Given the parametric equations x = 7cos θ and y = 5sin θ, which of the following represents the curve and its orientation?
Answers
We have the following parameters
[tex]\begin{gathered} x=7cos\theta \\ y=5sin\theta \end{gathered}[/tex]the general equation of a circle with center (0,0) is the following,
[tex]x^2+y^2=r^2[/tex]Let's use the following tigonometric identity,
[tex]sin^2\theta+cos^2\theta=1[/tex]solving for cos and sin in the equations we are given,
[tex]cos\theta=\frac{x}{7},sin\theta=\frac{y}{5}[/tex]replace,
[tex](\frac{y}{5})^2+(\frac{x}{7})^2=1[/tex]Since we have two different numbers in the denominator, this is not a circle equation but an elipse, of the form,
[tex]\frac{y^2}{a^2}+\frac{x^2}{b^2}=1[/tex]where,
a is the vertex and,
b is the covertex
thus, in the x axis, the vertex is 7 and the y-axis the covertex is 5
Now, let's determine the direction by replacing
when Θ = 0 , then x = 7*cos0 = 7*1 = 7 , and y = 5*sin0 = 5*0 = 0
when Θ = 90° or π/2 , then x = 7*cos90° = 7*0 = 0 , and y = 5sin90° = 5*1 = 5
If we draw this, we can see that the direction is counterclockwise as in the bottom right image.
Related Questions
Part of a manufacturing plant packages tissues in boxes. Each box contains 250 tissues. Part A: Write an algebraic expression that can be used to find the total number of tissues packaged one day. Describe what the variable stands for in your expression. Part B: In one hour, 87,500 tissues are packaged into boxes. How many boxes of tissues are packaged? Show your work. Answer: boxes
Answers
Given
A manufacturing plant packages tissues in boxes and each box contains 250 tissues.
Required
We need to find an algebraic expression that illustrates the number of tissues packed per day.
Explanation
Let x be the number of boxes manufactured in one day
Then total number of tissues manufactured on that day is 250x
This answers our first part.
Now in one hour 87500 tissues are manufactured
Let the number of boxes packed in one hour be y
Then
[tex]y=\frac{number\text{ }of\text{ }tissues\text{ }in\text{ }one\text{ }hour}{number\text{ }of\text{ }tissues\text{ }in\text{ }each\text{ }box}=\frac{87500}{250}=350\text{ boxes}[/tex]So the answer to second part is 350 boxes.
A normal distribution has a mean of 101 and a standard Deviation of 12. find the probability that a value selected at random is in the following interval.at most 13
Answers
Answer:
84.134%
Explanation:
First, determine the value of the z-score.
[tex]\begin{gathered} Z=\frac{X-\mu}{\sigma} \\ =\frac{113-101}{12} \\ =\frac{12}{12} \\ z-score=1 \end{gathered}[/tex]Next, we determine the probability that a value selected at random is at most 113:
[tex]\begin{gathered} P(X\le113)=P(x\le1)_{} \\ =0.84134 \\ =84.134\% \end{gathered}[/tex]Thus, the probability that a value selected at random is in the given interval is 84.134%.
If f(x)=2x+1, what is f(2)?
Answers
f(2) means that we must substitute the value 2 in the place of x, that is
[tex]f(2)=2\cdot2+1[/tex]which gives f(2)=5.
Solve the equation by working backward through the number trick.
Answers
x = 3
Explanations:The given equation is:
[tex]\frac{4(x+3)-6}{2}=\text{ 9}[/tex]Step 1: Cross multiply
4 ( x + 3) - 6 = 9(2)
Step 2: Remove the brackets by expanding the equation
4x + 12 - 6 = 18
4x + 6 = 18
Step 3: Collect like terms
4x = 18 - 6
4x = 12
Step 4: Divide both sides by 4
4x / 4 = 12 / 4
x = 3
Help math help math
What is this answer?
