Let us have four distinct collinear points $a,$ $b,$ $c,$ and $d$ on the cartesian plane. the point $c$ is such that $\dfrac{ab}{cb} = \dfrac{1}{2}$ and the point $d$ is such that $\dfrac{da}{ba} = 3$ and $\dfrac{db}{ba} = 2.$ if $c = (0, 4),$ $d = (4, 0),$ and $a = (x, y),$ what is the value of $2x + y$?
Answers
Start with a line segment connecting two points, A and B. [tex]\dfrac{DA}{BA}=3[/tex] means DA is 3 times longer than BA. Clearly, D cannot fall between A and B because that would mean DA is shorter than BA. So there are two possible locations where D can be placed on the line relative to A and B.
But with [tex]\dfrac{DB}{BA}=2[/tex], or the fact that DB is 2 times longer than BA, we can rule out one of these positions; referring to the attachment, if we place D to the left of A, then DB would be 4 times longer than BA.
Finally, [tex]\dfrac{AB}{CB}=\dfrac12[/tex], so that CB is 2 times longer than AB. Again we have two possible locations for point C (it cannot fall between A and B), but one of them forces C to occupy the same point as D. However, A, B, C, D are distinct, so C must fall to the left of A.
Now let [tex]d[/tex] be the length of AB. Then the length of CD in terms of [tex]d[/tex] is [tex]4d[/tex]. We have the coordinates of C and D, and the distance between them is [tex]\sqrt{(4-0)^2+(0-4)^2}=4\sqrt2[/tex]. So
[tex]4d=4\sqrt2\implies d=\sqrt2[/tex]
The slope of the line through C and D is
[tex]\dfrac{0-4}{4-0}=-1[/tex]
and so the equation of the line through these points is
[tex]y-4=-(x-0)\implies x+y=4[/tex]
So the coordinates of A are [tex](x,y)=(x,4-x)[/tex]. The distance between C and A is [tex]d=\sqrt2[/tex], so we have
[tex]\sqrt{(x-0)^2+(4-x-4)^2}=\sqrt{2x^2}=|x|\sqrt2=\sqrt2\implies|x|=1[/tex]
Since A falls to the right of C (in the [tex]x,y[/tex] plane, not just in the sketch), we know to take the positive value [tex]x=1[/tex]. Then the [tex]y[/tex] coordinate is [tex]y=4-1=3[/tex].
All this to say that A is the point (1, 3), so
[tex]2x+y=2+3=5[/tex]
The value 2x + y of is 5.
Start with a line segment connecting two points, A and B. [tex]\frac{DA}{BA} =3[/tex] means DA is 3 times longer than BA. Clearly, D cannot fall between A and B because that would mean DA is shorter than BA. So there are two possible locations where D can be placed on the line relative to A and B.
But with [tex]\frac{DB}{BA} =2[/tex], or the fact that DB is 2 times longer than BA, we can rule out one of these positions; referring to the attachment, if we place D to the left of A, then DB would be 4 times longer than BA.
Finally, [tex]\frac{AB}{CB} =\frac{1}{2}[/tex], so that CB is 2 times longer than AB. Again we have two possible locations for point C (it cannot fall between A and B), but one of them forces C to occupy the same point as D. However, A, B, C, D are distinct, so C must fall to the left of A.
Now let be the length of AB. Then the length of CD in terms of is . We have the coordinates of C and D, and the distance between them is
[tex]\sqrt{(4-0)^2+ (4-0)^2} =4\sqrt{2}[/tex]
So
4d= 4√2 = d= √2
The slope of the line through C and D is
[tex]\frac{0- 4}{4-0} = -1[/tex]
and so the equation of the line through these points is
y- 4 = -(x- 0) =x + y =0
So the coordinates of A are (x, y) = (x,4 -x). The distance between C and A is d= √2, so we have
[tex]\sqrt{(x-0)^2+(4- x - 4)^2} = \sqrt{2x^3} = |x| \sqrt{2} = \sqrt{2} = |x| = 1[/tex]
Since A falls to the right of C (in the x, y plane, not just in the sketch), we know to take the positive value . Then the coordinate is .
