Find the area of each. 'Use your calculator's value of r. Round your answer to the nearest tenth.
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To find the area of a circle given the radius;
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convert r= 5/ 1+3sinθ to a rectangular equation
Answers
Given:
[tex]r=\frac{5}{1+3\sin \theta}[/tex]Find: Rectangular equation.
Sol:
[tex]r^2=x^2+y^2[/tex][tex]\begin{gathered} y=r\sin \theta \\ \sin \theta=\frac{y}{r} \end{gathered}[/tex][tex]\begin{gathered} r=\frac{5}{1+3\sin \theta} \\ r=\frac{5}{1+\frac{3y}{r}} \end{gathered}[/tex][tex]\begin{gathered} r=\frac{5r}{r+3y} \\ r+3y=5 \\ r=5-3y \\ r^2=(5-3y)^2 \end{gathered}[/tex]Put the value in rectangular equation:
[tex]\begin{gathered} x^2+y^2=r^2 \\ x^2+y^2=(5-3y)^2 \end{gathered}[/tex]Find y if the point (5,y) is on the terminal side of theta and cos theta = 5/13
Answers
For this problem we have a point given (5,y) and we know that this point is on a terminal side of an angle, we also know that:
[tex]\cos \theta=\frac{5}{13}[/tex]If we know the cos then we can find the sin on this way:
[tex]\sin \theta=\frac{y}{13}[/tex]Then we can apply the following identity from trigonometry:
[tex]\sin ^2\theta+\cos ^2\theta=1[/tex]Using this formula we got:
[tex](\frac{5}{13})^2+(\frac{y}{13})^2=1[/tex]And we can solve for y:
[tex]\frac{y^2}{169}=1-\frac{25}{169}=\frac{144}{169}[/tex]And solving for y we got:
[tex]y=\sqrt{169\cdot\frac{144}{169}}=\sqrt{144}=\pm12[/tex]And the two possible solutions for this case are y=12 and y=-12
The hypotenuse of a right triangle is 1 centimeter longer than the longer leg. The shorter leg is 7 centimeters shorter than the longer leg. Find the length of the shorter leg of the triangle.
Answers
Answer:
[tex]\text{Shorter leg= 5cm}[/tex]Step-by-step explanation:
As a first step to go into this problem, we need to make a diagram:
Let x be the measure of the longer leg.
Now, understanding this we can apply the Pythagorean theorem to find x, it is represented by the following equation:
[tex]\begin{gathered} a^2+b^2=c^2 \\ \text{where,} \\ a=\text{longer leg} \\ b=\text{shorter leg} \\ c=\text{hypotenuse } \end{gathered}[/tex]Substituting a,b, and c by the expressions corresponding to its sides:
[tex]\begin{gathered} x^2+(x-7)^2=(x-1)^2 \\ \end{gathered}[/tex]apply square binomials to expand and gather like terms, we get:
[tex]\begin{gathered} x^2+x^2-14x+49=x^2-2x+1 \\ 2x^2-x^2-14x+2x+49-1=0 \\ x^2-12x+48=0 \end{gathered}[/tex]Now, factor the quadratic equation into the form (x+?)(x+?):
[tex]\begin{gathered} (x-4)(x-12)=0 \\ x_1=4 \\ x_2=12 \end{gathered}[/tex]This means, the longer leg could be 4 or 12, but if we subtract 7 to 4, we get a negative measure for the shorter leg, that makes no sense.
Therefore, the long leg is 12 cm.
Hence, if the shorter leg is 7 centimeters shorter than the longer leg:
[tex]\begin{gathered} \text{Shorter leg=12-7} \\ \text{Shorter leg=}5\text{ cm} \end{gathered}[/tex]25. A group of students were asked how many movies they had watched the previous week. The results are shown below.Number of MoviesFrequency0818253547Find the mean and median for the number of movies watched per student. Round your answers to the nearest hundredth.Mean = Median =
Answers
Answer:
Explanation:
Given the results of the number of movies watched by the group of students and the frequency, we're asked to determine the mean and median for the number of movies watched per student.