Answers
Answer:
24/25
Step-by-step explanation:
We are dividing 3/10 by 5/16
Gourmet Eatery has a policy of automatically adding a 18% tip to every restaurant Bill if a restaurant bill is $12 how much is it
Answers
Let:
B = Bill
C = Cost of the meal
T = Tip
[tex]undefined[/tex]Sobczak,€8(.8((8.8(.;77;.;&
Answers
just text
okiji
Convert 145 to base 4
Answers
Answer:
Converting 145 to base 4 will give;
[tex]2101_4[/tex]Explanation:
We want to convert;
[tex]145_{ten}\text{ to base 4}[/tex]Converting, we have;
[tex]\begin{gathered} 145\text{ }\div\text{ 4 } \\ 36\text{ }\div\text{ 4 R 1} \\ 9\text{ }\div\text{ 4 R 0} \\ 2\text{ }\div\text{ 4 R 1} \\ 0\text{ R 2} \end{gathered}[/tex]Therefore, converting 145 to base 4 will give;
[tex]2101_4[/tex]Ms wash investdd $22000 in two accounts, one yielding 8% interest and the other yielding 11%. if she recieved a total of $1910 in interest at the end of the year, how much did she invest in each accouny
Answers
Take into account the following formula for the simple interest:
[tex]I=P\cdot r\cdot t[/tex]where:
P: principal investment
r: interest rate
t: time
In order to determine the investments for both accounts, proceed as follow:
-Consider that both investments are represented by P1 and P2 respectively, then, you have:
[tex]\begin{gathered} P_1+P_2=22000 \\ P_2=22000-P_1 \end{gathered}[/tex]- Next, use the given values for parameters r and t for each investment:
8% = 0.08
11% = 0.11
t = 1 year
[tex]\begin{gathered} I_1=P_1\cdot0.08\cdot1=0.08P_1 \\ I_2=P_2\cdot0.11\cdot1=0.11P_2 \end{gathered}[/tex]- Next, consider that the sum of the total earnings is $1910, then:
[tex]I_1+I_2=1910[/tex]- Replace I1 and I2 by the expressions in terms of P1 and P2 and write down the resultant expression in terms of P1, as follow:
[tex]\begin{gathered} 0.08P_1+0.11P_2=1910 \\ 0.08P_1+0.11(22000-P_1)=1910 \\ 0.08P_1+2420-0.11P_1=1910 \\ -0.03P_1=-510 \\ P_1=\frac{510}{0.03}=17000 \end{gathered}[/tex]And for P2:
[tex]\begin{gathered} P_2=22000-P_1 \\ P_2=22000-17000=5000 \end{gathered}[/tex]Hence, the amount of money invested in each account was $5000 and $17000
subtract (7u^2+10u+6) from (3u^2_5u+4).
Answers
Given:
[tex]\mleft(3u^2-5u+4\mright)-(7u^2+10u+6)[/tex]The objective is to subtract both the terms.
[tex]\begin{gathered} \mleft(3u^2-5u+4\mright)-(7u^2+10u+6) \\ 3u^2-5u+4-7u^2-10u-6 \\ -4u^2-15u-2 \end{gathered}[/tex]Hence the subtraction of the given term is,
[tex]-4u^2-15u-2[/tex]The perimeter of a rectangular poster is 14 feet and the length is 4 feet. Describe how to use the perimeter formula to find the width.This calculator has a tray why the answer is not 3.2
Answers
Explanation
We are told that the perimeter of a rectangular poster is 14 feet and the length is 4 feet.
Perimeter simply means the total sum of all the sides of the rectangle
[tex]\begin{gathered} From\text{ the above} \\ let\text{ the length = y} \\ width\text{ =x} \end{gathered}[/tex]So, the perimeter is
[tex]x+x+y+y=2x+2y[/tex]Since the perimeter is 14 then
[tex]2x+2y=14[/tex]Also, the length is 4 feet
Therefore y = 4, so that
[tex]\begin{gathered} 2x+2(4)=14 \\ 2x+8=14 \\ collecting\text{ like terms} \\ 2x=14-8 \\ 2x=6 \end{gathered}[/tex]Making x the subject of the formula
[tex]\begin{gathered} x=\frac{6}{2}=3 \\ \\ x=3 \end{gathered}[/tex]Therefore, the width of the rectangle is 3 feet
The rectangle is
[tex]4+3+4+3=14[/tex]
what is the simplified ratio of 32:24
Answers
Answer:
4/3
Step-by-step explanation:
The simplest form of
32: 24
is 43
Steps to simplifying fractions
Find the GCD (or HCF) of numerator and denominator
GCD of 32 and 24 is 8
Divide both the numerator and denominator by the GCD
32 ÷ 8
24 ÷ 8
Reduced fraction:
4/3
Therefore, 32/24 simplified to lowest terms is 4/3.
