Answers
Given
The path of particle 1 is,
[tex]x(t)=3t-6,\text{ }y(t)=t^2-2t[/tex]And, the path of second particle is,
[tex]x(t)=\sqrt{t+6},\text{ }y(t)=-3+2t[/tex]To model the path of the two particles in cartesian form and to find whether, the two particles collide.
Explanation:
It is given that,
The path of the first particle is,
[tex]x(t)=3t-6,\text{ }y(t)=t^2-2t[/tex]That implies,
[tex]x=2t-6,\text{ }y=t^2-2t[/tex]Consider,
[tex]\begin{gathered} x=2t-6 \\ 2t=x+6 \\ t=\frac{x+6}{2} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} y=(\frac{x+6}{2})^2-2(\frac{x+6}{2}) \\ y=\frac{x^2+12x+36}{4}-\frac{2x+12}{2} \\ y=\frac{x^2+12x+36-2(2x+12)}{4} \\ y=\frac{x^2+12x+36-4x-24}{4} \\ y=\frac{x^2+8x+12}{4}\text{ \_\_\_\_\_\_\_\_\_\_\lparen1\rparen} \end{gathered}[/tex]Also, the path of second particle is,
[tex]x(t)=\sqrt{t+6},\text{ }y(t)=-3+2t[/tex]That implies,
[tex]x=\sqrt{t+6},\text{ }y=-3+2t[/tex]Consider,
[tex]\begin{gathered} y=-3+2t \\ 2t=y+3 \\ t=\frac{y+3}{2} \end{gathered}[/tex]Therefore,
[tex]\begin{gathered} x=\sqrt{t+6} \\ \Rightarrow x^2=(t+6) \\ \Rightarrow x^2=(\frac{y+3}{2})+6 \\ \Rightarrow x^2=\frac{y+3+12}{2} \\ \Rightarrow2x^2=y+15 \\ \Rightarrow y=2x^2-15\text{ \_\_\_\_\_\lparen2\rparen} \end{gathered}[/tex]Hence, y=(x^2+8x+12)/4, y=2x^2-15 are the paths of the two particles respectively.
The graph of the path of the two particles are,
From, this it is clear that the particle collide at the points (-2.686, -0.568) and (3.829, 14.324).
Related Questions
State whether the given information is enough to prove that ABCD is a parallelogram.
Answers
From the image given, the data shows that
[tex]\begin{gathered} <1\cong<3 \\ \text{and} \\ AD\cong BC \end{gathered}[/tex]We can observe that
[tex]\begin{gathered} \Delta BDA\cong\Delta DBC\text{ (SAS)} \\ \text{ Reasons:} \\ AD\cong BC\Rightarrow side \\ \measuredangle1\cong\measuredangle3\Rightarrow\text{angle} \\ BD\cong DB(common\text{ sides or reflexive)}\Rightarrow\text{side} \end{gathered}[/tex]Thus, from the above we can say that;
[tex]\begin{gathered} AB\cong DC\text{ (corresponding parts of congruent triangles are congruent)} \\ \text{Therefore, } \\ \measuredangle2\cong\measuredangle4 \end{gathered}[/tex]Hence
Yes, the given information is enough to prove that ABCD is a parallelogram.
Factor the expression. 12y + 14
Answers
Answer:
2(6y+7)
Explanation:
To factor the expression:
[tex]12y+14[/tex]First, find the greatest whole number that divides 12 and 14.
The number = 2, therefore:
[tex]\begin{gathered} 12y+14=2\mleft(\frac{12y}{2}+\frac{14}{2}\mright) \\ =2(6y+7) \end{gathered}[/tex]What is the greatest common factor of 9 and 72?
Answers
The Greatest Common Factor of 9 and 72 is: 9
SOLUTION
Problem Statement
The question asks us to find the greatest common factor of 9 and 72.
Method
In order to solve this question, we just need to follow these steps:
1. Write out the prime factors of 9 and 72
2. Choose the common factors from both expressions.
3. Multiply the common factors.
Implementation
1. Write out the prime factors of 9 and 72:
[tex]\begin{gathered} 9=1\times3\times3 \\ \text{The common factors of 9 are: 3 and 3} \\ \\ 72=1\times2\times2\times2\times3\times3 \\ \text{Common factors of 72 are: 1,2, 2, 2 and 3, 3} \end{gathered}[/tex]2. Choose the common factors from both expressions.:
We need to examine the two expressions for 9 and 72 above. Choose the common values.
