A logarithm is the inverse operation of exponentiation, that is to say, log(a^b) = b, where log is logarithm, a is the base and b is the exponent.
A logarithmic function is called such because it relates the logarithm of a number to the number itself.
In other words, it is a function that expresses the value of the logarithm for a given number. This function is represented by the notation y = logb(x) where y is the logarithm, b is the base, and x is the number.
The logarithmic function is used to solve exponential equations, it is also useful in many fields such as mathematics, physics, engineering, and finance.
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A. To wash the first window, Roger extends the
ladder to 20 feet long. He places the base of
the ladder 6 feet from the side of the house.
Sketch a labeled picture of this to the right.
B. How far up the house will the 20-foot ladder
reach, to the nearest tenth of a foot? Show your
work in the space to the right.
ESSENTIAL QUESTION CHECK-IN
17. Tell how you can determine whether a number is a solution of an
equation.
324 Unit 4
Answer:
Step-by-step explanation:
1. Substitute the number for the variable in the equation.
Which two ratios form a proportion?
Responses
A 12/15 and 5/4
B 15/12 and 5/4
C 6/5 and 4/5
D 5/6 and 4/5
Answer:
Step-by-step explanation: We will determine if ratios are proportional using cross multiplication.
first, we will check option (a)
12/15 =5/4
12*4 = 15*5
48 is not equal to 75. So, these are not proportional.
now, we will check option (b)
15/12 = 5/4
15*4 = 12*5
60 = 60.
as the left-hand side is equal to the right-hand side.
So, option (b) is correct.
Answer:
[tex]\textsf{B)} \quad \dfrac{15}{12}\; \textsf{and}\;\dfrac{5}{4}[/tex]
Step-by-step explanation:
If two fractions are equal, they form a proportion.
Therefore, to determine if two ratios form a proportion, rewrite the fractions so that their denominators are the same, then compare.
[tex]\textsf{A)}\quad \dfrac{12}{15}\; \textsf{and}\;\dfrac{5}{4} \implies \dfrac{12 \times 4}{15\times 4}\; \textsf{and}\;\dfrac{5\times 15}{4\times 15}\implies \dfrac{48}{60}\; \textsf{and}\;\dfrac{75}{60}[/tex]
As the two fractions are not equal, they do not form a proportion.
[tex]\textsf{B)} \quad \dfrac{15}{12}\; \textsf{and}\;\dfrac{5}{4}\implies \dfrac{15}{12}\; \textsf{and}\;\dfrac{5 \times 3}{4 \times 3}\implies \dfrac{15}{12}\; \textsf{and}\;\dfrac{15}{12}[/tex]
As the two fractions are equal, they form a proportion.
[tex]\textsf{C)} \quad \dfrac{6}{5}\; \textsf{and}\;\dfrac{4}{5}[/tex]
The denominators of these two fractions are already the same.
Therefore, as the two fractions are not equal, they do not form a proportion.
[tex]\textsf{D)} \quad \dfrac{5}{6} \; \textsf{and}\;\dfrac{4}{5}\implies \dfrac{5\times 5}{6\times 5} \; \textsf{and}\; \dfrac{4\times 6}{5\times 6}\implies \dfrac{25}{30}\; \textsf{and}\;\dfrac{24}{30}[/tex]
As the two fractions are not equal, they do not form a proportion.
Ali found out the number of rooms in each of 40 houses in a town. He used the information to complete the frequency table. Number of Rooms Frequency 4 4 5. 7 6 10 7 12 8 5 9 2 Calculate the mean number of rooms.
The mean number of rooms = 6
We know that the formula for the mean.
mean = sum of all observations / total number of observations
First we find thr total number of rooms, from given frequency table.
Number of Rooms Frequency Total number of rooms
4 4 16
5 7 35
6 10 60
7 12 84
8 5 40
9 2 18
So, the total number of rooms of 40 houses in a town would be,
16 + 35 + 60 + 84 + 40 + 18 = 253
And the total number of houses = 40
Using above formula of mean, the mean number of rooms would be,
x = 253/40
x = 6.325
Therefore, on an average there are 6 rooms in each house.
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Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix. -2 1 1 - 4 3 4 ; 2 = -1,4 -2 2 1 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. For P=___ D= 0 4 0 0 0 4 -1 0 0 O B. For Pa = ___ D = 0 -1 0 0 04 OC. O The matrix cannot be diagonalized.
The correct answer is option A. For P= -1/2 0 1/2 1/2 0 -1/2 0 0 1 D= 0 4 0 0 0 4 -1 0 0. The matrix can be diagonalized using the eigenvectors.