We'll follow the below steps to solve for the mean and median;
1. Find the product of the number of movies and frequency;
[tex]\begin{gathered} 0\times8=0 \\ 1\times8=8 \\ 2\times5=10 \\ 3\times5=15 \\ 4\times7=28 \end{gathered}[/tex]2. Find the sum of the product of the number of movies and frequency;
[tex]0+8+10+15+28=61[/tex]3. Find the sum of the frequency;
[tex]8+8+5+5+7=33[/tex]The mean can now be determined using the below formula;
[tex]\begin{gathered} \text{Mean}=\frac{\Sigma(f\cdot x)}{\Sigma f} \\ \text{where} \\ \Sigma(f\cdot x)=\text{ sum of the product of the number of movies and frequency} \\ \Sigma f=\text{ sum of the frequency} \end{gathered}[/tex]Therefore, our mean is;
[tex]\text{Mean}=\frac{61}{33}=1.85[/tex]We can go ahead and determine the median using the below formula;
[tex]undefined[/tex]Elena is organizing her craft supplies. She estimatesthat her jars will fit 1,000 buttons or 50 large beads.They actually fit 677 buttons or 22 large beads. DoesElena's estimate about the buttons or her estimateabout the large beads have less percent error? To thenearest percent, how much less?
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Step 1
Given;
[tex]\begin{gathered} Elena-\text{ estimates her jar will take 1000 buttons or 50 large beads} \\ Her\text{ Jar actually takes 677 buttons or 22 large beads} \end{gathered}[/tex]Required; To find if Elena's estimates have percentage error, to which percent, and how much less
Step 2
State the formula for percentage error
[tex]\text{ \% error=}\frac{|Approximate-exact|}{exact}\times100[/tex][tex]Elena^{\prime}s\text{ estimate about the button has a percentage error }[/tex][tex]\begin{gathered} For\text{ buttons} \\ Approximate=1000 \\ Exact=677 \end{gathered}[/tex][tex]\text{ \%error=}\frac{|1000-677|}{677}\times100=47.71048744\text{\%}[/tex][tex]\begin{gathered} For\text{ large beads} \\ \operatorname{\%}\text{error=}\frac{\text{\lvert50-22\rvert}}{22}\times100 \end{gathered}[/tex][tex]\text{ \% error=}\frac{28}{22}\times100=127.272727...\text{\%}[/tex]Percent errors tells you how big your errors are when you measure something in an experiment. Smaller values mean that you are close to the accepted or real value. For example, a 1% error means that you got very close to the accepted value, while 45% means that you were quite a long way off from the true value.
The percentage error for buttons with about 47.71% is less than that of the large beads which is about 127.273%.
How much less of the percentage error to the nearest percent will be;
[tex]\begin{gathered} =79.56223986 \\ \approx80\text{\%} \end{gathered}[/tex]Estimate the square root to the nearest whole numberV450question 1
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We need to solve the next square root
[tex]\sqrt[]{450}=21.21[/tex]The nearest whole number is 21
Graph two full periods, highlighting the first period using bold marking and analyze each function.Y = 2 sin (1/2 (x + pi/2) ) + 1
Answers
Given
[tex]y=2\sin(\frac{1}{2}(x+\frac{\pi}{2}))+1[/tex]
Procedure
Period: 4pi
Interval length: In the graph 2 periods 8pi
Phase shift: -pi/2
1st Per. begins: -pi/2
1st Per. ends: 7pi/2
Amplitude: 2
Domain:
(-∞, ∞)
Range:
[-1,3]
y-intercep:
(0,2.414)
x-intercept:
[tex]x=\frac{11\pi}{6}+4\pi n,\frac{19\pi}{6}+4\pi n,\text{ for any integer of n }[/tex]write an equation to represent"three consecutive integers is 12 less than 4 times the middle integer'
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Consider that the three consecutive integers are:
least integer = n
middle integer = n + 1
greatest integer = n + 2
THe expression "three consecutive integers is 12 less than 4 times the middle integer" can be written as follow:
n + (n + 1) + (n + 2) = 4(n +1) - 12
In order to find the numbers, proceed as follow:
n + (n + 1) + (n + 2) = 4(n +1) - 12 cancel parenthesis
n + n + 1 + n + 2 = 4n + 4 - 12 simplify like terms
3n + 3 = 4n - 8 subtract 4n and 3 both sides
3n - 4n = - 8 - 3
-n = -11
n = 11
Hence, the three consecutive integers are:
n = 11
n + 1 = 12
n + 2 = 13
Lincoln Middle School plans to collect more than 2,000 cans of food in a food drive. So far, 668 cans have been collected. Write and solve an inequality to find numbers of cans the school can collect on each of the final 7 days of the drive to meet this goal.Which inequality represents the solution to this situation?