[tex]\begin{gathered} 3\times3\text{ is common to both 9 and 72} \\ i\mathrm{}e\text{.} \\ 9\text{ is common to both 9 and 72} \\ 3\text{ is common to both 9 and 72 as well} \\ 1\text{ is also common to both 9 and 72} \end{gathered}[/tex]3. Multiply the common factors.:
[tex]\begin{gathered} \text{Thus, choosing the greatest values from 1,3 and 9.} \\ \therefore\text{The Greatest Common Factor = 9} \end{gathered}[/tex]Final Answer:
The Greatest Common Factor of 9 and 72 is: 9
Permutation/combination In how many ways can five people line up to get on a bus? (Hint: all 5 people are lining up)
Answers
We have to calculate in how many ways can five people line up to get on a bus.
This is a permutation of 5 in 5 without repetition, so it can be calculated as:
[tex]P(5,5)=5!=120[/tex]Answer: in 120 ways.
X) *11.4.14 Find the volume of the cylinder in terms of it and to the nearest tenth. 2 in 1 in The volume in terms of it is V= in3
Answers
We can calculate the volume of the cylinder as the product of the area of the base and the height of the cylinder.
[tex]V=A_b\cdot h[/tex]The area of the base is equal to:
[tex]A_b=\pi r^2=\pi\cdot2^2=4\pi[/tex]Then, the volume becomes:
[tex]V=A_b\cdot h=4\pi\cdot1=4\pi\approx12.6\text{ in}^3[/tex]Answer:
The volume in function of π is V = 4π in^3.
The volume rounded to the nearest tenth is 12.6 in^3.
Robin is saving money to buy a 720$ phone. she has 105$ saved, and each week she adds 30$ to her savings. write an equation to find the number of weeks (w) until she has enough savings to buy the phone.
Answers
Let call w the number of weeks she has been saving.
Then, we can write the expression for her saving in function of the number of weeks as:
[tex]S(w)=105+30\cdot w[/tex]We now have to find the number of weeks it will take for her savings to reach $720.
We can find it by calculating w for S(w)=720:
[tex]\begin{gathered} S(w)-105+30w=720 \\ 105+30w=720 \\ 30w=720-105 \\ 30w=615 \\ w=\frac{615}{30} \\ w=20.5\approx21 \end{gathered}[/tex]Answer: it will take her 21 weeks to have enough savings.
Real numbersA stock lost 8 and 3/8 on Monday, 1 and 5/8 points on Tuesday. On Wednesday, it gained 13 points. What was the net gain or loss of the stocks for these three days?
Answers
Answer: The net gain was 3 stocks for these three days.
Explanation
Given
• Monday: lost 8 and 3/8.
,• Tuesday: lost 1 and 5/8 points.
,• Wednesday: gained 13 points.
Procedure
To calculate the net gain, we have to add all the values we are giving, considering that the stock lost will have a negative sign:
[tex]\text{Net gain}=Gains-losses[/tex]0. Calculating the losses
We have to mixed numbers, for which we can add the whole numbers and the fractions separately.
• Whole numbers
[tex]-8-1=-9[/tex]• Fractions
[tex]-\frac{3}{8}-\frac{5}{8}=-\frac{8}{8}=-1[/tex]At last, we add both results and we get the losses:
[tex]losses=-9-1=-10[/tex]2. Calculating the gains
As we only have one quantity, there is no need for calculations:
[tex]Gains=13[/tex]3. Calculating net gain
Finally, we put both quantities in the formula:
[tex]\text{Net gain}=13-10=3[/tex]The graph shows the projections in total enrollment at degree granting institutions from Fall 2003 to Fall2012The linear model, y= 0.2145x + 15.79, provides the approximate enrollment, in millions, between the years 2003 and 2012, where x = 0 corresponds to 2003, x = 1to 2004, and so on, and y is in millions of students.(a) Use the model to determine projected enrollment for Fall 2008.The projected enrollment for Fall 2008 is millions.(Type an integer or decimal rounded to the nearest tenth as needed.)
Answers
In order to find the projected enrollment for 2008, we need to use the value of x equal to 5, because x represents the number of years after 2003.
Then, using the linear model with x = 5, we have:
[tex]\begin{gathered} y=0.2145\cdot5+15.79\\ \\ y=1.0725+15.79\\ \\ y=16.8625 \end{gathered}[/tex]Rounding to the nearest tenth, we have y = 16.9.
11-3x75-1 what is the answer?
