Answer:
550 gallons
Step-by-step explanation:
Let [tex]x[/tex] be the number of gallons for the 90% antifreeze solution and [tex]x+100[/tex] be the total number of gallons that will contain 80% antifreeze solution:
[tex]\displaystyle \frac{0.90x+0.25(100)}{x+100}=0.80\\\\0.90x+25=0.80x+80\\\\0.10x+25=80\\\\0.10x=55\\\\x=550[/tex]
Therefore, you would need 550 gallons of the 90% antifreeze solution.
a 4-foot length of ribbon costs $1.32. how much will it cost to buy 10 yards of ribbon?
Answer:
$3.3
Step-by-step explanation:
4x = 1.32
x = 0.33$ / ft
10x = 10 * 0.33 = 3.3$
Does the mapping diagram represent a function? Why or why not?
-5
8
9
y
-8
A. Yes; each input pairs with only one output.
B. No; the input value x = -5 pairs with two output values.
C. No; each input pairs with only one output.
D. No; each output value pairs with two input values.
The correct answer is: A. Yes; each input pairs with only one output.
To determine whether the given mapping diagram represents a function, we need to analyze the relationship between the input values (x) and the corresponding output values (y).
Looking at the mapping diagram, we can see that the input value -5 is paired with the output value 8. This implies that when x = -5, the corresponding y value is 8. Similarly, when x = 9, the corresponding y value is -8.
Since each input value has a unique and specific output value, we can conclude that the mapping diagram represents a function. In other words, for every input value, there is only one output value associated with it.
This is consistent with the definition of a function, where each input has a single and distinct output. Therefore, the mapping diagram satisfies the criteria for a function, and the correct answer is:
A. Yes; each input pairs with only one output.
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The mapping diagram does not represent a function that reflects this is: No; each output value pairs with two input values. D.
To determine whether the mapping diagram represents a function, we need to assess whether each input value pairs with only one output value.
Looking at the given mapping diagram:
-5 -> 8
9 -> y
-8 -> y
We can see that the input value -5 maps to the output value 8.
There are two different input values, 9 and -8, that map to the same output value, y.
In a function, each input should correspond to exactly one output, but in this case, the output value y is associated with two different input values.
In this case, y, the output value, pairs with both 9 and -8, indicating that the diagram does not meet the criteria for a function.
We must examine if each input value couples with only one output value in order to evaluate whether the mapping diagram reflects a function.
Observing the provided mapping diagram:
-5 -> 8 9 -> y -8 -> y
As can be seen, the input value of -5 corresponds to the output value of 8.
The identical output value, y, is mapped to two distinct input values, 9 and -8.
A function should have only one output for each input, however in this situation, the output value y is connected to two separate input values.
This figure does not fit the definition of a function because the output value, y, couples with both 9 and -8.
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Roger can run one mile in 9 minutes. Jeff can run one mile in 6 minutes. If Jeff gives Roger a 1 minute head start, how
long will it take before Jeff catches up to Roger? How far will each have run?
They each will have run of a mile.
Both Roger and Jeff will have run a distance of 1/3 mile when Jeff catches up to Roger after 2 minutes.
To solve this problem, we can determine the relative speeds of Roger and Jeff.
Since Roger runs one mile in 9 minutes, his speed is 1/9 miles per minute.
Similarly, Jeff runs one mile in 6 minutes, so his speed is 1/6 miles per minutes.
Let's assume that Jeff catches up to Roger after t minutes.
In that time, Roger would have run (1/9) [tex]\times[/tex] t miles, and Jeff would have run (1/6) [tex]\times[/tex] t miles.
Since Jeff gives Roger a 1-minute head start, we can express their distances covered as:
Distance covered by Roger = (1/9) [tex]\times[/tex] (t+1) miles
Distance covered by Jeff = (1/6) [tex]\times[/tex] t miles
For Jeff to catch up to Roger, their distances covered must be equal. So we can set up the equation:
(1/9) [tex]\times[/tex] (t+1) = (1/6) [tex]\times[/tex] t
To solve for t, we can cross-multiply and simplify:
6(t+1) = 9t
6t + 6 = 9t
6 = 9t - 6t
6 = 3t
t = 2
Therefore, it will take 2 minutes for Jeff to catch up to Roger.
Substituting t = 2 back into the equations, we can find the distances covered by each:
Distance covered by Roger = (1/9) [tex]\times[/tex] (2+1) = 1/3 mile
Distance covered by Jeff = (1/6) [tex]\times[/tex] 2 = 1/3 mile
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please help quick
Which of the following are solutions to the quadratic equation? Check all that
apply.
The correct solutions to the quadratic equation are:
c. -8
e. 2
To determine the solutions to the quadratic equation 2x^2 + 6x - 10 = x^2 + 6, we need to solve for x.