The graph below shows the number of snowballs, y, needed to make x snowmen.
Number of Snowballs
15
10
S
(1,3)
(3,9)
(4, 12)
+
2
Number of Snowmen
3 4 5
How many snowballs are needed to make 2 snowmen?
Answers
The number of snowballs that are needed to make 2 snowmen is equal to 6.
How to write a proportional equation?Mathematically, a proportional relationship can be represented by the following equation:
y = kx
Where:
k is the constant of proportionality.y and x represent the variables in a proportional relationship.Next, we would determine the constant of proportionality (k) for the data points on this graph as follows:
k = y/x
k = 3/1 = 9/3 = 12/4 = 3
When the number of snowmen, x = 2, the number of snowballs, y is given by:
y = kx
y = 3 × 2
y = 6.
Therefore, the ordered pair is equal to (2, 6).
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A pizza restaurant is offering a special price on pizzas with 2 toppings. They offer the toppings
below:
Pepperoni
Sausage
Chicken Green pepper
Mushroom Pineapple
Ham
Onion
Suppose that Rosa's favorite is sausage and onion, but her mom can't remember that, and she is
going to randomly choose 2 different toppings.
What is the probability that Rosa's mom chooses sausage and onion?
Choose 1 answer:
Answers
The Probability that Rosa's mom chooses sausage and onion is [tex]\frac{1}{^{8} C_{2} }[/tex]
What is Probability?Probability is the likelihood of an event occurring, measured by the ratio of the favorable cases to the whole number of cases possible.
The probability of an event happening = number of possible outcomes/total number of outcomes.
The number of possible outcomes is 8 exactly 1 of the total possible groups of toppings is sausage and onion.
The total number of outcome is 8 ways, because she has to choose the 2 toppings from possible 8 toppings
So the probability that Rosa's mom will chooses sausage and onion is [tex]\frac{1}{^{8} C_{2} }[/tex]
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Jerry takes out a 30-year mortgage for $170,000.00 to buy a condo. His monthly mortgage payment is $939.00. How much interest will he pay over the life of the loan? Round your answer to the nearest whole dollar.
Answers
Okay, here we have this:
Considering the provided information we obtain the following:
Mortgage capital=$170,000
Total payment = Monthly payment * 12 months of the year * number of years
Total payment = $939*12*30
Total payment = $338,040
Total payment = Mortgage capital + Interest
Replacing we obtain:
Total payment = Mortgage capital + Interest
$338,040=$170,000+interest
Interest= $338,040-$170,000
Total Interest=$168,040
Finally we obtain that the total interest is $168040.
100 Points.
A rectangle has sides measuring (2x + 5) units and (3x + 7) units.
Part A: What is the expression that represents the area of the rectangle? Show your work.
Part B: What are the degrees and classifications of the expression obtained in Part A?
Part C: How does Part A demonstrate the closure property for the multiplication of polynomials?
Answers
The expression that represents the area of the rectangle is [tex]6x^{2}[/tex]+29x + 35 square units , the degree of the obtained expression is 2.
According to the question,
We have the following information:
A rectangle has sides measuring (2x + 5) units and (3x + 7) units.
A) We know that following formula is used to find the area of rectangle:
Area = length*breadth
Area = (3x+7)(2x+5)
Area = [tex]6x^{2}[/tex] + 15x +14x + 35
Area = [tex]6x^{2}[/tex] +29x + 35 square units
B) The degree of an expression is the highest power of the expression. In this case, the highest power is 2. Hence, the degree of the expression obtained is 2.The expression can be classifies as a quadratic polynomial.
