What is the solution to the equation below? Log6 4x^2-log6x=2
Combine the left hand side, then write both sides as powers of 6:
[tex]\log_64x^2-\log_6x=\log_6\dfrac{4x^2}x=2\implies6^{\log_6\frac{4x^2}x}=6^2[/tex]
[tex]\implies\dfrac{4x^2}x=36[/tex]
[tex]\implies x^2=9x[/tex]
[tex]\implies x^2-9x=x(x-9)=0[/tex]
[tex]\implies x=0,x=9[/tex]
However, for any base [tex]b[/tex], [tex]\log_bx[/tex] is undefined if [tex]x=0[/tex], so the only solution is [tex]x=9[/tex].
Answer:
D. x=9
Step-by-step explanation:
Given: ∆ABC, AB = CB BD − median to AC E∈ AB ,F∈ BC AE = CF Prove: △ADE ≅ △CDF ΔBDE ≅ ΔBDF
Answer:
1) By SAS theorem, ΔADE≅ΔCDF
2) By SSS theorem, ΔBDE≅ΔBDF
Step-by-step explanation:
Consider isosceles triangle ABC (see diagram).
1. In triangles ADE and CDF:
AD≅DC (since BD is median, then it divides side AC in two congruent parts);AE≅CF (given);∠A≅∠C (triangle ABC is isosceles, then angles adjacent to the base are congruent).By SAS theorem, ΔADE≅ΔCDF.
2. In triangles BDE and BDF:
side BD is common;DE≅DF (ΔADE≅ΔCDF, then congruent triangles have congruent corresponding sides);BE≅FB (triangle ABC is isosceles, AB≅BC, AE≅CF, then BE=AB-AE, FB=BC-CF).Be SSS theorem, ΔBDE≅ΔBDF.
The point (0, 5) lies on circle A with the center at the origin. Does the point (0, −5 ) lie on the circle? A. Yes, because both points are equidistant from the center of the circle. B. Yes, because the distance between the two points is half the distance from the center to one of the points. C. No, because both points are not equidistant from the center of the circle. D. No, because the distance between the two points is twice the distance from the center to one of the points.
A. Yes, because both points are equidistant from the center of the circle.
Step-by-step explanation:The point (0, 5) is 5 units from the point (0, 0).
The point (0, -5) is 5 units from the point (0, 0).
The given points are equidistant from the circle center at (0, 0), so both will lie on the circle—along with any other points that are 5 units from (0, 0).
why is 3 * 1/10 less than 3*10
Multiplying 3 by a smaller number gives a smaller product.
1/10 is smaller than 10, so 3·(1/10) is smaller than 3·10.
i will give the brainliest thank you
Answer:
option 2
Step-by-step explanation:
in the given triangle,
tan45° = 8/x
=> 1 = 8/x
=>x = 8
for y,
cos45° = 8/y
=>1/√2 = 8/y
=>y = 8√2
Answer:
x=8, [tex]y=8 \sqrt2[/tex]
Step-by-step explanation:
Consider the given right triangle,
As, [tex]\tan \Theta = \frac{Perpendicular}{Base}[/tex]
Consider [tex]\tan 45^{\circ}=\frac{x}{8}[/tex]
[tex]1=\frac{x}{8}[/tex]
So, x=8
As, [tex]\sin \Theta = \frac{Perpendicular}{Hypotenuse}[/tex]
Consider [tex]\sin 45^{\circ}=\frac{8}{y}[/tex]
[tex]\frac{1}{\sqrt2}=\frac{8}{y}[/tex]
So, [tex]y=8 \sqrt2[/tex]
Therefore, x = 8 and [tex]y=8 \sqrt2[/tex]
Max is installing a 12-ft swimming pool slide at a 50° angle of elevation. The bottom of the slide will be 1 foot off the ground and the top of the slide will be fixed to a platform. Find the height of the platform. (round to nearest tenth) A) 9.2 ft B) 10.0 ft C) 10.1 ft D) 10.2 ft
D) 10.2 ft
Step-by-step explanation:The mnemonic SOH CAH TOA reminds you that
... Sin = Opposite/Hypotenuse
You want to find the length of the side Opposite the 50° angle, given the Hypotenuse is 12 ft. Then the relation is ...