Answers
Inequalities
Let's call c the number of cans of food.
The school wants to collect more than 2,000 cans in a food drive.
668 cans have been collected so far.
The number of cans needed to reach the goal is 2,000 - 668.
These cans will be collected in 7 days, thus:
7c > 2,000 - 668
Operating
7c > 1,332
Dividing by 7:
c > 1,332 / 7
c > 190.29
This is the average number of cans needed to collect each day.
The first choice is correct
A parabola can be drawn given a focus of (-7,3) and a directrix of x = 9. What canbe said about the parabola?
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The focus of a parabola is given by:
[tex]F(h,k+p)[/tex]and the directrix is given by:
[tex]y=k-p[/tex]since the directrix is x = 9, we can conclude it is a horizontal parabola, so:
[tex]\begin{gathered} x=9=k-p \\ so\colon \\ k=9+p \end{gathered}[/tex][tex]\begin{gathered} F(-7,3)=(h,k+p) \\ h=-7 \\ k+p=3 \\ 9+p+p=3 \\ 9+2p=3 \end{gathered}[/tex]solve for p:
[tex]\begin{gathered} 2p=3-9 \\ 2p=-6 \\ p=-\frac{6}{2} \\ p=-3 \end{gathered}[/tex][tex]\begin{gathered} k=3-p \\ k=3-(-3) \\ k=6 \end{gathered}[/tex]We can write the parabola in its vertex form:
[tex]\begin{gathered} x=\frac{1}{4p}(y-k)^2+h \\ so\colon \\ x=-\frac{1}{12}(y-6)^2-7 \end{gathered}[/tex]It is a horizontal parabola that opens to the left, and has vertex located at (-7,6)
The fire department is having a BBQ fundraiser. The hot dogs costs $1.50 each and cans ofsoda cost $0.75 each. The department uses the algebraic expression 1.50x+0.75y to calculatecustomers' total expenses.a. What does the x variable represent?b. What does the y variable represent?c. A family buys 7 hot dogs and 4 sodas. What are their total expenses?
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a) As the expression represents the total expense for a family, the term 1.50x represents how much the familiy spends in hot dogs.
This term is the product of the price (1.50) and the number of hot dogs purchased (x).
Then, x is the number of hot dogs bought by the familiy.
b) In the same way, 0.75y represent how much the family spends in soda: 0.75 is the price and y represents the number of soda cans purchased by the family.
c) If a family buys x=7 hot dogs and y=4 sodas, we can calculate the expenses as:
[tex]\begin{gathered} E=1.50x+0.75y \\ E=1.50\cdot7+0.75\cdot4 \\ E=10.5+3 \\ E=13.5 \end{gathered}[/tex]Their total expenses are $13.5.
What is the slope of the line containing (-2,5) and (4,-4)?A.3/2B.-3/2C. -2D. 2
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Answer:
B.-3/2
Step-by-step explanation:
To find the slope, we need to take two points from a line. I am going to call them:
(x1,y1) and (x2,y2).
The slope is:
[tex]a=\frac{y2-y1}{x2-x1}[/tex]In this question:
(x1,y1) = (-2,5)
(x2,y2) = (4, -4)
So
[tex]a=\frac{y2-y1}{x2-x1}=\frac{-4-5}{4-(-2)}=\frac{-9}{4+2}=-\frac{9}{6}=-\frac{3}{2}[/tex]So the correct answer is:
B.-3/2
How many kilometers could the red car travel in 12 hours? Write an equation to show your work.
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The kilometers that the red car travel in 12 hours is 2604 kilometers.
What is an equation?An equation is the statement that illustrates that the variables given. In this case, two or more components are taken into consideration to describe the scenario. It is vital to note that an equation is a mathematical statement which is made up of two expressions that are connected by an equal sign.
From the diagram, it should be noted that the red car has a speed of 217 km per hours.