Answers
Answer:
Step-by-step explanation:
-215
Answer:
-215
Step-by-step explanation:
the answer to 11-3x75-1 is -215
there are 14 square and 18 rectangles. what is the simplest ratio of squares to rectangles?
Answers
The simplest ratio of squares to rectangles can be obtained as follows:
There are 14 squares and 18 rectangles. The ratio of squares to rectangles is:
[tex]\frac{14}{18}=\frac{7}{9}[/tex]Then, the simplest ratio is 7/9 because 7 is a prime number and the ratio cannot be simplified any more. To obtain 7/9 we divided the numerator by 2 and the denominator also by 2.
A buoy oscillates in simple harmonic motion as waves go past. At a given time it is noted that the buoy moves a total of 3.9 feet from its low point to its high point, and that it returns to its high point every 16 seconds
Answers
Concept
What is a simple harmonic motion ?Repetitive movement back and forth through an equilibrium, or central, position, so that the maximum displacement on one side of this position is equal to the maximum displacement on the other side. The time interval of each complete vibration is the same
a) y= 1.95ftcos(π/2)
b) v(t)= -0.76sin(π/5t)
Part aSince the buoy oscillates in simple harmonic motion the equation to model this is given by: y= A cos(ωt+θ)
For this case from the info given we know that:
2A= 3.9 , A= 3.9/2= 1.95ft
It returns to its high point every 16 seconds. That means period = 16 , and the angular frequency can be founded like this:
ω=2π/16
= π/8
Assuming that the value for the phase is (θ=0°) our model equation is given by
y= 1.95ftcos(π/2)
Part bFrom definition we can obtain the velocity with the derivate of the position function and if w calculate the derivate we got this,
dy/dt= v(t)= -1.95ft(π/8)sin(π/8t)
v(t)= -0.76sin(π/8t)
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160 is what percent of the sum of 100 and 120 and 160
Answers
42.105%
1) Let's first add thosse numbers up: 100 +120+160 =380
2) Now we can write out the following ratio, to find out what percentage is equivalent to 160 out of that 380:
[tex]\begin{gathered} 160----x \\ 380----100 \\ \frac{160}{380}=\frac{x}{100} \\ 380x=16000 \\ \frac{380x}{380}=\frac{16000}{380} \\ x=42.105\% \end{gathered}[/tex]Note that we had to cross multiply that.
3) Hence, the answer is 160 is approximately 42.105% of 380
Find two functions f and g such that (f O g) (x) =h(x) [tex]h(x) = (9x + 7)^{2} [/tex]
Answers
f(x) = x² and g(x) = 9x + 7
Explanation:(fog) (x) = h(x)
h(x) = (9x + 7)²
To get the two functions, we need to understand that the function g(x) will be inserted in function f(x) to get (fog)(x)
g(x) = 9x + 7
This is the function that will be inserted into f(x)
Since we have a square, the function of f(x) will have a square
f(x) = x²
Putting both together, replace x with (9x + 7) in f(x)
(fog)(x) = (9x + 7)²
Hence, f(x) = x² and g(x) = 9x + 7
Question 2(Multiple Choice Worth 2 points)
(03.01 LC)
Jordan compared 10 books at the school library. The following table shows the number of chapters and the total number of pages for each book.
Number of Chapters 3 4 8 10 16
Total Pages 25 38 85 76 180
Which data display would you use to represent this data?
O Histogram
Scatter plot
O Line graph
O Line plot
Answers
To represent the data, histogram would have been used.
Jordan compared 10 books at the school library. The following table shows the number of chapters and the total number of pages for each book.
Number of Chapters 3 4 8 10 16
Total Pages 25 38 85 76 180
In a histogram, a graphical representation of the distribution of data is done. The histogram is represented by a set of rectangles, adjacent to each other and each bar represent a kind of data.
Here the number of chapters can be kept in x axis and the total number of pages can be kept in the y axis.
Therefore, histogram would be used to display the data.
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3 A free diver can dive at a rate of -0.75 meters per second. About how long would it take to reach a depth of -145 meters?
Answers
Answer:193.33s or 3.2 mins
Step-by-step explanation:
Here the diver dives at -0.75 constant velocity. So we can use the formula s = vt
from there we can find out the time when s = -145 m and v = -0.75m/s
Laverne created a painting with an area of 72 square inches and a length of 9 inches. They create a second painting with an area of 56 square inches. It has the same width as the first painting. What is the length of the second painting?
Answers
Answer: 7
Step-by-step explanation:
As our first step, we can divide 72 by 9 to discover the width of the first AND second painting. 72÷9=8
Now that we have discovered the width of the second painting, we will now solve the mathematical expression 56÷8.