... sin(50°) = height/(12 ft)
Multiplying by 12 ft gives
... height = (12 ft)·sin(50°) ≈ 9.2 ft
The height of the platform is 1 ft up from this value:
... 9.2 ft + 1 ft = 10.2 ft
Ramona went to a theme park during spring break. She was there for 7 hours and rode 14 rides. At what rate did Ramona ride rides in rides per hour?
Final answer:
Ramona rode rides at a rate of 2 rides per hour during her visit to the theme park.
Explanation:
Ramona's Rate of Riding Rides:
Rate = Number of rides ÷ Number of hours
Rate = 14 rides ÷ 7 hours
Rate = 2 rides per hour
Suppose that a six-sided die is "loaded" so that any particular even-numbered face is three times as likely to be observed as any particular odd-numbered face. (a) what are the probabilities of the six simple events? (hint: denote these events by o1,..., o6. then p(o1) = p, p(o2) = 3p, p(o3) = p,..., p(o6)= 3p. now use a condition on the sum of these probabilities to determine p. answer as an exact fraction or round your answers to three decimal places.)
Answer:
p = 1/12 = p(o1) = p(o3) = p(o5)
3p = 1/4 = p(o2) = p(o4) = p(o6)
Step-by-step explanation:
p(o1) +p(o2) +... +p(o6) = 1 . . . . the condition on the sum of probabilities
... p +3p +p +3p +p +3p = 1 . . . . substitute values
... 12p = 1 . . . . simplify
... p = 1/12 . . . . divide by 12
Then ...
... 3p = 3/12 = 1/4
Final answer:
The probabilities of rolling each face of a loaded six-sided die, where even numbers are three times more likely than odd numbers, are 1/9 for odd numbers (1, 3, 5) and 1/3 for even numbers (2, 4, 6), satisfying the condition that the total probability sums to 1.
Explanation:
The student is dealing with a probability question involving a "loaded" die. To find the probabilities for each face of the die, we assign a variable p to represent the probability of rolling an odd-numbered face and 3p for an even-numbered face. The sum of all probabilities must equal 1 because one of the outcomes must occur when the die is rolled. We can set up the following equation: p + 3p + p + 3p + p + 3p = 1.
Combining like terms gives us 9p = 1, so p = 1/9. Thus, the probabilities of rolling each face are as follows:
P(Odd) = 1/9 for faces 1, 3, and 5P(Even) = 3/9 or 1/3 for faces 2, 4, and 6This system satisfies the condition that all probabilities sum to 1, making it the correct distribution for this loaded die.
cos(−θ)=√3/4 , sinθ<0
What is the value of sinθ ?
Answer:
as written, sin(θ) = -√13/4perhaps, sin(θ) = -1/2Two answers are given because the question is "unexpected." cos(θ) = √(3/4) is more commonly seen in such problems than is cos(θ) = (√3)/4, which is what you have written here. Choose the answer that matches your intent.
Step-by-step explanation:
The cosine function is an even function, so cos(θ) = cos(-θ).
The relationship between sin(θ) and cos(θ) is ...
... sin(θ) = ±√(1 -cos(θ)^2)
For sin(θ) < 0 and cos(θ) = (√3)/4, ...
... sin(θ) = -√(1 -3/16) = -√(13/16)
... sin(θ) = -(√13)/4
For sin(θ) < 0 and cos(0) = √(3/4), ...
... sin(θ) = -√(1 -3/4) = -√(1/4)
... sin(θ) = -1/2
Answer:
-√13/4 took the test.
−10x+3y=5 ASAP
x=y−4
x ?
y?
Answer:
x = 1 and y = 5
Step-by-step explanation:
Use substitution because you know that x = y - 4, and plug this into the first equation to get -10(y - 4) + 3y = 5, or -10y + 40 + 3y = 5. This is -7y = -35 so y = 5. Plug this into the 2nd equation to get that x = 1 and y = 5.
Given: ∆AFD, m ∠F = 90°
AD = 14, m ∠D = 30°
Find: Area of ∆AFD
I need an answer asap thanks
We have the triangle 30° - 60° - 90°.
The sides are in proportion: 2 : √3 : 1
(look at the picture).
[tex]2a=14[/tex] divide both sides by 2
[tex]a=7[/tex]
[tex]a\sqrt3=7\sqrt3[/tex]
The area of a trinagle AFD:
[tex]A_{\Delta}=\dfrac{1}{2}(7)(7\sqrt3)=\boxed{\dfrac{49\sqrt3}{2}}[/tex]
The area of a triangle ADF is 42.46 square units.
What are trigonometric ratios?The sides and angles of a right-angled triangle are dealt with in Trigonometry. The ratios of acute angles are called trigonometric ratios of angles. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
Given that, in ∆ADF, m∠F=90°, AD = 14 and m∠D=30°
We know that, sinθ= Perpendicular/Hypotenuse
Now, sin A=FD/AD
sin60°=FD/14
⇒ √3/2 = FD/14
⇒ FD=7√3
sin30°=FA/AD
⇒ 1/2 =FA/14
⇒ FA=7
We know that, area of a triangle is 1/2 ×Base×Height
Area of ∆ADF
= 1/2 ×FD×FA
= 1/2×7√3×7
= 42.46 square units
Therefore, the area of a triangle ADF is 42.46 square units.
Learn more about the trigonometric ratios here:
brainly.com/question/25122825.
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At Frank's work he can produce 2 widgets every hour. Which of the following expresses how many widgets Frank can make during an 8 hour day?
Answer:
8x2 or (8)(2)
Step-by-step explanation:
The value of 1 hour is 2 widgets, in 8 hours simplified would lead to 8x2 which is 16 widgets.
This leaves you with 16 widgets made in 8 hours.
Frank can make 16 widgets by working 8 hours a day.
What is the concept of Unitary method ?The unitary method is a technique for solving a problem by first finding the value of a single unit, and then finding the necessary value by multiplying the single unit value.
Solving the given problem using the Unitary method -Given that at Frank's work he can produce 2 widgets every hour.
Thus if he produce 2 widgets in one hour, using Unitary method
In 1 hour Frank can make 2 widgets.
Thus in 8 hour work, Frank can make (2 * 8) widgets.
In 8 hour work, Frank can make 16 widgets.
To learn more about Unitary method, refer -
https://brainly.com/question/26548578
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“Camilla makes and sells jewelry. She has 8,160 silver beads and 2,880 black beads to make necklaces. Each necklace will contain 85 silver beads and 30 black beads. How many necklaces can she make?” ^^ please help and show work, due in 10 minutes
96 necklaces
Step-by-step explanation:To find out how many necklaces Camilla can make, we need to see how many times the required number of beads "goes into" the available number of beads.
Silver: 8160/85 = 96
Black: 2880/30 = 96
Camilla has enough beads to make 96 necklaces.
_____
Comment on this answer
Obviously, if one of the quotients is smaller than the other, Camilla can only make as many necklaces as are supported by the constraining resource.
Even if she had 3000 black beads (way more than 2880), she could still only make 96 necklaces (for example) because she would run out of silver beads making that 96th necklace. (There would be 120 black beads left over in that scenario.)
Jeff answered all 25 questions on his chemistry test. For each right answer, he got 4 points and for each wrong answer he lost 2 points. If he got a score of 70 points, how many questions did he get right?
Answer:
Jeff got 20 answers right.
Step-by-step explanation:
The maximum number of points Jeff can get is 100. (25 * 4) However, for each question he gets wrong, he's actually losing 6 points because he doesn't get the four points for getting right, and has the additional -2 points for getting it wrong. Since he only got a score of 70, that means he lost 30 points.
30/6 = 5. So he missed 5 question and got 20 question right.
As per the given values, Jeff got 20 questions right.
Explanation:Total questions answered = 25
Total points for each correct answer = 4
Total points for each incorrect answer = 2
To find the number of questions Jeff got right, we can set up the equation:
4x - 2(25 - x) = 70,
where x represents the number of questions he got right.
Simplifying the equation, we get -
6x - 50 = 70.
Adding 50 to both sides, we have 6x = 120.
Dividing both sides by 6, we get x = 20.
Therefore, Jeff got 20 questions right.
Learn more about Finding the number of questions answered correctly here:https://brainly.com/question/40214390
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Find the midpoint between (-1+9i) and B=(5-3i)
The midpoint between (-1+9i) and B=(5-3i) is (2+3i).
Explanation:The midpoint between (-1+9i) and B=(5-3i) can be found by taking the average of the real and imaginary parts separately.
For the real part, we take the average of -1 and 5 which is 2. For the imaginary part, we take the average of 9i and -3i which is 3i.
Therefore, the midpoint between (-1+9i) and B=(5-3i) is (2+3i).
Aaron want to buy sod for his backyard. What is the amount of area he will need to cover?
Measurements in picture
(I know the square's area but not the triangle's)
Answer:
(16 + 8√3) in²
Step-by-step explanation:
The ratio of side dimensions for a 30°-60°-90° triangle are 1 : √3 : 2. If we call the horizontal dimension of the triangle its base, then its height (vertical dimension) will be √3×4 in. Of course the area of the triangle is ...
... A = (1/2)bh = (1/2)(4 in)(4√3 in) = 8√3 in²
The total sod area is the sum of the square area (16 in²) and the triangle area, so is ...
... area to cover = (16 +8√3) in²
_____
Comment on problem dimensions
The area involved here is not much larger than the size of your hand. It would make more sense for the dimensions to be in feet or yards or meters, rather than inches.
Answer: 29.9 in.
Step-by-step explanation:
16+(8*√3)
The mean of a set of data is 4.68 and its standard deviation is 2.83. find the z score of the value
Answer:
z=-0.35 ( Depending on value given)
Step-by-step explanation:
Let the value be x. For calculation purposes, let us take x as 3.68. ( as mean is 4.68). We may assume a higher value as well. Depends on the given value.
The formula for z score is given as-
z=(value-mean)/Standard Deviation
z=(3.68-4.68)/2.83
z=-0.35
Which set of points matches the line above?
A.
(0,9) (6,8)
B.
(8,6) (3,11)
C.
(0,9) (11,3)
D.
(9,0) (3,11)
Answer:
C
Step-by-step explanation:
When x = 0, y = 9
When x = 11, y = 3
Find 11 on the horizontal line and go up to the crossing and the follow the line till the y-axis(going to the left direction(. Did you see the 3?
That is the (11 | 3) point
The sum of two numbers is 67 and the difference is 13 . What are the numbers?
Answer:
27 and 40
Step-by-step explanation:
Final answer:
The sum of two numbers is 67 and the difference is 13. The two numbers are 40 and 27.
Explanation:
Let's solve this problem step by step:
Let's call one number x and the other number y.We know that x + y = 67 and x - y = 13.To find the numbers, we can solve this system of equations.Adding the two equations together, we get 2x = 80.Dividing both sides by 2, we find that x = 40.Substituting x = 40 into either equation, we find that y = 27.Therefore, the two numbers are 40 and 27.
an investment double in 10 years what was its exponential growth rate?
Answer:
So, our rate of exponential growth is is 6.93 %
Step-by-step explanation:
The formula for exponential growth rate is
A = P[tex]e^{rt}[/tex] .... (1
Here
P = is the initial investment
A = amount after certain time
As we are given after 10 years (t = 10) the investment doubles
so plugging A=2P and t= 10 into equation (1)
2P = P[tex]e^{10r}[/tex]
cancelling P on both sides
2 = [tex]e^{10r}[/tex]
taking natural log on both sides
ln (2) = ln ([tex]e^{10r}[/tex]
ln(2) = 10r(lne)
as ln(e) = 1
ln (2) = 10r
r = [tex]\frac{ln(2)}{10} [/tex]
solving the natural log
r = 0.0693
or
r =6.93 %
So, our rate is 6.93 %
What is the effect on the graph of the function f(x) = x when f(x) is replaced with -1/2 f(x)?
A) vertical reflection over x-axis and vertical stretch
B) vertical reflection over x-axis and vertical compression
C) horizontal reflection over y-axis and horizontal stretch
D) horizontal reflection over y-axis and horizontal compression
Answer:
C) horizontal reflection over y-axis and horizontal stretch
Step-by-step explanation:
1a- A vertical reflection over the x-axis occurs when a function [tex]f(x)[/tex] is transformed into [tex]f(-x)[/tex]
1b- A horizontal reflection over the y-axis occurs when a function [tex]f(x)[/tex] is transformed into [tex]-f(x)[/tex]
2a- A function is being compressed if [tex]f(x)[/tex] is multiplied by a positive factor k: [tex]k f(x)[/tex] with [tex]k>1[/tex]
2b- A function is being stretched if [tex]f(x)[/tex] is multiplied by a positive factor k: [tex]k f(x)[/tex] with [tex]k<1[/tex]
In our problem, the original function [tex]f(x)[/tex] is:
- Multiplied by 1/2, so by a factor which is smaller than 1, so we are in case 2b
- Transformed from [tex]f(x)[/tex] into [tex]-f(x)[/tex] (due to the negative sign in front of it), so we are in case 1b
So, overall, we had a horizontal reflection over the y-axis and a stretch of the function.
26.4w = 285.12. What is the width (w) of this rectangle? 10.8 units 11.0 units 258.7 units 7527.2 units
Answer:
10.8 units
Step-by-step explanation:
As the equation represents the area of a rectangle
i.e.
length * width = Area
Now area is 285.12 units square
while length is 26.4
and width is represented by w
now putting in the formula it will become
26.4 * w = 285.12
dividing both sides by 26.4
[tex]\frac{26.4w}{26.4}=\frac{285.12}{26.4}[/tex]
this will become
w = 10.8 units
Which is the required width of the Rectangle
so answer is 10.8 units
Which values from the specified set make up the solution set of the inequality?
4n<16 ; {1,2,3,4}
Select ALL OF THE correct answers.
A. 1
B. 2
C. 3
D. 4
You may solve this problem in two ways:
If you solve the inequality explicitly (divide both sides by 4), you get
[tex] \dfrac{4n}{4} < \dfrac{16}{4} \iff n < 4 [/tex]
So, if [tex] n [/tex] has to be stricktly less than 4, you can only choose 1, 2 and 3 as answers.
Alternatively, you can plug in all of the values you're proposed and check if the inequality holds:
If [tex] n=1 [/tex], you have [tex] 4<16 [/tex], which is true.
If [tex] n=2 [/tex], you have [tex] 8<16 [/tex], which is true.
If [tex] n=3 [/tex], you have [tex] 12<16 [/tex], which is true.
If [tex] n=4 [/tex], you have [tex] 16<16 [/tex], which is false.
So, again, only 1, 2 and 3 are solutions.
Can someone help me out with this? I’m not sure what I’m doing wrong.
Which graph best represents the solution to this system of inequalities?