All this to say that A is the point (1, 3), so 2x + y= 2+3=5
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Question
Let us have four distinct collinear points a, b, c, and d on the cartesian plane. the point c is such that [tex]\dfrac{ab}{cb} = \dfrac{1}{2}[/tex] and the point d is such that [tex]\dfrac{da}{ba} = 3[/tex] and [tex]\dfrac{db}{ba} = 2[/tex] c = (0, 4), d = (4, 0), and a = (x, y), what is the value of 2x + y?
Related Questions
Geometry help! Will give brainliest!! The slope of line b is -1/6. What is the slope of the line parallel to line b?
1/6
-1/6
-6
6
Answers
Answer:
-1/6
Step-by-step explanation:
Parallel lines are lines which have the exact same slope but different y-intercepts. This means any line which is parallel and does not cross will have the exact same slope of -1/6.
Todd is saving for a vacation. The cost of his vacation is 1,089. Todd has a year to save the money. About how much does he need to save each month to reach his goal
Answers
Answer:
He needs to save $90.75 each month to reach his goal.
Step-by-step explanation:
We have the cost of his vacation that is $1089.
We know that Todd has a year to save the money.
We know that a year in terms of time is twelve months (12 months).
If we want to calculate how much does he need to save each month we need to divide the total cost of his vacation by the amount of months in a year ⇒
[tex]\frac{1089}{12}=90.75[/tex]
We find that Todd needs to save $90.75 per month to reach his goal.
Dan Williams took out a simple interest loan at 14.5% interest for 12 months. His previous balance is $119.42. What is the final payment if the loan is paid off with the next payment?
Answers
Answer:
$132.90
Step-by-step explanation:
Answer:
$120.86
Step-by-step explanation:
Sharma is building a shed and wants to determine the measurements for the roof. The span of the roof will be 10 feet and she plans to use a 5:12 roof pitch. This means that the roof will rise 5 inches for every 12 horizontal inches. Complete her diagram by determining the unknown angle measures, to the nearest degree, and side lengths, to the nearest tenth of a foot.
Answers
Answer:
In the figure below :
d = 2.08, e = 5, c = 5.42 , a = 22.78°, b = 67.22°
Step-by-step explanation:
Side lengths :
[tex]2\cdot e=10\\ \Rightarrow e=5 feet\\\frac{d}{e}=\frac{5}{12}\\\Rightarrow d=\frac{25}{12}\approx 2.08 feet[/tex]
By pythagoras theorem,
[tex]c^2=d^2+e^2\\c^2=29.33\\c\approx 5.42 feet[/tex]
Now to find angles,
[tex]a+b+90^{\circ}=180^{\circ}\\\Rightarrow a+b=90^{\circ}\\\tan a=\frac{d}{e}\\\Rightarrow \tan a =0.42\\\Rightarrow a=22.78^{\circ}\\\Rightarrow b=67.22^{\circ}[/tex]
The unknown side lengths of the shed's roof are:
Angle measure at the top of the roof: 22.62 degrees
Vertical length of the roof (rise): 4.17 feet
Horizontal length of the roof (run): 5 feet
To determine the unknown angle measures and side lengths of the shed's roof, we can use the given information about the roof pitch and span.
Angle Measures
The roof pitch is 5:12, which means that for every 12 horizontal inches, the roof rises 5 inches. This corresponds to an angle of approximately 22.62 degrees.
Side Lengths
The span of the roof is 10 feet, which represents the horizontal distance from one side of the shed to the other.