First, let's simplify the equation by combining like terms:
2x^2 + 6x - 10 - x^2 - 6 = 0
x^2 + 6x - 16 = 0
Now, we can solve this quadratic equation by factoring or by using the quadratic formula.
By factoring:
(x + 8)(x - 2) = 0
Setting each factor equal to zero, we get:
x + 8 = 0 --> x = -8
x - 2 = 0 --> x = 2
So, the solutions to the quadratic equation are x = -8 and x = 2.
Now, let's check the given options:
a. -2: This value is not a solution to the equation.
b. 1/3: This value is not a solution to the equation.
c. -8: This value is a solution to the equation.
d. -1/2: This value is not a solution to the equation.
e. 2: This value is a solution to the equation.
f. 8: This value is not a solution to the equation.
The following are the proper answers to the quadratic equation: c. -8 e.2
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Select the correct text.
Regina’s teacher recently gave her a homework assignment on solving equations. Since she has been thinking about saving for a new cell phone, she decided to use the assignment as an opportunity to model a savings plan.
She created this equation to model the situation. In it, y represents the total amount saved for the new cell phone, 74 is the amount of money she has now, 40 is the amount of money she saves each month for the phone, and x represents the number of months since she started saving a regular amount:
74 + 40x = y.
She then solved the equation to determine how many months she’d need to save to have enough to purchase the new cell phone. Review her work, and select the error.
Justification
1: given
2: subtraction property of equality
3: simplification
4: multiplication property of equality
5: simplification
6: substitution, y = 834
7: simplification
Step 1: 74 + 40x = y
Step 2: 74 + 40x − 74 = y − 74
Step 3: 40x = y − 74
Step 4:
=
Step 5: x =
Step 6: x =
Step 7: x = 19
Roger can run one mile in 9 minutes. Jeff can run one mile in 6 minutes. If Jeff gives Roger a 1 minute head start, how
long will it take before Jeff catches up to Roger? How far will each have run?
They each will have run of a mile.
It will take approximately 5.33 minutes for Jeff to catch up to Roger. Both Jeff and Roger will have run a distance of 1 mile.
Let's calculate the time it takes for Jeff to catch up to Roger and the distance each will have run.
First, we need to determine how much distance Roger covers during the 1-minute head start. Since Roger runs one mile in 9 minutes, in 1 minute, he would cover 1/9 of a mile.
Now, let's set up an equation to represent the time it takes for Jeff to catch up to Roger:
Distance covered by Jeff = Distance covered by Roger + Distance covered during the head start
Let's denote the time it takes for Jeff to catch up as t minutes. During this time, Jeff runs a distance of 1 mile, while Roger runs a distance of 1/9 of a mile (from the head start).
Distance covered by Jeff = Distance covered by Roger + Distance covered during the head start
1 mile = (1/9) mile + (6 minutes) × t × (1 mile/6 minutes)
Now, let's solve this equation to find the time it takes for Jeff to catch up:
1 = (1/9) + (1/6)t
Multiplying the equation by the least common multiple (LCM) of 9 and 6, which is 18, to clear the fractions:
18 = 2 + 3t
Subtracting 2 from both sides:
16 = 3t
Dividing by 3:
t = 16/3 = 5.33 minutes (rounded to two decimal places)
Therefore, it will take approximately 5.33 minutes for Jeff to catch up to Roger. Both Jeff and Roger will have run a distance of 1 mile.
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Which table of values represents a function? Step by step.
Answer:
A
Step-by-step explanation:
For a table of values to be a function, all inputs must have one unique output. The only table that doesn't violate this is table A.
Notice that while two different inputs (-4 and 1) have the same output of 7, it is still a function because both outputs of 7 are associated with two different inputs.
Write the correct measurement for A-F (Example 2.2, 5.9, 9)
Answer:
9
Step-by-step explanation:
Please help! :')
Prove that the two circles shown below are similar. (10 points)
Circle X is shown with a center at negative 2, 8 and a radius of 6. Circle Y is shown with a center of 4, 2 and a radius of 3.
6. For each final matrix, state the solution.
Please answer this picture
The solutions associated with the final matrices representing a system of linear equations are listed below:
First case: x = 3, y = 1, z = 8
Second case: No solution
Third case: No solution
What are the solutions contained in each final matrix?
In this problem we find three of final matrices, each of them representing a system of linear equations. There are two rules to be considered:
A system has no solution if there is a row with only zeroes in the dependent coefficients matrix.A system has a solution if there is a singular matrix, there is, the dependent coefficients matrix has only ones in the main diagonal and the rest of elements are zeroes.Now we proceed to determine the solution associated with each matrix:
First matrix:
x = 3, y = 1, z = 8
Second matrix:
No solution
Third matrix:
No solution
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Step-by-step explanation:
For the left matrix, since the matrix is already in reduced row form,
The solutions to the matrix is (3,-1,8)
For the middle matrix, we need to convert the -6 to a zero,
notice how in the third row, every entry except the last one is 0, this implies that
0=-2, which is not the case thus the middle matrix has no solution.