Because 56 divided by 8 equals 7, the length of the second painting is 7 inches.
9) The temperature outside feels like - 3°C on Thursday. When the temperature is taken, it is actually 16°C. Howmany degrees lower does the temperature feel?
Answers
The temperature feels
[tex]16-(-3)=19[/tex]degrees lower from the actual.
I really need help solving thisIt’s from my trig prep bookIt asks to answer (a) and (b)
Answers
The sum in summation notation is:
[tex]\sum ^4_{r\mathop{=}0}(-1)^r(4Crx^{4-r}y^r)[/tex]The expansion is:
[tex]81x^{20}-12x^{15}y^3+\frac{2}{3}x^{10}y^6-\frac{4}{243}x^5y^9+\frac{1}{6561}y^{12}[/tex]Explanation:Given the expression:
[tex](3x^5-\frac{1}{9}x^3)^4[/tex]In summation notation, this can be written as:
[tex]\sum ^4_{r\mathop=0}(-1)^r(4Crx^{4-r}y^r)[/tex]The simplified terms of the expression is:
[tex]\begin{gathered} 4C0(3x^5)^{\mleft\{4-0\mright\}}(\frac{1}{9}y^3)^0-4C1(3x^5)^{\mleft\{4-1\mright\}}(\frac{1}{9}y^3)^1+4C2(3x^5)^{\mleft\{4-2\mright\}}(\frac{1}{9}y^3)^2-4C3(3x^5)^{\mleft\{4-3\mright\}}(\frac{1}{9}y^3)^3+4C4(3x^5)^{\mleft\{4-4\mright\}}(\frac{1}{9}y^3)^4 \\ \\ =(3x^5)^4-4(3x^5)^3(\frac{1}{9}y^3)+6(3x^5)^2(\frac{1}{9}y^3)^2-4(3x^5)^{}(\frac{1}{9}y^3)^3+(\frac{1}{9}y^3)^4 \\ \\ =81x^{20}-12x^{15}y^3+\frac{2}{3}x^{10}y^6-\frac{4}{243}x^5y^9+\frac{1}{6561}y^{12} \end{gathered}[/tex]A reflection across which line(s) carries the trapezoid onto itself?
Answers
If we reflect about x =2, it is a mirror image on each side ( left and right). There is no other line where we can have a mirror image on each side.
7 singles, 13 fives, 4 twenties, and 3 hundred dollar bills are all placed in a hat. If a player is to reach into the hat and randomly choose one bill, what is the fair price to play this game?
Answers
So,
The probability to obtain one bill of the four, is 1/4.
So, the expected value is:
[tex]\begin{gathered} \frac{1}{4}(7)+\frac{1}{4}(13)+\frac{1}{4}(4)+\frac{1}{4}(3) \\ \\ =6.75 \end{gathered}[/tex]Therefore, the fair price to play this game is 6.75.
Assume that 5 cards are drawn from a standard deck of 52 cards. How many ways can I get 3 sevens, 1 six and 1 five?
Answers
Answer
64 ways
Explanation
In a standard deck of 52 cards, there are four 'sevens', four 'sixes' and four 'fives'.
Using Combination formula, the number of ways to pick 3 sevens, 1 six and 1 five is given as
⁴C₃ × ⁴C₁ × ⁴C₁
= 4 × 4 × 4
= 64
Hope this Helps!!!
Sean johnson 1. The angles of a triangle are described as follows: angle A is the largest angle: its measure is twice the measure of angle B. The measure of angle C is 2 less than half the measure of angle B. Find the measures of the three angles in degrees.