Since the roof pitch is 5:12, we can calculate the rise of the roof by dividing the span by 12 and multiplying by 5:
Rise = (10 feet * 12 inches/foot) * (5/12) = 41.667 inches ≈ 4.17 feet
Therefore, the unknown side lengths of the shed's roof are:
Angle measure at the top of the roof: 22.62 degrees
Vertical length of the roof (rise): 4.17 feet
Horizontal length of the roof (run): 5 feet
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Brandy is trying to factor the quadratic $3x^2 - x - 10.$ She starts by assuming that the quadratic factors as \[3x^2 - x - 10 = (x + B)(3x + D),\]for some integers $B$ and $D.$ After some work, Brandy successfully factors the quadratic. Find the ordered pair $(B,D).$
Answers
Answer:
(B , D) = (-2,5)
Step-by-step explanation:
The given quadratic expression is
[tex]3x^{2} -x-10[/tex]
compare the given expression with [tex]ax^{2} +bx+c[/tex]
so we have [tex]a = 3 , b = -1 ,c = -10[/tex]
now [tex]ac=3 (-10)=- 30[/tex] and [tex]b= -1[/tex]
we need to find two numbers such that their product is -30 and sum is -1
-6 and 5 are such numbers
so we have
[tex]3x^{2} -6x+5x-10[/tex]
grouping first two terms and last two terms
[tex](3x^{2} -6x)+(5x-10)[/tex]
factor out 3x from first two terms and 5 from last two terms
[tex]3x(x-2) +5(x-2)[/tex]
[tex](x-2)(3x+5)[/tex] ( factor out (x-2))
[tex](x+(-2))(3x+5)[/tex]
hence B =-2 and D= 5
(B,D)= (-2,5)
Hailey ran a few laps. D(n) models the duration (in seconds) of the time it took for Hailey to run her n lap. When did her lap duration increase
Answers
Answer:
2 hours
Step-by-step explanation:
Answer:
during the 7th and the 9th lap
Step-by-step explanation:
I just did it
The vertex of the function graph is given. Find c. Y= c-x^2-2x;(-1,3)
Answers
Answer:
c = 2
Step-by-step explanation:
(x, y) = (-1, 3)
3 = c - (-1)^2 - 2(-1)
3 = c - 1 + 2
3 = c + 1
2 = c
What are the zeroes of the polynomial function f(x)=x^3+2x^2-x-2
Answers
[tex]f(x)=x^3+2x^2-x-2\\\\f(x)=0\iff x^3+2x^2-x-2=0\\\\x^2(x+2)-1(x+2)=0\\\\(x+2)(x^2-1)=0\iff x+2=0\ \vee\ x^2-1=0\\\\x+2=0\qquad\text{subtract 2 from both sides}\\\\\boxed{x=-2}\\\\x^2-1=0\qquad\text{add 1 to both sides}\\\\x^2=1\to x=\pm\sqrt1\\\\\boxed{x=-1\ \vee\ x=1}\\\\Answer:\ -2,\ -1,\ 1[/tex]
Answer: -2, -1, 1
Step-by-step explanation:
f(x) = x³ + 2x² - x - 2
0 = x³ + 2x² - x - 2
= x²(x + 2) -1(x + 2)
= (x² - 1) (x + 2)
= (x - 1) (x + 1) (x + 2)
0 = x - 1 0 = x + 1 0 = x + 2
1 = x -1 = x -2 = x
(a+b+c)(d+e+f)(a+b+c)(d+e+f). The expression consists of 3 factors, and each factor contains 2 terms.
Answers
Answer:
Not true
Step-by-step explanation:
It is in fact an expression with 4 factors, each with 3 terms.
Factors are things you multiply, terms are things you add.
Determine between which consecutive integers the real zeros of f(x)=2x^2-5x+1 are located a. between 0&1 and 1&2 c. between 0&1 and 2&3 b. between -1&0 and 2&3 d. between 1&2 and 2&3
Answers
Answer:
c. between 0&1 and 2&3
Step-by-step explanation:
The roots of the given quadratic functions lie between 0&1 and 2&3 thus option (C) is correct.
What is a quadratic equation?An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, and a must not be zero.
As per the given quadratic equation,
f(x) = 2x² - 5x + 1
The value of x will be as,
x = [-(-5) ± √((-5)² - 4 ×2 × 1)]/(2 × 2)
x = (5 ± √17)/4
x = 0.219 and x = 2.28
So, 0 < 0.219 < 1 and 2 < 2.28 < 3.
Hence "The roots of the given quadratic functions lie between 0&1 and 2&3".
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5a + 5b + 5c + 5d
Which expression is another way to write the expression shown here?
A) 5abcd
B) 5a + bcd
C) 5(a + b + c + d)
D) (5a)(5b)(5c)(5d) what the answer
Answers
The correct way to rewrite the expression 5a + 5b + 5c + 5d using the distributive property is option C) 5(a + b + c + d).
The expression given is 5a + 5b + 5c + 5d. This is a sum where each term is multiplied by 5. According to the distributive property, also known as the distributive law of multiplication, a common factor can be factored out of a sum. The distributive property is represented as (A + B)C = AC + BC. In this case, the common factor is 5, and the sum of the variables a, b, c, and d is grouped, giving us 5(a + b + c + d).
