-4,-5).
We can now calculate two points to the right of the parabola by giving values to x as x = 0 and x = -2:
[tex]y=0^2+8\cdot0+11=11[/tex][tex]\begin{gathered} y=(-2)^2+8\cdot(-2)+11 \\ y=4-16+11 \\ y=-1 \end{gathered}[/tex]We now know two points to the right of the parabola: (0, 11) and (-2, -1).
As the line x = -4 is the axis of symmetry, we will have the same value for y when the values of x are at the same distance from this line.
Then, we can write:
[tex]\begin{gathered} y(0)=y(-8)=11 \\ y(-2)=y(-6)=-1 \end{gathered}[/tex]Then, we have two points to the left: (-8, 11) and (-6, -1).
We can graph the parabola as:
If the measure of one complementary angle is 30° more than twice the other angle measure, writean equation and find the measure of each angle.
For this problem we kow that the measure of one complementary angle is 30º more than twice the other angle measure
If our original angle is xthe complement would be 90-xº. then using the statement we can write the following equation:
[tex]x=2(30+90-x)[/tex]And from this equation we can solve for x like this:
[tex]x=240-2x[/tex]Adding 2x in both sides we got:
[tex]3x=240[/tex]And dividing both sides by 3 we got:
[tex]x=\frac{240}{3}=80º[/tex]And the final answer for this case would be 80º
A random number generator is programmed to produce numbers with a Unif (−7,7) distribution. Find the probability that the absolute value of the generated number is greater than or equal to 1.5.
We are given the following uniform distribution:
The probability that the absolute value of the number is in the following interval:
[tex]\begin{gathered} -7The probability is the area under the curve of the distribution. Therefore, we need to add both areas. The height of the distribution is:[tex]H=\frac{1}{b-a}[/tex]Where:
[tex]\begin{gathered} a=-7 \\ b=7 \end{gathered}[/tex]Substituting we get:
[tex]H=\frac{1}{7-(-7)}=\frac{1}{14}[/tex]Therefore, the areas are:
[tex]P(\lvert x\rvert>1.5)=(-1.5-(-7))(\frac{1}{14})+(7-1.5)(\frac{1}{14})[/tex]Simplifying we get:
[tex]P(\lvert x\rvert>1.5)=2(7-1.5)(\frac{1}{14})[/tex]Solving the operations:
[tex]P(\lvert x\rvert>1.5)=0.7857[/tex]Therefore, the probability is 0.7857 or 78.57%.
Write down the domain of f-1 according to the following figure. A. {4, 5, 6, 7} B. {4, 3, 2, 7} C. {1, 2, 4, 5} D. {1, 2, 3, 4}
given to find down the domain of f inverse of the function.
An inverse function is found interchanging the first and second coordinate of each ordered pair.
thus the answer is, option A. {4,5,6,7}
The Elkhart Athletic Departments sells T-shirts and Hats at a big game to raise money. They sale the T-shirts for $12 and the Hats for $5. At the last football game they sold a total of 32 items and raised $265. How many T-shirts and Hats were sold at the game?
Let x represent the number of T shirts that they sold
Let y represent the number of hats that they sold
They sold the T-shirts for $12 and the Hats for $5. This means that the cost of x T shirts and y hats would be
12 * x + 5 * y
= 12x + 5y
The total amount raised was $265. It means that
12x + 5y = 265 equation 1
Also, the total number of t shirts and hats sold was 32. It means that
x + y = 32
x = 32 - y
Substituting x = 32 - y into equation 1, it becomes
12(32 - y) + 5y = 265
384 - 12y + 5y = 265
- 12y + 5y = 265 - 384
7y = 119
y = 119/7
y = 17
x = 32 - y = 32 - 17
x = 15
15 T shirts and 17 hats
In the figure, k//l, find the values of z and y.
Answer:
• z=113°
,• y=67°
Explanation:
In the diagram below, by the principles of vertical and corresponding angles:
[tex](6y-113)\degree=67\degree\text{ (Corresponding angles)}[/tex]We solve for y:
[tex]\begin{gathered} 6y=67+113 \\ 6y=180 \\ y=\frac{180}{6} \\ y=30 \end{gathered}[/tex]Next, angles z and (6y-113) are on a straight line. Therefore:
[tex]z+(6y-113)\degree=180\degree[/tex]However, recall we stated earlier that (6y-113)°=67°, therefore:
[tex]\begin{gathered} z+67\degree=180\degree \\ z=180\degree-67\degree \\ z=113\degree \end{gathered}[/tex]The values of z and y are 113° and 67° respectively.