Therefore, the distance traveled in 12 hours will be:
Distance = Speed × Time
= 217 × 12
= 2604 km
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A class of 20 students wants to form a committee to fundraise for cancer research. If the committee is formed with four students, how many possible committees can be made?A: 116,280B: 24C: 4,845D: 9,690
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1) Since there are 20 students, and each committee is formed by 4 people.
The order does not matter, and there can't be repetition. Just one person can be, let's say president, VP, secretary, and treasurer.
2) So we can write, the possibilities on the numerator of people filling in and on the denominator the number of vacancies for that committee, we can set this Combination simply as:
[tex]\frac{20}{4}\times\frac{19}{3}\times\frac{18}{2}\times\frac{17}{1}=4845[/tex]3) So there are 4845 possibilities to form a Committee with 20 people for 4 vacant lots.
Translate the following phrase into an algebraic expression. Do not simplify. Use the variable names "x" or "y" to describe the unknowns.six subtracted from a number
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If a golden rectangle has a length of 1 cm, what is its width (shorter side) rounded to the NEAREST TENTH?
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In any golden rectangle the following poreperty should hold:
[tex]\frac{a+b}{a}=\frac{a}{b}[/tex]where a+b is the length and a is the width. We know that the length of the rectangle is 1, then:
[tex]\begin{gathered} a+b=1 \\ b=1-a \end{gathered}[/tex]Plugging this values in the first equation we have:
[tex]\frac{1}{a}=\frac{a}{1-a}[/tex]Solving this equation for a:
[tex]undefined[/tex]Lesson 6.02: Finn's fish store has 5 tanks of goldfish; each tank holds 40 fish. He collects andinspects 5 fish from each tank and finds that 4 fish have fin rot. Find the estimated numbergoldfish in the store that have fin rot. SHOW ALL WORK!
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Answer:
Explanation:
We are told that of each 5 fish inspected in the tank, 4 have fin rot, therefore, the probability o getting a fin rot is
[tex]\frac{4}{5}\times100\%=80\%[/tex]This means 80% of the fish in a tank have fin rot.
Now, for one tank 80% of 40 fish is
[tex]\frac{80\%}{100\%}\times40=32[/tex]Now, since there are 5 fish tanks in the store and 32 fish in each have fin rot; therefore, the total number of fish that have fin rot will be
[tex]32fish\times5=160\text{fish}[/tex]Hence, the estimated number of fish with fish rot in the store is 160.
Which set of parametric equations represents the function y=x^2+4x-5? Select all that apply.
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Solution
- The way to solve the equation is to take the expression for x i.e. x = 2t, and substitute into the expression for y(x).
- The result must be the corresponding y-value in terms of t.
- This is done below:
Option A:
[tex]\begin{gathered} x=2t \\ y(x)=x^2+4x-5 \\ \\ \text{ put }x(t)=2t \\ \\ y(x(t))=(2t)^2+4(2t)-5 \\ y(x(t))=y(t)=4t^2+8t-5 \\ \\ \therefore y(t)=4t^2+8t-5\text{ \lparen OPTION A\rparen} \end{gathered}[/tex]Option B:
[tex]\begin{gathered} x=t+1 \\ y=x^2+4x-5 \\ \\ y(x(t))=y(t)=(t+1)^2+4(t+1)-5 \\ t^2+2t+1+4t+4-5 \\ y(t)=t^2+6t\text{ \lparen NOT IN THE OPTIONS\rparen} \end{gathered}[/tex]Option C:
[tex]\begin{gathered} x=t-3 \\ y=x^2+4x-5 \\ \\ y(x(t))=(t-3)^2+4(t-3)-5 \\ =t^2-6t+9+4t-12-5 \\ =t^2-2t-8\text{ \lparen NOT IN THE OPTIONS\rparen} \end{gathered}[/tex]Option D:
[tex]\begin{gathered} x=t^2 \\ y=x^2+4x-5 \\ \\ y(x(t))=(t^2)^2+4(t^2)-5 \\ =t^4+4t^2-5\text{ \lparen NOT IN THE OPTIONS\rparen} \end{gathered}[/tex]Option E:
[tex]\begin{gathered} x=t+1 \\ y=x^2+4x-5 \\ \\ y(x(t))=(t+1)^2+4(t+1)-5 \\ =t^2+2t+1+4t+4-5 \\ =t^2+6t\text{ \lparen OPTION E IS CORRECT\rparen} \end{gathered}[/tex]Final Answer
The answers are OPTIONS A AND E
7x +4 for x =9 The solution is ?