So, the correct representation of the expression is 5(a + b + c + d), which corresponds to option C).
H values from the specified set make up the solution set of the inequality? 5w–10>20 ; {5,6,7,8} Select each correct answer. A.5 B.6 C.7 D.8 plz help
Answers
Answer: (C) 7 and (D) 8
Step-by-step explanation:
5w - 10 > 20
+10 +10
5w > 30
÷5 ÷5
w > 6
Anything GREATER than 6 will work.
solve for x
kx + c = 9
Answers
kx + c = 9
Subtract C from both sides:
kx = 9-c
Divide both sides by k:
x = (9-c)/k
Statewide, there are 20 full-time employees to every 3 part-time employees. If there are 250,000 full-time employees, how many part-time employees are there statewide?
Answers
Divide total full time employees by 20 to find how many groups of 20 there are, then multiply that number by 3 to find total part time employees:
250,000 / 20 = 12,500
12,500 x 3 = 37,500 part time employees.
A pair of jeans was in a 15% off sale but, when Bill came to buy them a month later, the sale was over and he had to pay the regular price of $70. How much money did Bill lose?
Answers
Answer:
10.5
Step-by-step explanation: if he got the pants on sale the price would be 59.5 becuase 70*15%=10.5 and u would subtract 70-10.5=59.5
70-59.5=10.5
Answer:
$10.5
Step-by-step explanation:
In first gear, or low gear, an automobile's engine runs about three times as fast as the drive shaft. In second gear, the engine does not have to run as fast; usually it runs about 1.6 times faster than the drive shaft. Finally, in third, or high gear, the engine runs at the same speed as the drive shaft. Engine speed = 4,100 r.p.m. Transmission in first gear Drive-shaft speed (to nearest r.p.m.) =
a) 1367
b) 12,300
c) 6560
Answers
Answer:
A 1367
Step-by-step explanation:
What is the solution to the equation 3c/8 = c+4/4
Answers
Answer:
c=8
Step-by-step explanation:
3c/8 = (c+4)/4
We can use cross products to solve
3c c+4
----- = ----------
8 4
3c*4 = 8 *(c+4)
12c = 8c+32
Subtract 8c from each side
12c-8c = 32
4c = 32
Divide by 4
4c/4 = 32/4
c =8
The solution to the equation 3c/8 = c+4/4 is found by first simplifying the equation to 3c = 4c + 16, then isolating 'c' gives -c = 16, and finally multiplying both sides by '-1' gives the solution, c = -16.
Explanation:To solve the equation 3c/8 = c+4/4, firstly we'll simplify it, eliminating fractions by multiplication. The equation becomes 3c = 4c + 16.
Next, we rearrange the equation to isolate 'c', by subtracting '4c' from both sides which results in -c = 16.
Finally, to get the value of 'c', we multiply both sides by '-1'. Hence, the solution to the equation is c = -16.
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HELP PLEASE!! which of the following represents a function?
please explain too i don’t understand this.
Answers
im really bad at explaining but thats the simplest way i learned
Answer:
I think it's c.
Step-by-step explanation:
A graph usually stands as a representation for functions.
Who knows please help me I legit love you guys ASAP NEED ANSWERS
Answers
Answer:
I think the answer would be B because volume is length times width times height. Therefore, if you multiply the base which is length time width and the height you will get your answer. Let me know if this helps, if not I can further explain it to you.
Step-by-step explanation:
Flo told her coach that she swam 5.4 kilometers in the pool. The chart shows the length of a standard-sized pool and the distance around a standard-sized running track.
Venue
Kilometers
Pool
0.05
Running Track
0.4
Which is the best method to find the number of laps Flo swam?
divide 5.4 by 0.05
multiply 5.4 by 0.05
divide 5.4 by 0.4
multiply 5.4 by 0.4
Answers
Answer:
Divide 5.4 by 0.5
Step-by-step explanation:
It's says to find the number of laps she swam. It doesn't ask us to figure out how many laps she could have run with the amount of laps she did.
To find the number of laps Flo swam, divide the total distance Flo swam (5.4 km) by the length of one lap in the pool (0.05 km). This results in 108 laps.