To diagonalize the matrix, the eigenvectors must be found first. The eigenvectors can be found by solving the characteristic equation and finding the eigenvalues. The eigenvalues of the matrix are 2 and -1. The eigenvectors associated with each eigenvalue are determined by solving the eigenvalue equation.
The eigenvectors for the matrix are [1, -2], [1, 2], [1, 0], and [0, 1]. After the eigenvectors are found, the matrix can be diagonalized by constructing the transformation matrix P. The transformation matrix P is composed of the normalized eigenvectors. The transformation matrix P is: P=[-1/2, 0, 1/2, 1/2, 0, -1/2, 0, 0, 1], and the diagonal matrix D is composed of the eigenvalues.
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Solve this problem
Step by step
Answer:
√259 or 16.09
Step-by-step explanation:
Hello!
I'll be labeling the vertices so that it is easier to understand the sides.
These are both right triangles, and we can find the measures of right triangles (given two other sides) by using the Pythagorean theorem:
[tex]a^2 + b^2 = c^2[/tex]
a = legb = legc = hypotenuse (leg opposite to 90°)In Triangle BCD, we have the measures of the hypotenuse and a leg. We can use that to find the missing side: CB
Since DB is the hypotenuse, it would be side c, and we can say that CD can be side a.
Solve for CB:[tex]a^2 + b^2 = c^2[/tex][tex]10^2 + b^2 = 25^2[/tex][tex]100 + b^2 = 625[/tex][tex]b^2 = 525[/tex][tex]b = 5\sqrt21[/tex]CB in simplest radical form is 5√21.
We can now solve for x, because now in triangle ABC, we have the measures of two sides: hypotenuse AC and leg CB.
Solve for AB:[tex]a^2 + b^2 = c^2[/tex][tex](5\sqrt{21})^2 + b^2 = 28^2[/tex][tex]525 + b^2 = 784[/tex][tex]b^2 = 259[/tex][tex]b = \sqrt{259}[/tex]AB in simplest radical form is √259, which is approximately 16.09.
The solution for x is √259 or 16.09.
6
2/3
15
13
Hit
Miss
2/5
3
5
215
2
Not score
Score
Find the probability that Ryan misses the target and does not score a goal,
(2 marks)
Not score
Answer:
Step-by-step explanation:
6
2/3
15
13
Hit
Miss
2/5
3
5
215
2
Not score
Score
Find the probability that Ryan misses the target and does not score a goal,
(2 marks)
A student researcher compares the ages of cars owned by students and cars owned by faculty at a local state college. A sample of 224 cars owned by students had an average age of 5.06 years. A sample of 233 cars owned by faculty had an average age of 7.19 years. Assume that the population standard deviation for cars owned by students is 3.42 years, while the population standard deviation for cars owned by faculty is 2.81 years. Determine the 95% confidence interval for the difference between the true mean ages for cars owned by students and faculty. Step 2 of 3 : Calculate the margin of error of a confidence interval for the difference between the two population means. Round your answer to six decimal places.
The difference between the population means - 3.4 and -1.84 when calculated is found to be -2.62 years.
The 95% confidence interval for the difference in true mean ages of cars owned by students and faculty is between -3.4 and -1.84 years.
We know that the sample of 138 cars which belongs to the students have an average age of 5.13 years. The standard deviation = 3.45 years.
Then again,
A sample of 111 cars which belongs to the faculty have an average age of 7.75 years. The standard deviation= 2.08 years.
So the difference between them will be , s - f
= 3.45 - 2.08 = 1. 37
and the mean will be , mean of students - mean of faculty =
-2.62.
This is the point of difference between their means.
The SD is found to be as 0.3961.
and MOE is 0.78
with these details now we will calculate the required intervals,
The lower end is found to be as -2.62 - 0.78 = -3.4
The upper end is found to be as -2.62 + 0.78 = -1.84
Therefore, the mean difference between the cars is - 3.4 and -1.84.
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What is the domain of the function y=5√6−2x
The domain is all values of x that makes the expression defined.
Set-Builder Notation: {x| x ≤ 3 }
Interval Notation: (-∞,3]
What is the domain of the given function?Given the function in the question;
y = 5√(6−2x)
To find the domain, set the radicand greater than or equal zero and solve for x.