For the last one, notice that we have 3 colums but only two non zero rows, this means that this matrix has infinite solutions.
Please awnser asap I will brainlist
The row operation on the matrix [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex] is [tex]\left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]
How to perform the row operation on the matrixFrom the question, we have the following parameters that can be used in our computation:
[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]
The row operation is given as
1/2R₁
This means that we divide the entries on the first row by 2
Using the above as a guide, we have the following:
[tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]
Hence, the row operation on the matrix is [tex]\left[\begin{array}{ccc|c}2&0&0&16\\0&8&0&3\\0&0&5&6\end{array}\right] = \left[\begin{array}{ccc|c}1&0&0&8\\0&8&0&3\\0&0&5&6\end{array}\right][/tex]
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25 ÷ 5+7-(4 x 3) Solve the problem is fast as possible
Answer:
Step-by-step explanation:
0
Answer:
0
Step-by-step explanation:
25 ÷ 5+7-(4 x 3)
25 ÷ 5+7-(4 x 3)
5+7-(4 x 3)
5+7-(4 x 3)
5+7-12
12-12
0
I wrote in bold the steps you need to follow using PEMDAS (Parentheses, Exponents, Multiplication and Divison left to right, and Addition and Subtraction left to right).
Let A be the point (7,4) and D be (-5, -3). What is the length of the shortest path ABCD, where B is a point (x, 2) and C is a point (x,0)? This path consists of three connected segments, with the middle one vertical.
The length of the shortest path ABCD is 7 units.
To find the length of the shortest path ABCD, we need to determine the coordinates of points B and C and then calculate the distances between these points.
Given that B has a y-coordinate of 2, it lies on a horizontal line. Therefore, the y-coordinate of B is 2, and the x-coordinate is the same as the x-coordinate of point A, which is 7. So, B is the point (7, 2).
Similarly, C lies on a vertical line, and its x-coordinate is the same as the x-coordinate of point D, which is -5. So, C is the point (-5, 0).
Now, we can calculate the distances between the points. The distance between A and B can be found using the distance formula:
AB = √[tex]((x2 - x1)^2 + (y2 - y1)^2[/tex])
Substituting the coordinates of A and B, we have:
AB = √[tex]((7 - 7)^2 + (2 - 4)^2) = √(0^2 + (-2)^2[/tex]) = √4 = 2
The distance between B and C is simply the difference in their y-coordinates:
BC = |y2 - y1| = |2 - 0| = 2
Finally, the distance between C and D can be calculated using the distance formula:
CD = √[tex]((-5 - (-5))^2 + (0 - (-3))^2)[/tex] = √[tex](0^2 + 3^2)[/tex] = √9 = 3
Therefore, the length of the shortest path ABCD is the sum of the distances AB, BC, and CD:
Shortest path ABCD = AB + BC + CD = 2 + 2 + 3 = 7
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complete the table of values for y=3/x
The table of values for the equation y = 3/x can be completed as follows:
x | y
1 | 3/1 = 3
2 | 3/2 = 1.5
3 | 3/3 = 1
4 | 3/4 = 0.75
5 | 3/5 = 0.6
To complete the table of values for the equation y = 3/x, we substitute different values of x into the equation and calculate the corresponding values of y.
When x = 1:
Substitute x = 1 into the equation: y = 3/1 = 3
So, when x = 1, y = 3.
When x = 2:
Substitute x = 2 into the equation: y = 3/2 = 1.5
So, when x = 2, y = 1.5.
When x = 3:
Substitute x = 3 into the equation: y = 3/3 = 1
So, when x = 3, y = 1.
When x = 4:
Substitute x = 4 into the equation: y = 3/4 = 0.75
So, when x = 4, y = 0.75.
When x = 5:
Substitute x = 5 into the equation: y = 3/5 = 0.6
So, when x = 5, y = 0.6.
By substituting different values of x into the equation y = 3/x, we can complete the table of values as shown above.
Hence, the completed table of values for y = 3/x is:
x | y
1 | 3
2 | 1.5
3 | 1
4 | 0.75
5 | 0.6
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Jake drives a tractor from one town to another, a distance of 120 kilometers. He drives 6 kilometers per hour faster on the return trip, cutting 1 hour off the time. How fast does he drive each way?
The speed of Jake's initial trip is x = 24 kilometers per hour, and the speed of the return trip is x + 6 = 30 kilometers per hour.