Explanation:To find out the number of laps Flo swam, we must take the total distance she swam and divide it by the length of one lap in the pool. According to the chart, the length of one lap in a standard-sized pool is 0.05 kilometers. Therefore, the best method would be to divide 5.4 (the total distance Flo swam in kilometers) by 0.05 (the length of one lap in kilometers).
So, we take 5.4/0.05 = 108. This means that Flo swam 108 laps in the pool. Remember, when you want to find how many times a certain distance fits into another, you should utilize division.
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An airplane with a speed of 160 knots is headed east while a 24-knot wind is blowing from 240°. Explain how you would use the result of finding the ground speed to find the course of the airplane.
Answers
Question: Explain how you would use the result of finding the ground speed to find the course of the airplane.
Answer: 181.2
Explanation: vertical speed of 20.78 knots
new course is (-148 , 20.78) = inverse tan ( 20.78 / -148) = 172° or 8° N of East at 149.45 knots
New speed (- 148)² + ( 20.28)² = 149.45 knots round and you get 181.2
question answered by
(jacemorris04)
The ground speed of the airplane is 181.18 knots (approx).
What is the formula to find ground speed of airplane ?
The formula of ground speed of a airplane is given below,
[tex]v_{g} =\sqrt{}[/tex][tex](v_{a} ^{2}+v_{w} ^{2}-2v_{a} v_{w}cos\alpha)[/tex]
where, [tex]v_{g}[/tex] = Ground speed of airplane
[tex]v_{a}[/tex] = Speed of the airplane relative to the air
[tex]v_{w}[/tex] = Wind speed
[tex]\alpha[/tex] = Internal angle
How to find the ground speed of given airplane ?Given, [tex]v_{a}[/tex] = 160 knots, [tex]v_{w}[/tex] = 24 knots
Here, the angles increase clockwise and east is 90° heading.
The airplane is flying a heading of 90° at 160 knots speed & wind is blowing from 240° at 24 knots speed.
So, the internal angle[tex](\alpha)[/tex] = 240°-90° = 150°
∴ Ground speed, [tex]v_{g} =\sqrt{}[/tex][tex](v_{a} ^{2}+v_{w} ^{2}-2v_{a} v_{w}cos\alpha)[/tex]
⇒ [tex]v_{g}[/tex] [tex]=\sqrt{}[/tex][tex](160^{2}+24^{2}-2*160*24*cos(150))[/tex]
⇒ [tex]v_{g}[/tex] = 181.18 knots (approx)
Hence the ground speed of airplane is 181,18 knots (approx).
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Brad puts an equal amount of money in his savings account once a month. He started with $27. The next month, he had $44 in his account. Two months after that, he had $78 in his account. How much money will Brad have in his account after 6 months?
Answers
Brad deposits $17 per month. Hence, after 6 months, Brad will have $115 in his account.
Explanation:The subject of the question is a simple arithmetic problem involving Brad's savings account. The first month, he started with $27. The next month, he had $44 in his savings account, so he adds ($44-$27) = $17 to his account each month. In the third and fourth months, he added the same amount twice, so he had $78 ($44 + $17*2). Hence, after 6 months, he will have $115 in his savings account because for the fifth and sixth months he will add ($17*2 = $34) to his account of $78 from the fourth month.
In simpler terms, we can break down these steps into the following:
Determine the monthly deposit amount ($44-$27)= $17.Add the monthly deposit to the current amount for each subsequent month.Calculate for the 6th month: $27 + $17 * 5 = $115.Learn more about Arithmetic progression here:
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Given the fact that Brad saves an equal amount of $17 each month, after 6 months, his total saved amount will be $129.
Explanation:The question is about calculating the amount of money Brad will have saved in 6 months, given that he saves an equal amount each month. From the information provided, we can determine that Brad saves $17 each month ($44 - $27 = $17). After two months, we can see that $78 - $44 = $34, which confirms that Brad is indeed saving $17 each month, since $34 / 2 = $17. Now, to find how much Brad will have saved after 6 months, we need to multiply the monthly saving amount by 6 and add that to the initial amount. This gives us: $27 + ($17 * 6) = $27 + $102 = $129.
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A baseball team loses twice as many games as it wins.The baseball season is 162 games long.How many games will the team loses?