6 - 2x ≥ 0
Subtract 6 from both sides
6 - 6- 2x ≥ 0 - 6
-2x ≥ -6
Divide both sides by -2
-2x/-2 ≤ -6/-2
x ≤ -6/-2
x ≤ 3
Therefore, the domain is;
Set-Builder Notation: {x| x ≤ 3 }
Interval Notation: (-∞,3]
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a paint roller has a width of 12 inches and a radius of 3 inches, what is the surface area that can be painted with one complete rotation of the roller
The surface area that can be painted with one complete rotation of the roller is 226.19 inches².
Surface area is defined as the total amount of area that covers the surface or outside of a three-dimensional figure.
A paint roller is in the shape of a cylinder. To determine the surface area that can be painted with one complete rotation of the roller, solve for the surface area of a cylinder without the circular bases.
SA = 2πrh
where SA = surface area
r = radius of the base = 3 inches
h = height of the cylinder = width = 12 inches
Plug in the values and solve for the surface area.
SA = 2πrh
SA = 2π(3 inches)(12 inches)
SA = 226.19 inches²
Hence, the surface area that can be painted with one complete rotation of the roller is 226.19 inches².
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HELP MEEEEEEEE PLEASEEEEEEEE
Answer: 75p
Step-by-step explanation:
So, if one kilogram of bacon is £1.50, then half a kilogram would cost half of that, so 75p
What is an intractable problem?
Answer: A problem is said to be intractable if there is no efficient algorithm to solve it.
Intractable problems are common. We need to discuss how we approach it when we actually encounter it.
Step-by-step explanation:
How many colours do you need to colour a graph such that no adjacent vertices are of the same colour showed in the image below?
Conclusion : Provided the graph with a large number of vertices, we see that we are again faced with resorting to a systematic tracing of all paths, comparison of the neighbour's colours, backtracking, etc resulting in exponential time complexity once again.
Find the equation of the line that goes through ( 0, 2 ) and ( 3, 4 ).
Select one:
a.
2x + y - 3 = 0
b.
- 2x + 3y - 6 = 0
c.
x + y - 2 = 0
d.
2x + 3y + 6 = 0
Answer:
Option b. -2x + 3y - 6 = 0
Step-by-step explanation:
Let's look for a linear equation of the form y=mx+b, where m is the slope and b is the y-intercept (the value of y when x = 0).
Calculate the slope first. Slope is the Rise/Run of any two points on a straight line. Using the given points, (0,2) and (3,4) we find:
Rise = 4 - 2 = 2
Run = 3 - 0 = 3
Slope, m, Rise/Run = (2/3)
The equation becomes y = (2/3)x + b
To find b, enter either of the two given points.
y = (2/3)x + b
2 = (2/3)(0) + b : for point (0,2)
b = 2
The equation becomes: y = (2/3)x + 2
Reformat this to match the format of the answer options.
y = (2/3)x + 2
3y = 2x + 6 [Multiplied by 3]
3y - 2x - 6 = 0
-2x + 3y - 6 = 0 [rearrange]
This matches option b.
Can you help me with this
Answer:
I have graphed a line with a slope of 3/5 and attached a screenshot in the explanation.
Step-by-step explanation:
6(x+4)=30
how do i solve this?
the sum of 1/4 a number and 5 is 3. What is the number?
Answer:
x = -2/4 = -1/2
Step-by-step explanation:
The sum of 4 times a number and 5 is 3.
Let sum = +
times = *
is means =
a number = x
So, 4x + 5 = 3. Then we solve it !
Subtract 5 to both sides to get 4x = - 2.
Next, we divide 4 to both sides to get x = -2/4 = -1/2
Answer:
The number is -8
Step-by-step explanation:
You can set up the question as (x/4)+5 = 3
x/4 = -2
x = -8
How many real solutions of 2x 3y 5 are possible?
The equation 2x + 3y = 5 has one real solution.
To find it, you can solve the equation for one of the variables (either x or y) and then substitute the result into the equation to find the other variable.
For example, you can solve for y to get y = (5 - 2x)/3, and then substitute this into the equation to get 2x + 3((5 - 2x)/3) = 5, which you can then solve for x. Doing this will give you x = 5/4, which means that the solution is (5/4, 5/3).
Linear equations are equations that involve two variables and can be expressed in the form ax + by = c, where a and b are constants and x and y are the variables. These equations have one real solution, which can be found by solving for one of the variables and then substituting the result into the equation to find the other variable. Once both variables are known, the solution to the equation is the point (x,y) that satisfies both equations.
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shown below PLEASE ANSWER FAST I ONLY HAVE 4 MORE MINUTES!
Answer:
15 lessons
Step-by-step explanation:
18. Solve for b:
3 (23-2b) + 3b = 48
A. 10
B. 13
C. 7
D. -10
Answer:
Step-by-step explanation:
3(23-2b) + 3b = 48
1. Use the distributive property and distribute the 3 before the parenthesis
3(23-2b) (make sure you read the (-) as part of the 2b so it's -2b
69 - 6b
2. 69 - 6b +3b = 48 (combine like terms...the bs: -6b + 3b)
69 -3b=48
3. 69 - 3b = 48 (subtract 69 from both sides)
-69 -69
4. -3b = -21 (divide both sides by negative three...we want the b to be +
/-3 /-3
Answer: b = 7
So, C is your answer
Darius has a cylindrical can that is completely full of sparkling water. He also has an empty cone-shaped paper cup. The height and radius of the can and cup are shown.
Darius pours sparkling water from the can into the paper cup until it is completely full. Approximately, how many centimeters high is the sparkling water left in the can?
Answer: 6.2
Step-by-step explanation:
Christopher has 95 toy cars. He arranges his cars in 5 equal rows. Write numbers in the boxes to complete the partial quotient model and the equation to find the number of cars in each row. 95 + 5 = cars
The number of boxes to complete the partial quotient model is found to be as 19 cars .
On dividing 95 from 5 we will get 19 as the quotient.
95 / 5 = 19
The quotient is the number that is produced when two integers are divided. It is the outcome of the division process , when we divide a number from another the result is known as quotient.
In arithmetic division, four primary terms are used which are divisor, dividend, quotient, and remainder. In this post, each phrase will be defined and shown using examples.
Dividend = Divisor + Quotient + Remainder
If the divisor is not able to completely divide the dividend then it leaves a remainder.
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The goal for the size of the Santa on a Christmas Santa cup is 3. 5 cm with an acceptable tolerance of ± 0. 9 cm. The grand mean of the size of the Santa from the samples that were taken is 3. 4 cm and the standard deviation is 0. 28 cm. What is CP? (rounded to three decimals)
The processing capability for the size of the Santa rounded to three decimal places is 1.07.
CP (Process Capability) is a statistical measure that compares the performance of a process to the specifications of that process. It can be calculated using the formula:
CP = (USL - LSL) / (6 * standard deviation)
Where:
USL = Upper Specification Limit (3.5 + 0.9) = 4.4 cm
LSL = Lower Specification Limit (3.5 - 0.9) = 2.6 cm
Standard deviation = 0.28 cm
Substituting these values into the formula:
CP = (4.4 - 2.6) / (6 * 0.28) = 1.8 / 1.68 = 1.07
So, CP rounded to three decimal places is 1.07
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Complete the table with values for x or y that make this equation true : 3x+y=15.
The table with values for x or y that make this equation true : 3x + y = 15 are
x 2 4 6 0 3 5 7/3
y 9 3 -3 15 6 0 8
What is solving an equation?
One-step equations can be solved in four different ways: by adding, subtracting, multiplying, and dividing. In an equation, both sides will remain equal if we add the same amount to each side.
Consider, the given equation: 3x + y = 15
We have to complete the given table for the values of x and y.
Plug x = 2 in the given equation.
⇒ 3(2) + y = 15
6 + y = 16
y = 9
Plug y = 3 in the given equation.
3x + 3 = 15
3x = 12
x = 4
Plug x = 6 in the given equation.
3(6) + y = 15
18 + y = 15
y = -3
Plug x = 0 in the given equation.
3(0) + y = 15
y = 15
Plug x = 3 in the given equation.
3(3) + y = 15
9 + y = 15
y = 6
Plug y = 0 in the given equation.
3x + 0 = 15
3x = 15
x = 5
Plug y = 8 in the given equation.
3x + 8 = 15
3x = 7
x = 7/3
Hence, the table with values for x or y that make this equation true : 3x+y=15 are
x 2 4 6 0 3 5 7/3
y 9 3 -3 15 6 0 8
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A hiker maintains an average speed of 3 3/4 km/h. How many kilometers will the hiker walk in:5hrs
help
multiple 5x 3 3/4 then change to an improper fraction
What is translation in coordinate geometry?
In the coordinate geometry the translation represents one type of transformation to change the location without changing the shape, size.
As given in the question,
In the coordinate geometry,
The translation is one type of transformation.Translation represent the change in the location of the given geometrical figure in the coordinate plane.No change in the shape and size of the geometrical figure.Change in the location vertical and horizontal direction.Right and left direction represents the horizontal direction.Up and down represents the vertical direction.Therefore, in the coordinate geometry the translation represents one type of transformation to change the location of the geometrical figure.
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The scale on a map of Florida is 2 inches = 50 miles. What is the distance in miles
represented by a length on the map of 7 inches?
The distance in miles represented by a length on the map of 7 inches is 0.28
What is scale factor?The ratio between comparable measurements of an item and a representation of that thing is known as a scale factor in arithmetic.
Given;
The scale of a map says that 2 inches represents 50 miles.
2 inches = 50 miles
1 Mile = 2/50 inches
Hence, 7 Mile = 7 * 2/50 inches
7 Mile = 14/50 inches
7 Mile = 0.28 inches
Therefore, the scale factor of 7miles in the map will be 0.28inches
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How do you find the parent function of a line?
The parent function of a line (usually of the form y = ax + b) is y = x.
The most basic form of a function is a parent function. Its essential shape is not changed in any manner. In mathematics, some graphs are frequently seen. Because of this, the graphs of quadratic functions, square roots, absolute values, cubics, and cube roots are among the original, widespread functions that are referred to as parent graphs.
For instance, you should still be able to identify the parabola that has undergone various transformations when you view a u-shaped graph that has been inverted and stretched vertically.
Linear FunctionLinear functions have x as the term with the highest degree and a general form of y = a + bx. All linear functions have a straight line as a graph. The parent function of linear functions is y = x, and it passes through the origin. The domain and range of all linear functions are all real numbers.
These functions represent relationships between two objects that are linearly proportional to each other.
Therefore, the parent function of a line (usually of the form y = ax + b) is y = x.
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Select the correct answer from each drop-down menu.
The boundary of a park is shaped like a circle. The park has a rectangular playground in the center and 2 square flower beds, one on each side
of the playground. The length of the playground is /and its width is w. The length of each side of the flower beds is a. Which two equivalent
expressions represent the total fencing material required to surround the playground and flower beds? Assume that the playground and beds
do not overlap
2(l+w) + 4a and 2l+2w + 2(2a) are two equivalent expressions which represent the total fencing material required to surround the playground and flower beds.
What are mathematical expressions?Mathematical expressions are a way of representing mathematical ideas, concepts, and operations using symbols and notation. They can take many forms, including numbers, variables, and operators. They can also be combined to create more complex expressions, equations, and formulas.
Explain:
The total fencing material required to surround the playground and flower beds would be the sum of the fencing material for the playground and the fencing material for the flower beds.
So, one possible expression for the total fencing material required to surround the playground and flower beds would be: 2(l+w) + 4a
Another possible expression for the total fencing material required to surround the playground and flower beds would be: 2l+2w + 2(2a)
Please note that these expressions are based on the information provided, and the actual total fencing material required will depend on the specific dimensions of the park and playground, and may differ from these expressions.
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How do you find the missing side of a triangle?
The missing sides of the triangle can be found by applying the Pythagoras Theorem .
What is Pythagoras Theorem ?
The Pythagoras Theorem states that square of hypotnuse (that is longest side) is equal to sum of the square of perpendicular and base .
that means : [tex]Hypotnuse^{2} = Perpendicular^{2} +Base^{2}[/tex] ;
let missing side of the Right triangle be hypotnuse ;
So , missing side is written as : [tex]Hypotnuse = \sqrt{Perpendicular^{2} +Base^{2}}[/tex] .
For Example : Let a right triangle has sides lengths of perpendicular = 6 and hypotnuse = 10 .
let , the length of side "base" be unknown.
So , in order to find length of side b (base) , we substitute the known values into Pythagoras theorem:
[tex]10^{2} =6^{2} +Base^{2}[/tex] ;
[tex]Base = 100-36[/tex] ;
So , missing side "base" is = 8 ;
Therefore , the missing side can be found by using the Pythagoras Theorem .
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What are 5 ways to solve problems?
The 5 ways to solve problems are identifying the problem, generating different solutions, choosing one solution, implementing the chosen solution and evaluating the result.
A problem is any type of disruption from normality that is impeding progress. A problem can be time-consuming and requires too much energy to solve it.
There are 5 different steps to solving a problem
Step 1. Identify the Problem
Step 2. Generate potential solutions
Step 3. Choose one solution
Step 4. Implement the solution you’ve chosen
Step 5. Evaluate results
Sometimes it is required to start the process entirely over. To make the problem-solving process easier, it’s best to facilitate the solution as much as possible. Try to concentrate on the solution instead of the problem. Finally, be in the correct psyche to want to change. This contains having the right mindset language, both are positive and open-minded. With sufficient practice, any problem can be solved.
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