C) Part A demonstrates the closure property for the multiplication of polynomials because the expression within the brackets are polynomials and the result obtained is also a polynomial.
Hence, the area of the rectangle is [tex]6x^{2}[/tex] +29x + 35 square units and the degree of the obtained expression is 2.
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From 1999 to 2009, the number of dogs [tex]D[/tex] and the number of cats [tex]C[/tex] (in hundreds) adopted from animal shelters in the United States are modeled by the equations [tex]D = 2n+3[/tex] and [tex]C = n +4[/tex], where [tex]n[/tex] is the number of years since 1999.
a. Write a function that models the total number [tex]T[/tex] of adopted dogs and cats in hundreds for that time period.
b. If this trend continues, how many dogs and cats will be adopted in 2013?
Answers
The functions that models the number of adopted dogs and cat is T = 3n + 7.
If the trend continues, the number of cats and dog that will be adopted by 2013 is 4600.
How to find the function that models a problem?From 1999 to 2009, the number of dogs D and the number of cats C (in hundreds) adopted from animal shelters in the United States are modelled by the equations D = 2n + 3 and C = n + 4, where n is the number of years since 1999.
Therefore, the functions that models the total number T of the adopted dogs and cats in hundreds for that time period can be represented as follows:
T = D + C
where
D = 2n + 3
C = n + 4
where
n = number of yearsT = 2n + 3 + n + 4
T = 3n + 7
b. If the trends continues the number of cats and dogs that will be adopted in 2013 can be calculated as follows:
n = 2013 - 1999 = 13Hence,
T = 3(13) + 7
T = 39 + 7
T = 46
Recall it's represented in hundred's
Therefore, 4600 dogs and cat will be adopted by 2013
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The area of the unshaded region is 22.5cm2 . What is the area of the rectangle?A) 11.25cm2B) 22.5cm2C) 45cm2D) 90cm2
Answers
Let's use the variable b to represent the white triangle base (which is the width of the rectangle), and the variable h to represent the white triangle height (which is the length of the rectangle).
The white triangle area is given by:
[tex]A=\frac{b\cdot h}{2}=22.5\text{ cm}^2[/tex]The area of the rectangle is given by the product of its length and its width, so we have:
[tex]A_2=b\cdot h=2\cdot(\frac{b\cdot h}{2})=2\cdot A=2\cdot22.5=45\text{ cm}^2[/tex]Therefore the correct option is C.
The table of values represents a quadratic function.What is the average rate of change for f(x) from x=−10 to x = 0?Please help me with this problem so that my son can understand better. Enter your answer in the box.xf(x)−10184−5390−654910204
Answers
We are given a quadratic function and the rather than the equation for this function we already have the outputs at each given input as shown in the table provided. This means, for example, for the function given, when the input is -10, the output is 184. Thus the table includes among other values;
[tex]x=-10|f(x)=184[/tex]To calculate the average rate of change we shall apply the formula for the slope (which is also the average rate of change). This is given below;
[tex]\text{Aerage Rate of Change}=\frac{f(b)-f(a)}{b-a}[/tex]Note that the variables are;
[tex]\begin{gathered} f(a)=\text{first input value} \\ f(b)=\text{second input value} \end{gathered}[/tex]The first input value is -10 and the function at that value is 184
The second input value is 0 and the function at that value is -6
We now have;
[tex]\begin{gathered} a=-10,f(a)=184 \\ b=0,f(b)=-6 \end{gathered}[/tex]We can now substitute these into the formula shown nearlier and we'll have;
[tex]\begin{gathered} \text{Ave Rate Of Change}=\frac{f(b)-f(a)}{b-a} \\ =\frac{-6-184}{0-\lbrack-10\rbrack} \end{gathered}[/tex][tex]\begin{gathered} =\frac{-190}{0+10} \\ \end{gathered}[/tex][tex]=\frac{-190}{10}[/tex][tex]\text{Average Rate of Change}=-19[/tex]ANSWER:
The average rate of change over the given interval is -19
In a nearby park, a field has been marked off for the neighborhood Pop Warner football team. If the field has a perimeter of 310 yd and an area of 4950 yd', what are the dimensions of the field?