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Find the area of the parallelogram. 6.5 cm 3.1 cm 3 cm O 9.3 cm^2 O 19.5 cm^2 O 20.15 cm^2 O 60.45 cm^2
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Data
length = 6.5 cm
height = 3 cm
side = 3.1 cm
Formula
Area = base x height
Substitution
Area = (6.5 x 3)
Result
Area = 19.5 cm^2
Next
The right answer is the second choice
12 + 1.5x >= 20
x-(7.65 + 3.18)=4 solve for x
Answers
Answer:
The value of x is;
[tex]x=14.83[/tex]Explanation:
Given the equation;
[tex]x-(7.65+3.18)=4[/tex]Solving for x;
[tex]\begin{gathered} x-(10.83)=4 \\ x=4+10.83 \\ x=14.83 \end{gathered}[/tex]Therefore, the value of x is;
[tex]x=14.83[/tex]In one month, Jason eams $32.50 less than twice the amount Keyin earns, Jason earns $212.50write and solve an equation to solve for the amount of money that kevin earns
Answers
Let the amount Kevin earns be represented with K
Let the amount Jason earns be represented with J
Jason earns $32.50 less than twice Keyin earns can be represented by
J = K - 32.5 ----- equation 1
Jason earns $ 212.5
J = 212.5 ----- equation 2
From equation 1, we can write the equation to solve for what Kevin earns
J = K - 32.5
Making K the subject of the formula
K = J + 32.5
Putting J = 212.5 into the equation above
K= $ 212.5 + $ 32.5
K = $ 245
Kevin earns $245
Answer please the picture scanner deal won’t scan over this and i don’t know how to type it out
Answers
Solution
We are given the arithmetic sequence
[tex]\begin{gathered} a_1=5 \\ a_n=a_{n-1}-4 \end{gathered}[/tex]To find an explicit formula
[tex]\begin{gathered} First\text{ }Term=5 \\ a=5 \end{gathered}[/tex]From the second recursive formula
[tex]\begin{gathered} a_n-a_{n-1}=-4 \\ Common\text{ }Difference=-4 \\ d=-4 \end{gathered}[/tex]The nth term of an Arithmetic sequence is given by
[tex]\begin{gathered} a_n=a+(n-1)d \\ a_n=5+(n-1)(-4) \end{gathered}[/tex]Therefore, the answer is
[tex]a_{n}=5+(n-1)(-4)[/tex]Tim is building a model of a castle with small wooden cubes. So far Tim has constructed part of a security world castle,as shown below. Each wooden cube has a side length of 1/8ft
Answers
From the given model, the length of the wall is 9/8 ft, the width of the walk is 1/2 ft, and the height of the wall is 11/8. The volume of the portion of security wall that Tim has constructed so far is 99/128 cu ft.
What is the Volume of the Block?From the given image of the building model we see that part of a security world castle is shown.
We see that the length has 9 blocks.
Since the length has a total of 9 blocks and each side length is 1/8 ft, then we say that;
Length = 9*(1/8) = 9/8 ft
We also observe that the height has 11 blocks and as such;
height = 11*(1/8) = 11/8 ft
Meanwhile the width will have a length of: 1/2 ft
Formula for volume is;
Volume = length * height * width
Thus;
Volume = (9/8) * (11/8) * (1/2)
Volume = (9 * 11 * 1)/(8 * 8 * 2)
Volume = 99/128 cu ft
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Complete question is;
Tim is building a model of a castle with small wooden cubes. So far Tim has constructed part of a security world castle,as shown below. Each wooden cube has a side length of 1/8ft
Based on the model,the length of the wall is ___ft, the width of the walk is 1/2 ft, and the height of the wall is ___. The volume of the portion of security wall that Tum has constructed so far is ___ cu ft.
Find the domain of f(x) = 3x/x-1 and discuss the function behavior of f near any excluded x-values.