Answers
Answer:
11
Step-by-step explanation:
I guessed
Dentify the constant of proportionality in the equation. 4y = 48x A) 1 B) 4 C) 12 D) 48
Answers
4y=48x
______
4 4
Y=12x
Hope this helps
The constant of proportionality in the equation 4y = 48x is 12 (option C) after dividing both sides of the equation by 4 to get the equation into the form y = kx, with k representing the constant of proportionality.
Explanation:The question is asking for the constant of proportionality in the equation 4y = 48x. In an equation of the form "y = kx", "k" is the constant of proportionality. Here, to make this equation look like "y = kx", we need to divide both sides of the equation by 4. Doing this gives us:
y = 12x
So, the constant of proportionality in this equation is 12 which corresponds to option C.
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Determine which of the following graphs represent the equation below. Also determine the y-intercept of the graph.
Answers
Answer:
Option (a)
Step-by-step explanation:
The given equation is
y = 4 (2ˣ)
To find y-intercept, substitute x=0 in the equation.
y = 4 (2⁰) = 4 (1) =4
Hence y-intercept is 4.
Since y-intercept is 4, the equation should pass through the point (0,4).
The only graph passing through the point (0,4) is the graph in option (a).
Hence the answer is option (a).
Answer:
a
Step-by-step explanation:
Suppose logbx = logcx, where b ≠ c. Then the value of x can be
A) 0 only
B) 1 only
C) b^c or c^b only
D) any positive real number
Answers
Not A: We can't have [tex]x=0[/tex] because [tex]\log0[/tex] is undefined for a logarithm of any base.
B is true: [tex]\log1=0[/tex] for any base.
Not C: If [tex]x=b^c[/tex], then [tex]\log_bb^c=c[/tex], but [tex]\log_cb^c=c\log_bc[/tex] which only reduces to [tex]c[/tex] if [tex]\log_bc=1[/tex]. This can only happen if [tex]b=c[/tex], however, but we've assumed otherwise.
Not D: The reasoning for C not being correct is enough to rule out this possibility.
Which of these figures represents a line segment?
Answers
A police officer investigating an accident finds a skid mark 115 feet long approximately how fast was the car traveling when the driver applied the brakes
Answers
The speed that describes how fast the car was traveling when the driver applied the brakes is 49 mph.
What was the speed?
The formula that is often used by police officers to determine the speed from a skid mark is this s = √21d.
So, in this case, the length or distance of the skid mark is 115 feet. We can substitute this value in the formula to get;
Speed of car = √21(115)
Speed of car = √2,415
Speed of car = 49.1426 approximately 49 mph.
So, the speed that describes how fast the car was traveling when the driver applied the brakes is 49 mph.
Complete Question:
A police officer investigating an accident finds a skid mark 115 feet long. Approximately how fast was the car traveling when the driver applied the brakes?
which type of correlation is suggested by the scatter plot
Answers
Answer:
Negative Correlation
Step-by-step explanation:
As you can see in the image I added, a negative correlation happens when the variables move in opposite directions, in other words, one variable increases as the other decreases. We can see this correlation represented by a value of -1 slope.
I hope you find this information useful and interesting! Good luck!
Answer: That's negative.
Step-by-step explanation: hope this helps.
Number 19 only! 100 points!
Rewrite each expression in terms of sine and cosine, and simplify as much as possible.
Answers
Answer:
The answer is sin^2 (theta) cos (theta) .
Step-by-step explanation:
We need to rewrite tan (theta) as sin (theta) /cos (theta)
tan^2 (theta) = sin^2 (theta)/ cos^2 (theta)
We also need to rewrite sec (theta) as 1/ cos(theta)
sec^3 = 1 / cos^3( theta)
tan^2 (theta) sin^2 (theta)
------------------ = ----------------
sec ^3 (theta) cos ^ 2(theta)
---------------------------
1
-----------
cos ^3 (theta)
Remember when dividing fractions, we can use, copy dot flip
sin^2 (theta) cos^3 (theta)
------------------ * -------------------
cos ^ 2(theta) 1
Canceling a cos^2(theta) from the top and bottom,
sin^2 (theta) cos (theta)
------------------ * -------------------
1 1
we are left with
sin^2 (theta) cos (theta)
The answer is sin^2 (theta) cos (theta) .
The answer is sin^2