Answers
Answer:
The dimension of the field is ( 110 x 45)
Exolanations:
Perimeter of the field, P = 310 yd
Area of the field, A = 4950 yd²
Note that the shape of a field is rectangular:
Perimeter of a rectangle, P = 2(L + B)
Area of a rectangle, A = L x B
Substituting the values of the perimeter, P, and the Area, A into the formulae above:
310 = 2(L + B)
310 / 2 = L + B
155 = L + B
L + B = 155...............................................(1)
4950 = L x B...............(2)
From equation (1), make L the subject of the formula:
L = 155 - B...................(3)
Substitute equation (3) into equation (2)
4950 = (155 - B) B
4950 = 155B - B²
B² - 155B + 4950 = 0
Solving the quadratic equation above:
B² - 110B - 45B + 4950 = 0
B (B - 110) - 45(B - 110) = 0
(B - 110) ( B - 45) = 0
B - 110 = 0
B = 110
B - 45 = 0
B = 45
Substitute the value of B into equation (3)
L = 155 - B
L = 155 - 45
L = 110
The dimension of the field is ( 110 x 45)
What is the standard form of the equation of a line passing through points (2,3) and (2,-5)?
Answers
Answer:
[tex]x\text{ = 2}[/tex]Explanation:
Here, we want to find the standard form of the equation
We have the standard form as:
[tex]Ax\text{ + By = C}[/tex]We can arrive at this using the two-points form:
This is:
[tex]\frac{y_2-y_1}{x_2-x_1}\text{ = }\frac{y-y_1}{x-x_1}[/tex](x1,y1) = (2,3)
(x2,y2) = (2,-5)
Now, as we can see, the line is a vertical line since the x-value is the same
Thus, we have it that:
[tex]x\text{ = c}[/tex]where c will represent the x-intercept
Thus, we have the equation of the line as:
[tex]x\text{ = 2}[/tex]Crystal's favorite playlist has 80 rock songs, 40 jazz songs, 25 country songs, 30 hip hop songs, and 45 classical music songs. Which of thesestatements is true?
Answers
This problem tests the knowledge of the probability of a random event occuring: of playing a type of song from a variety of different song types
Thus, we have to compute the probability that each type of song is played.
To do this, we need to obtain the total number songs, as follows:
80 + 40 + 25 + 30 + 45 = 220
Thus, the probabilities are now easily computed as follows:
P(rock) = 80/220
P(jazz) = 40/220
P(country) = 25/220
P(hip hop) = 30/220
P(classical) = 45/220
Now:
Option 1 (the first statement in the options) claims that : P(rock) = 2 * P(hip hop)
However, 2 * P(hip hop)
A basketball player shooting from the foul line has a 40% chance of getting a basket. He takes five shots. Whether he scores on one shot is independent of what he does on another shot. What is the probability that he misses at most one basket (rounded off to three decimals)?
Answers
The probability that the basketball player misses at most one basket is 0.077 as it is a mutually exclusive event.
what are mutually exclusive events in probability?Two events are said to be mutually exclusive if they cannot occur at the same time or simultaneously. This implies they are disjoint events and the probability of both events occurring at the same time will be zero.
Let us represent the probability of the player getting a basket to be p(y) and that of not getting a basket to be p(x)
then p(y)=40%=40/100=2/5
p(x)=1-(2/5)=3/5
The probability the player misses at most one basket implies his highest miss is one out of the five shots he took
So, the probability that he missed the:
1st shot= (3/5)×(2/5)×(2/5)×(2/5)×(2/5)=48/3125
2nd shot= (2/5)×(3/5)×(2/5)×(2/5)×(2/5)=48/3125
3rd shot= (2/5)×(2/5)×(3/5)×(2/5)×(2/5)=48/3125
4th shot= (2/5)×(2/5)×(2/5)×(3/5)×(2/5)=48/3125
5th shot= (2/5)×(2/5)×(2/5)×(2/5)×(3/5)=48/3125
The probability that he misses at most one basket= (48/3125)+(48/3125)+(48/3125)+(48/3125)+(48/3125)+(48/3125)= 249/3125=0.0768.