Answers
Explanation
Let
angle A is the largest angle
[tex]m\measuredangle A=m\measuredangle A[/tex]its measure is twice the measure of angle B,( in other words you have to multiply angle b by 2, to get angle A)
[tex]2(m\measuredangle B)=m\measuredangle A[/tex]i need help solving -12=3-2K-3K
Answers
Solve
[tex]\begin{gathered} -12=3-2k-3k \\ \text{Collect like terms} \\ -12-3=-2k-3k \\ \text{Note that when a positive value crosses the equality sign, it becomes negative} \\ \text{Similarly when a negative value crosses the equality sign it becomes positive} \\ \text{Hence, 3 crosses over and becomes negative} \\ -12-3=-2k-3k \\ -15=-5k \\ \text{Divide both sides by -5} \\ \frac{-15}{-5}=\frac{-5k}{-5} \\ 3=k \end{gathered}[/tex]Consider the non-right triangle below.Suppose that m∠CAB=62∘, and that x=35 cm and y=17 cm. What is the area of this triangle? cm^2
Answers
Given that:
x=35 cm and y=17 cm
and angle CAB= 62 degree
[tex]\begin{gathered} A=\frac{1}{2}\times x\times y\times\sin (\angle CAB) \\ A=\frac{1}{2}(35)(17)\sin (62) \\ A=297.5\times\sin (62) \\ A=262.67cm^2 \end{gathered}[/tex]what is a unit rate for meter per second if a car travels 274 m in 17 seconds
Answers
The rate is 274m/17s = 16.1176m/s
The pie chart below shows the percentage of total revenue that a publisher receives from publications.A) which category accounts for approximately 1/5 of publishers total revenue type in which one (textbooks, cookbooks, magazines, paperbacks,poetry, nonfiction B)Approximately percentage total revenue from paperbacks poetry combined C) if revenue from nonfiction 10% total revenue approximately percentage total comes from poetry
Answers
a) To determine which category represents approximately one-fifth of the publisher's revenue, you can compare the given pie chart with a pie chart divided into 5 parts.
The category that is approximately one-fifth of the revenue, that is, 20% of the total revenue, is textbooks.
To solve parts b and c you can use the following pie chart to compare with the given chart for the publisher's revenue:
b) Looking at the pie charts, the categories "paperback" and "poetry" are approximately 20% of the total revenue.
c) The category "Nonfiction" is 10% of the total revenue, the category "poetry" is approximately half of the size of the category "nonfiction" so you can say that it corresponds to 5% of the total revenue.
Four friends had dinner at a restaurant. The bill was $45.40, and they added a 20% tip. The 4 friends shared the total cost equally. How much money did each person pay?
Answers
Answer:
$13.62
Step-by-step explanation:
To find the tip of 20%, multiply 45.40 by 0.20, and you get 54.48.
Now, since there are 4 friends, divide the amount by 4.
54.48/4=13.62
So each person would pay $13.62.
Hope this helps! :D
Andy spent 1/3 of his money on pastries and 3/4 of his remaining money on 2 pies. Each pie costs 6 times as much as each pastry. if all pastries cost the same how many did he buy
Answers
This means he originally had enough money to buy 4 pies or 24 pastries
1/3 of his money would buy 8 pastries
What is the area of the blue shape?
Answers
In rectangle , 38.5 sq units is the area of the blue shape.
What is rectangle?
A rectangle is a sort of quadrilateral with parallel sides that are equal to one another and four vertices that are all 90 degrees apart. As a result, it is sometimes referred to as an equiangular quadrilateral. Because the opposite sides of a rectangle are equal and parallel, it can also be referred to as a parallelogram.We can split the shape into two rectangles.
The small rectangle at the top has an area of 14 sq units (2 * 7).
The middle rectangle has an area of 49 sq units ( 7*7).
Since the blue part in the middle rectangle is half of the whole rectangle, the area of the blue part there is 24.5 sq units.
Adding that to 14 sq units will give us 38.5 sq units.
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determine whether the given by binomial is a factor of the polynomial p(x) . If so, find the remaining factors of p(x).
Answers
The given binomial is a factor of the polynomial p(x) with remaining factors of (x + 1)(x - 1).
What is termed as the factors of polynomial?Factorisation is the process of determining the factors of a given value as well as mathematical expression. Factors are integers which are multiplied together to create the original number.For the given question.
The polynomial is given as; x³ + 2x² -x - 2.
The binomial is given as; (x +2).
The, to get the remainder, divide the polynomial with the binomial.
= (x³ + 2x² - x - 2)/ (x +2)
Taking x² common from the first two terms of the numerator and (-1) from the last two terms.
= x²(x + 2) - (x + 2)/ (x +2)
Taking (x + 2) common from two terms.
= (x + 2)(x² - 1)/(x + 2)
Cancel (x + 2) from both.
= (x² - 1)
Now use the identity to open the square.
(a² + b² ) = (a + b) (a - b)
= (x + 1)(x - 1).
Thus, the given binomial is a factor of the polynomial p(x) with remaining factors of (x + 1)(x - 1).
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The correct question is-
Determine whether the given binomial is a factor of the polynomial p(x).
If so, find the remaining factors of p(x).
p(x) = x³ + 2x² -x - 2 ; (x +2)
Answer:
a
Step-by-step explanation: