Answer:
[tex]x = \displaystyle \frac{5 + \sqrt{17}}{2}[/tex].
Step-by-step explanation:
Because [tex]3\, x[/tex] is found in the input to a logarithm function in the original equation, it must be true that [tex]3\, x > 0[/tex]. Therefore, [tex]x > 0[/tex].
Similarly, because [tex](x - 4)[/tex] and [tex](2\, x - 1)[/tex] are two other inputs to the logarithm function in the original equation, they should also be positive. Therefore, [tex]x > 4[/tex].
Let [tex]a[/tex] and [tex]b[/tex] represent two positive numbers (that is: [tex]a > 0[/tex] and [tex]b > 0[/tex].) The following are two properties of logarithm:
[tex]\displaystyle \ln (a) + \ln(b) = \ln\left(a \cdot b\right)[/tex].
[tex]\displaystyle \ln (a) - \ln(b) = \ln\left(\frac{a}{b}\right)[/tex].
Apply these two properties to rewrite the original equation.
Left-hand side of this equation:
[tex]\begin{aligned}&\ln(3\, x) - \ln(x - 4)= \ln\left(\frac{3\, x}{x -4}\right)\end{aligned}[/tex]
Right-hand side of this equation:
[tex]\ln(2\, x- 1) + \ln(3) = \ln\left(3 \left(2\, x - 1\right)\right)[/tex].
Equate these two expressions:
[tex]\begin{aligned}\ln\left(\frac{3\, x}{x -4}\right) = \ln(3(2\, x - 1))\end{aligned}[/tex].
The natural logarithm function [tex]\ln[/tex] is one-to-one for all positive inputs. Therefore, for the equality [tex]\begin{aligned}\ln\left(\frac{3\, x}{x -4}\right) = \ln(3(2\, x - 1))\end{aligned}[/tex] to hold, the two inputs to the logarithm function have to be equal and positive. That is:
[tex]\displaystyle \frac{3\ x}{x - 4} = 3\, (2\, x - 1)[/tex].
Simplify and solve this equation for [tex]x[/tex]:
[tex]x^2 - 5\, x + 2 = 0[/tex].
There are two real (but not rational) solutions to this quadratic equation: [tex]\displaystyle \frac{5 + \sqrt{17}}{2}[/tex] and [tex]\displaystyle \frac{5 - \sqrt{17}}{2}[/tex].
However, the second solution, [tex]\displaystyle \frac{5 - \sqrt{17}}{2}[/tex], is not suitable. The reason is that if [tex]x = \displaystyle \frac{5 - \sqrt{17}}{2}[/tex], then [tex](x - 4)[/tex], one of the inputs to the logarithm function in the original equation, would be smaller than zero. That is not acceptable because the inputs to logarithm functions should be greater than zero.
The only solution that satisfies the requirements would be [tex]\displaystyle \frac{5 + \sqrt{17}}{2}[/tex].
Therefore, [tex]x = \displaystyle \frac{5 + \sqrt{17}}{2}[/tex].
Read the information below and answer the questions. Suppose a researcher wants to learn more about the mean attention span of individuals in some hypothetical population. The researcher cites that the attention span (the time in minutes attending to some task) in this population is normally distributed with the following characteristics: 20 36 . Based on the parameters given in this example, answer the following questions: 1. What is the population mean (μ)?
Answer:
The population mean (μ) = 26.
Step-by-step explanation:
a) Data and Calculations:
The population characteristics = 20 and 36
Total population of interest = 56 (20 + 36)
Number of groups in the population = 2
Therefore, the population mean (μ) = 56/2 = 28
b) Based on the parameters given in this example, the population mean equals the average of the group characteristic (attention span of individuals in minutes) or item of interest. The population mean is calculated by adding the values of the group characteristic and then dividing the result by the number of values.
PLEASE HELP!! I NEED THIS ANSWER TO GET A GOOD GRADE!!!!!
Answer:
-5-4
Step-by-step explanation:
Let me know if I am wrong, if I am I apologize.
Have a good one.
Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.
Answer:
Following are the solution to this question:
Step-by-step explanation:
please find the graph image in the attached file.
Given value:
[tex]AB= c= 1744 \ miles\\\\BC= a= 2451 \ miles\\\\AB= s= 714 \ miles[/tex]
using the law of cosines:
[tex]\cos A = \frac{s^2+c^2-a^2}{2bc}\\\\[/tex]
[tex]=\frac{714^2+1744^2-2451^2}{2 \times 714 \times 1744}\\\\= -0.9862[/tex]
[tex]A=cos^{-1} (-0.9862)\\\\A= 170.47^{\circ}[/tex]
[tex]\cos B = \frac{a^2+c^2-b^2}{2ac}\\\\=\frac{2451^2+1744^2-714^2}{2 \times 2451 \times 1744}\\\\= 0.9988\\\\B=cos^{-1} (0.9988)\\\\B= 2.7641^{\circ}\\\\[/tex]
[tex]\cos C = \frac{a^2+c^2-b^2}{2ac}\\\\=\frac{2451^2+1744^2-714^2}{2 \times 2451 \times 1744}\\\\= 0.9988\\\\C=cos^{-1} (0.9988)\\\\C= 2.7641^{\circ}[/tex]
(5x²-7)(5x²+7)
Help me
Answer:
[tex]25 {x}^{4 } - 49[/tex]
Please help on this question
Answer: bagles: 200 muffins: 164
Step-by-step explanation:
This list shows the lengths in feet of the 25 longest bridges in the United States.
1010
1470
1600
2000
2800
1053
1495
1632
2150
3500
1200
1596
1750
2150
3800
1207
1600
1800
2300
4200
1380
1600
1850
2310
4260
Jamie made a frequency table of the bridge data.
U.S. Bridges
Length (ft)
Tally
Frequency
(first interval)
|||| ||
7
(last interval)
||
2
Based on the tallies and frequencies for the first and last intervals, how many intervals of what length did Jamie use?
a.
4 intervals of 1000 feet
c.
6 intervals of 500 feet
b.
5 intervals of 1000 feet
d.
7 intervals of 500 feet
Answer:
D. 7 intervals of 500 feet
Step-by-step explanation:
Got it right
Jamie used 5 intervals of 1000 feet as per the given data. The correct option is b.
What are tallies and frequencies?A tally chart is a straightforward method of recording and counting frequencies. Each occurrence is represented by a tally mark, and every fifth tally is drawn diagonally to form a five-pointed star.
The first interval has a frequency of 7, which corresponds to the first 7 data points in the list:
1010
1470
1600
2000
2800
1053
1495
The last interval has a frequency of 2, which corresponds to the last 2 data points in the list:
2310
4260
Since there are 25 data points in total, and the first 7 and last 2 have been accounted for, that leaves 16 data points for the remaining intervals.
If Jamie used 1000-foot intervals, each interval would cover a 1000-foot range. The remaining data points range from 1200 to 4200 feet, so they are divided into 1000-2000, 2000-3000, and 3000-4000 foot intervals.
If Jamie used 500-foot intervals, each interval would cover a 500-foot range.
Thus, because the remaining data points range from 1200 to 4200 feet, they are divided into 1000-1499, 1500-1999, 2000-2499, 2500-2999, 3000-3499, 3500-3999, and 4000-4499 foot intervals.
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What's this? I don't understand
Answer:
There are two solutions
Step-by-step explanation:
The two equations intersect in two points therefore you willl have two solutions
Answer:
C. two solutions
Step-by-step explanation:
Hope this helped have an amazing day!
The lay in Burlington collected $3 930 in overdue fees last year. This year, the library
collectieted 3144 What is the percent of decrease in the amount collected?
Answer:
We conclude that the percent of decrease = 20%
Step-by-step explanation:
Given
Original amount = $3930Amount of decrease = $3930 - $3144 = $786Using the formula
Percent of decrease = ( amount of decrease / orignal amount × 100 ) %
= ( 786 / 3930 × 100 ) %
= ( 0.2 × 100 ) %
= 20%
Therefore, we conclude that the percent of decrease = 20%
Grade 6 - End of term assessment
Question 7
A seamstress needs to cut 15-inch pieces of ribbon from a roll of ribbon that is
9 feet in length. What is the greatest number of 15-inch pieces the seamstress
can cut from 5 of these rolls of ribbon?
Show your work.
Answer:
The greatest number of 15 inches pieces that can be cut from 5 rolls of length 9 feet is: 35
Step-by-step explanation:
Given
Total length of one roll of ribbon = 9 feet
As the pieces have to be cut into inches, we will convert the measurement in feet to inches
As there are 12 inches in one feet, 9 feet will be equal to:
9*12 = 108 inches
Now first of all, we have to see how many 15 inches pieces can be cut from one role
So,
[tex]=\frac{Length\ of\ roll}{Length\ of\ piece}\\=\frac{108}{15}\\=7.2[/tex]
So the seamstress can cut 7 15-inch long pieces from a roll.
Now given that he has to cut from 5 rolls, the total number of 15-inch pieces will be:
[tex]= 5 * 7 =35[/tex]
Hence,
The greatest number of 15 inches pieces that can be cut from 5 rolls of length 9 feet is: 35
A recipe for rice pudding calls for 2.5 cups of milk. The recipe makes 5 servings. How many cups of milk are needed to make 8 servings?
Step-by-step explanation
2.5 cups = 5servings
x cups = 8servings
2.5*8 = 5*x
20 = 5x
divide both sides by the coefficient of x
20/5 = 5x/5
4 = x
i.e x = 4
Which answer describes the transformation of g(x)=log2(x+3)+1 from the parent function f(x)=log2x ?
Answer:
g(x) is the function f(x) which is shifted 3 units left and 1 unit upStep-by-step explanation:
Given
g(x)=log2(x+3)+1andf(x)=log2xThe transformation is
g(x) = f(x + 3) + 1, which indicates thatg(x) is the function f(x) which is shifted 3 units left and 1 unit upg(x) is the function f(x) which is shifted 3 units left and 1 unit up
If you are given
g(x)=log2(x+3)+1 and f(x)=log2x
Then the transmission is g(x) = f(x + 3) + 1, which indicates that g(x) is the function f(x) which is shifted 3 units left and 1 unit up
ANSWER NOW PLEASE
for a singing contest in which 42,000 votes were cast, the winner received 3/5 of the votes. how many votes did the winner not receive
Given,
Total number of votes casted = 42,000
The winner received 3/5 of the votes.
To find,
How many votes did the winner not receive.
Solution,
Votes received by the winner is :
[tex]\dfrac{3}{5}\times 42000\\\\=25200[/tex]
Votes didn't received by the winner = Total votes - votes received
= 42,000 -25200
= 16800
Hence, 16800 votes is not received by the winner.
The Data set shows the ages of the members of two clubs.
CLUB A: 42, 38, 40, 34, 37, 46, 38, 45
CLUB B: 30, 44, 42, 61, 24, 27, 59, 65 m
(10 Points)
Find the mean age for CLUB A :
Answer:
it goes by +4 and -4
Step-by-step explanation:
Please find the median, mode, range, and the Mean Absolute Deviation for the following
data:
9,-1, 1, 3, 3, 8, 5, 3, -1, 10, 11
Answer:
The median is the first number the mode is middle number and rage is last number
If it represents a number does it always represent 40% of number ?
Answer:
No
Step-by-step explanation:
a circle has a diameter of 5 inches. What is the area of the circle in terms of pi
Answer:
The area of the circle is [tex]\mathbf{6.25\pi}[/tex]
Step-by-step explanation:
Area of a Circle
A circle of radius r has an area calculated by the following formula:
[tex]A=\pi r^2[/tex]
The diameter of a circle is twice the radius, thus the radius is half the value of the diameter.
A given circle has a diameter of 5 inches.
The radius of this circle is r = 5/2 = 2.5 inches.
Thus, the area is calculated below:
[tex]A=\pi 2.5^2 = 6.25\pi[/tex]
The area of the circle is [tex]\mathbf{6.25\pi}[/tex]
An ice cream shop offers 3 ice cream flavors 3 types of syrup and 2 toppings for sundaes
Answer: 18
Step-by-step explanation:
Which math property is used in the expression below?
5(6 - 4) = 5(6) - 5(4)
Identity
Commutative
Associative
Inverse
Distributive
Answer:
Distributive
Step-by-step explanation:
Distributive Property
Which statements are true about the ordered pair (−1, 5) and the system of equations?
{x+y=4x−y=−6
Select ALL correct answers.
The ordered pair (−1, 5) is a solution to the first equation because it makes the first equation true.
The ordered pair (−1, 5) is a solution to the second equation because it makes the second equation true.
The ordered pair (−1, 5) is a solution to the system because it makes both equations true.
The ordered pair (−1, 5) is not a solution to the system because it makes at least one of the equations false.
Answer:
you answer would be A,B,D there all the answer u cant just have one also I just took the test im from k12
Which of the following systems of equations is an example of one where substitution is the best method?
A.)5x+3y=15
-3x+6=12
B.)y=2/7y+11
3x-4y=-24
C.)4x-5y=20
-6x+2y=-42
D.)x=4
Y=11
Answer:
B
Step-by-step explanation:
B is the best example where substitution is the best method since you already have one where y = something and you can plug -2/7x + 11 into the bottom equation for y and solve it that way. A and C make you get x or y by itself first first substitution would work and D already gives you the answers. B is the best since its already set in that format.
(Pls help) A pie crust recipe calls for 2 cups of butter for each 3 cups of flour. Ella is making enough
crusts to use 9 cups of flour. How many cups of butter will she need?
.6 cups of flour Step-by-step explanation:
NEED HELP ASAP. FOR FINALS.
Classify the following statement as true or false.
Every quadratic function either has a maximum or minimum value.
A. False, became a quadratic equation does not have a maximum value.
B. True, because the graph of quadratic function is a curve that is either increasing or decreasing.
C. True, because the graph of a quadratic function is a parabola that either opens downwards or upwards.
D. False, because a quadratic function does not have a minimum value.
Answer:
C
Step-by-step explanation:
If the parabola opens upward, it has a minimum.
If it opens downwards, it has a maximum.
Every quadratic function either has a maximum or minimum value is true, because the graph of a quadratic function is a parabola that either opens downwards or upwards.
What is Quadratic Function?A quadratic function is a polynomial function with one or more variables in which the highest exponent of the variable is two. Since the highest degree term in a quadratic function is of the second degree, therefore it is also called the polynomial of degree 2.
What are properties of quadratic functions?Three properties that are universal to all quadratic functions:
1) The graph of a quadratic function is always a parabola that either opens upward or downward (end behavior).
2) The domain of a quadratic function is all real numbers.
3) The vertex is the lowest point when the parabola opens upwards; while the vertex is the highest point when the parabola opens downwards.
According to the question
Every quadratic function either has a maximum or minimum value.
As per the properties of Quadratic Function
This statement is true as
i> the graph of a quadratic function is a parabola
ii> The vertex is the lowest point when the parabola opens upwards or the vertex is the highest point when the parabola opens downwards.
Hence, Every quadratic function either has a maximum or minimum value is true, because the graph of a quadratic function is a parabola that either opens downwards or upwards.
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POSSIBLE POINTS : 13.33 If a wall has a diagonal beam holding it in place that is 12ft long and the height of the wall is 8ft and the base of the beam on the floor is 7ft from the wall is the wall perpendicular?
Answer:
Step-by-step explanation:
If the wall is perpendicular, then that means that this is a right triangle. We can find out if it is by using Pythagorean's Theorem and filling in the values for the sides. If the statement is true, then the triangle is right and the wall is perpendicular to the base.
[tex]a^2+b^2=c^2[/tex] and filling in:
[tex]7^2+8^2=12^2[/tex] Let's find out if this is true:
[tex]49+64=144[/tex] Is this true?
113 = 144
Definitely not true. So this is not a right triangle and the wall is not perpendicular to the floor.
Grace was given a large box of 54 chocolates for her birthday. If she eats exactly 6
chocolates each day, how many chocolates would Grace have remaining 6 days after
her birthday?
This is an arithmetic sequence question
Evaluate the expression 33 • (12 – 2) : 2.
33 • (12 – 2) = 2 =
?
Tina wants to run 123 miles over
7 days. She wants to run the same
distance each day. How far must she
run each day?
Answer:
17 4/7
Step-by-step explanation:
we'll need to divide 123 by 7
123/7= 17 R 4
But she can't have an extra 4 miles out so,
17 4/7
Find the measure of the angle indicated. I’ll give BRAINLIEST
Answer:It should be -46 if im right dont hate if im worng.
Step-by-step explanation:
Find the dot product of vectors u and v, where |u| = 12, |v| = 7, and the angle between is θ = 55°.
- u · v = 68.81
- u · v = 48.18
- u · v = 10.90
- u · v = 2.87
Answer:
u · v = 48.18
B on edge 2020
Step-by-step explanation:
Answer:
B on edge
Step-by-step explanation:
Instructions
Scientific notation is a tool that is used frequently in science, including astronomy, chemistry, biology, and more. In this activity, imagine that you are a student intern at NASA and are researching information about travel within our solar system. You will complete a series of tasks using what you know about scientific notation and what you discover about our solar system to calculate distances that astronauts may use for their next trip to space.
Part 1
Using technology or other resources, research the average distance that each planet is from the sun (in kilometers). Once you have found the distance, create a chart showing each planet's distance from the sun. The sun itself will not receive a label since it is your starting point. For example, Mercury is located an average of 57,909,000 km from the sun, so on the chart, Mercury would be labeled with a distance of 57,909,000. Use the example of the chart below to help you get started:
Planet Distance from sun (in km) Distance in scientific notation
Mercury 57,909,000
Answer:
We use scientific notation to lessen the load in terms of calculation that must be done. Its often good practice to go to the tenth place.In terms of your question please look at the provided image!Answer:
5.7909 * 10^7
Step-by-step explanation:
Scientific notation uses a number from 1 to under 10 and an integer power of 10.
57,909,000 = 5.7909 * 10,000,000 = 5.7909 * 10^7
the 7s in 7,700 what is the value of the given digits
Answer:
the first 7 has a value of 7,000 while the second 7 has a value of 700