Answers
The domains are all real numbers except the the values that makes the denominator zero
x - 1 = 0
x=1
That is; the domain is all real numbers except x=1
Find d and then find the 20th term the sequence. Type the value of d (just the number) in the first blank and then type the 20th term(just the number) in the second blank.a1=6 and a3=14
Answers
We have that an arithmetic sequence can be defined by the following explicit formula:
[tex]a_n=a_1+(n-1)\cdot d[/tex]where n represents the index of each term in the sequence and d represents the common difference beteen each term. a1 is the first term of the sequence.
In this case we have that the first term is a1 = 6, and also we have that a3=14. We can use the formula to find the common difference:
[tex]\begin{gathered} a_3=a_1+(3-1)d \\ \Rightarrow a_3=a_1+2d \\ \Rightarrow14=6+2d \end{gathered}[/tex]solving for d, we get:
[tex]\begin{gathered} 2d+6=14 \\ \Rightarrow2d=14-6=8 \\ \Rightarrow d=\frac{8}{2}=4 \\ d=4 \end{gathered}[/tex]therefore, the value of d is d = 4.
We have now the explicit formula for the sequence:
[tex]\begin{gathered} a_n=6_{}+4(n-1) \\ \end{gathered}[/tex]then, for the 20th term, we have to make n = 20 on the formula, and we get the following:/
[tex]\begin{gathered} a_{20}=6+4(20-1)=6+4(19)=6+76=82 \\ \Rightarrow a_{20}=82 \end{gathered}[/tex]therefore, the 20th term is 82
which statement is the contrapositive of the given statement statement if you play a sport then you wear a helmet
Answers
We will have that the contrapositive statement is:
*If you do not wear a helmet, then you do not play a sport.
Write a word problem that involves a proportional relationship and needs more than one step to solve.Show how to solve the problem
Answers
To write a word problem that involves a proportional relationship:
Sam bought 4kg of apples for $12. How many kilograms of apples, he can buy for $30?
Sam bought 4kg of apples for $12.
So, cost of 1 kg of apples is,
[tex]\frac{12}{4}=3[/tex]Let x be the number of kg apples.
Therefore, He can buy 3x kg apples for $30.
So,
[tex]\begin{gathered} 3x=30 \\ x=\frac{30}{3} \\ x=10 \end{gathered}[/tex]Therefore, He can buy 10 kg apples for $30.
A student in MAT110 this semester has the following grades at the end of the semester, after two quizzes and one lab grade are dropped: Quiz average: 95 Lab average: 89 Tests: 52, 82, 88 Final Exam: 73 Question 1 of 5 20 Points Find the student's course average rounded to the nearest whole number. You'll need to consult your syllabus (and/or the feedback given) for weighting percentages and other grading information.Weights:Quiz = 25%Lab = 25%Test = 35%Final = 15%
Answers
The overall course average is the average weight of all the quizzes, tests, finals, labs.
First we need the average for all the information:
Quiz Average: 95 (given)
Lab Average: 89 (given)
Test Average >> we have to find by summing and dividing by number of tests
[tex]\text{average}=\frac{52+82+88}{3}=74[/tex]Final Exam: 73 (1 exam, so this is the average).
Thus, the information we have:
Quiz Avg = 95
Lab Avg = 89
Test Avg = 74
Final Avg = 73
Now, we multiply the scores with the respective weightage(in decimal) and sum it. We get:
[tex]95\mleft(0.25\mright)+89\mleft(0.25\mright)+74\mleft(0.35\mright)+73\mleft(0.15\mright)=82.85[/tex]Rounded to nearest whole number: 83
Answer:Course Average = 83 (rounded to nearest whole number)
matthew worked 20 hours ar $10 a hour. Taxes were 12%. How much money was left?
Answers
Step 1. calculate the totay pay (not including taxes)
Since he worked 20 hours with an hourly pay of $10, the total was:
[tex]20\times10=200[/tex]Step 2. Calculate the taxes
We need to calculate the 12% of $200, to find the amount that he paid in taxes. For this, we divide $200 by 100 and multiply by 12%:
[tex]\frac{200}{100}\times12[/tex]solving this operations we get:
[tex]\frac{200}{100}\times12=24[/tex]He paid $24 in taxes
Step 3. Calculate the remaining amount
We substract $24 from the initial total amount $200:
[tex]200-24=176[/tex]Answer:
How much money was left? $176