Finally, from the workings the probability that the player misses at most one basket is 0.077 rounded up to three decimals
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StatusExam9 ft.15 ft.The volume ofthe figure iscubic feet.15 ft.15 ft.
Answers
Step 1:
The figure is a composite figure with a square base pyramid and a cube.
Step 1:
The volume of the composite shape is the sum of the volume of a square base pyramid and a cube.
[tex]\text{Volume = L}^3\text{ + }\frac{1}{2}\text{ base area }\times\text{ height}[/tex]Step 3:
Given data
Cube
Length of its sides L = 15 ft
Square base pyramid
Height h = 9 ft
Length of the square base = 15 ft
Step 4:
Substitute in the formula.
[tex]\begin{gathered} \text{Volume = 15}^3\text{ + }\frac{1}{3}\text{ }\times15^2\text{ }\times\text{ 9} \\ \text{= 3375 + 675} \\ =4050ft^3 \end{gathered}[/tex]Rewrite the following expression so it does not contain any radical term
Answers
Given:
The expression is given as,
[tex]\sqrt[]{36p^{10}m^6}[/tex]The objective is to rewrite the expression without any radical form.
Explanation:
The given expression can be written as,
[tex]\sqrt[]{36p^{10}m^6}=\sqrt[]{6^2p^{10}m^6}\text{ . . . . .(1)}[/tex]In general, the radical form of a square root can be written as,
[tex]\sqrt[]{x}=x^{\frac{1}{2}}[/tex]Then, the equation (1) can be written as
[tex]\sqrt[]{36p^{10}m^6}=(6^2p^{10}m^6)^{\frac{1}{2}}[/tex]On further solving the above expression,
[tex]\begin{gathered} \sqrt[]{36p^{10}m^6}=6^{2\times\frac{1}{2}}p^{10\times\frac{1}{2}}m^{6\times\frac{1}{2}} \\ =6p^5m^3 \end{gathered}[/tex]Hence, the simplified expression of the given term is,
[tex]6p^5m^3[/tex]Which inference about the man is best supported by the events in the text?
Answers
The following statements "If you are wearing a helmet, you are riding a bike." and "If you are not riding a bike, you are not wearing a helmet." are an example of a _____ statement.Select one:a.inverseb.conversec.contrapositive
Answers
Given:
The given statements are,
"If you are wearing a helmet, you are riding a bike."
"If you are not riding a bike, you are not wearing a helmet."
Required:
To identify the kind of statements.
Explanation:
We have the given statement:
"If you are wearing a helmet, you are riding a bike."
Here, p : you are wearing a helmet
q : you are riding a bike
Thus, taking negation of both the parts of the statement as follows:
If not q, then not p.
Hence, the statement formed is,
"If you are not riding a bike, you are not wearing a helmet."
This is the contrapositive statement.
Final Answer:
Given statements are an example of contrapositive.
Is the following sequence arithmetic, geometric, or neither?1, 5, 25, 125, 625
Answers
This is a geometric sequence
This is because we can find the common ratio and not common difference
Here is another riddle:•The sum of two numbers is less than 2.•If you subtract the second number from the first, the difference is greater than 1.What are the two numbers? Explain or show how you know.
Answers
Let the two numbers be A and B
Their sum is less than 2
Thus,
[tex]A+B<2[/tex]When the second number is subtracted from the first number, the difference is greater than 1.
Thus,
[tex]A-B>1[/tex]What is the slope and y-intercept?
y=3x-2
Options:
Blank # 1
Blank # 2
Answers
The value of slope is 3 and the value of y - intercept is -2.
Slope and y intercept:
The slope refers the rate of change in y per unit change in x.
The y-intercept states the y-value when the x-value is 0.
Given,
Here we have the equation
y = 3x - 2
Now, we need to find the slope and y intercept of the equation.
We know that, the standard form of the equation of the line is,
y = mx + b
Where
m represents the slope
b represents the y-intercept.
So, we have to rewrite the given equation as,
y = 3x + (-2)
So, while comparing the given equation with standard form, then we get,
the value of the slope is 3 and the value of the y intercept is -2.
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