2x + 3y (> with line underneath) 2
3x - 4y (< with like underneath) 3
Answer:
Step-by-step explanation:
This is a system of inequalities such that:
[tex]\left \{ {{2x+3y\geq 2} \atop {3x-4y\leq 3}} \right.[/tex]
Let's start by solving for y for both equations:
Equation 1:
[tex]2x+3y\geq 2\\\\3y\geq -2x+2\\\\y \geq \frac{-2x+2}{3}[/tex]
Equation 2:
[tex]3x-4y\leq 3\\\\-4y\leq -3x+3\\\\y\geq -\frac{(-3x+3)}{4}[/tex]
Now if we substitute the 2nd y into the first equation we obtain:
[tex]2x+3(\frac{3x-3}{4}) \geq 2\\\\2x+\frac{9x-9}{4}\geq 2\\\\\frac{8x+9x-9}{4}\geq 2\\\\17x-9\geq 8\\\\17x\geq 17\\\\x\geq 1[/tex]
Now we will solve for the second equation using the first result of y and we obtain:
[tex]3x-4(\frac{-2x+2}{3}\leq 3\\\\3x+\frac{8x-8}{3}\leq 3\\\\\frac{9x+8x-8}{3}\leq 3\\\\17x-8\leq 9\\\\17x\leq 17\\\\x\leq 1[/tex]
And so our solution for the system of equations is:
[tex]x \leq 1\\and \\y\geq \frac{-2x+2}{3}[/tex]
As well as:
[tex]x>1\\and\\y\geq \frac{3x-3}{4}[/tex]
Can you help me graph this equation?
A graph is attached
Step-by-step explanation:The equation is in slope-intercept form:
... y = mx + b
Here, the value of m (the slope) is 1/2, and the value of b (the y-intercept) is -3.
This means that point (0, -3) is a point on the graph. This point is on the y-axis, 3 units below the x-axis.
Slope is sometimes referred to as "rise over run." The slope of 1/2 means the line will rise 1 unit for each 2 units to the right. That is, for some point (x, y) on the ine, another point will be found at (x+2, y+1).
Given a point at (0, -3), another point will be (0, -3) +(2, 1) = (2, -2), and another point will be (2, -2) +(2, 1) = (4, -1).
You can plot as many points as you like, then draw the line through them.
A computer and printer cost a total of $1060 . The cost of the computer is three times the cost of the printer. Find the cost of each item.
Answer:
Printer cost 265
Computer cost 795
Step-by-step explanation:
1060 / 4 = 265
he cost of the printer is $265, and the cost of the computer is $795.
The question involves solving a system of equations to find the cost of two items based on their total cost and the ratio of their costs. To find the cost of a computer and a printer, we can set up the equations based on the given information: the total cost is $1060, and the cost of the computer is three times the cost of the printer.
Let's define C as the cost of the computer and P as the cost of the printer. According to the problem, we have two equations: C + P = $1060 (total cost) and C = 3P (cost relationship). We can substitute the second equation into the first to find P: 3P + P = $1060, so 4P = $1060. Dividing both sides by 4, we get P = $265. Using this value, we can find C by multiplying P by 3, giving us C = 3 * $265 = $795.
Thus, the cost of the printer is $265, and the cost of the computer is $795.
The least absolute deviation line equation for the data in the table is m = 0.1x + 2.9
What is the sum of the absolute deviations?
Answer:
24.5
Step-by-step explanation:
A calculator or spreadsheet is good for doing this sort of computation.
_____
You add up the magnitudes of the differences between the given y-value and the corresponding value you get from the equation. For the first couple of points, these values are ...
... |3 - (0.1·1 +2.9)| = 0
... |2.5 -(0.1·6 +2.9)| = |2.5 -3.5| = 1
The computation proceeds like this for the remaining 6 points, and the numbers added. The result is 24.5.
Danny gets paid $8 per hour. What is the rate of change?
A) m=4
B) m= 1/2
C) m=8
Pls I need help with this one. Find the value of y in the equation
Answer:
y = 2 3/8
Step-by-step explanation:
Multiply by the denominator. This gives a linear equation that can be solved in the usual way.
... 3 = 8(y -2)
Divide by 8.
... 3/8 = y - 2
Add 2
... 2 3/8 = y
_____
In problems involving rational expressions, it often works well to multiply by the product of the denominators, or by their least common multiple, if that is easy to find. Doing this eliminates fractions. Here there is only one denominator (y-2), so we multiply by that.
The sum of two numbers is 60 . The smaller number is 12 less than the larger number. What are the numbers?
Answer:
36 and 24
Step-by-step explanation
36 - 24 = 12
36 + 24 = 60