Let's assume that Jake's speed during the initial trip is represented by "x" kilometers per hour.
On the return trip, he drives 6 kilometers per hour faster, so his speed can be represented as "x + 6" kilometers per hour.
To find the time taken for each trip, we can use the formula Time = Distance / Speed.
For the initial trip, the time taken is 120 kilometers divided by x kilometers per hour, which gives us 120/x hours.
On the return trip, the time taken is 120 kilometers divided by (x + 6) kilometers per hour, which gives us 120/(x + 6) hours.
According to the problem, the return trip takes 1 hour less than the initial trip. So we can set up the equation:
120/x - 1 = 120/(x + 6)
To solve this equation, we can multiply both sides by x(x + 6) to eliminate the denominators:
120(x + 6) - x(x + 6) = 120x
Simplifying this equation:
120x + 720 - x² - 6x = 120x
Combining like terms:
x² + 6x - 720 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. By factoring, we find:
(x + 30)(x - 24) = 0
This gives us two potential solutions: x = -30 or x = 24.
Since speed cannot be negative, we discard the solution x = -30.
Therefore, the speed of Jake's initial trip is x = 24 kilometers per hour, and the speed of the return trip is x + 6 = 30 kilometers per hour.
So, Jake drives at a speed of 24 kilometers per hour on the initial trip and 30 kilometers per hour on the return trip.
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Question #3
Solve for x
OOO
O 10
07
5
8
6x+8
N
p
U
K
122°
L
M
194°
The calculated value o x in the circle is 7
How to determine the solution for xFrom the question, we have the following parameters that can be used in our computation:
The circle
The value of x can be calculated using teh equation of the intersection of chords
So, we have
122 = 1/2(6x + 8 + 194)
Evaluate
244 = 6x + 202
So, we have
6x = 42
Divide by 6
x = 7
Hence, the solution for x is 7
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Please answer ASAP I will brainlist
The solution to the system is (a) a = 4/5, b = 5, c = -4 and d =-4
How to determine the solution to the systemFrom the question, we have the following parameters that can be used in our computation:
The augmented matrix
Where, we have
[tex]\left[\begin{array}{cccc|c}1&0&0&0&4/5\\0&1&0&0&5\\0&0&1&0&-4\\0&0&0&1&-4\end{array}\right][/tex]
From the above, we have the diagonals to be 1
And other elements to be 0
This means that the equation has been solved
So, we have
a = 4/5, b = 5, c = -4 and d =-4
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Find the volume of this cylinder give your answer to one decimal place 11height 14 Length
To find the volume of the given cylinder, we need to use the formula of volume of a cylinder which is given by V = πr²h, where V is the volume of the cylinder, r is the radius of the cylinder and h is the height of the cylinder. So, the volume is 1078.1 cubic units.
Given, Length = 14 units, Height = 11 units. Now, we know that the formula for the volume of a cylinder is given as; V = πr²h.
Here, the radius of the cylinder is half the length of the cylinder, which is; Radius = Length/2
Radius = 14/2
Radius = 7 units
Substituting the value of the radius and height, we have; V = π(7)²(11)
V = 1078.1 cubic units. Therefore, the volume of the cylinder is 1078.1 cubic units. Rounded off to one decimal place, the volume is 1078.1 cubic units.
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A wooden board in the shape of a rectangle prism measures 0.3 m by 2.1 m by 0.1 m and has a mass of 0.17 kilogram. What is the density of the board?
Enter your answer as a decimal in the box. Round your answer to the nearest tenth.
Answer:
Step-by-step explanation:
To find the density of the wooden board, we need to divide the mass of the board by its volume. The volume of a rectangular prism is calculated by multiplying its length, width, and height.
Given:
Length (l) = 0.3 m
Width (w) = 2.1 m
Height (h) = 0.1 m
Mass (m) = 0.17 kg
Volume (V) = l × w × h
V = 0.3 m × 2.1 m × 0.1 m
V = 0.063 m³
Density (ρ) = mass / volume
ρ = 0.17 kg / 0.063 m³
ρ ≈ 2.7 kg/m³
The density of the wooden board is approximately 2.7 kg/m³.
Suppose we have two equations and they are both equal to each other. Equation A is "y = x^2 - 9" and Equation B is "y = x + 3". If we had to solve this system of equations, what quadratic equation do we have to solve in order to get our x values?
a. x^2 - x - 12 = 0
b. x^2 + x + 3 = 0
c. x^2 - x - 6 = 0
Answer:
a) x² - x - 12 = 0
Step-by-step explanation:
We have equation A = equation B
⇒ x² - 9 = x + 3
⇒ x² - 9 - x - 3 = 0
⇒ x² - x - 12 = 0
I need help with this problem a s a p.
The calculated vertex of the function y = 2(x + 4)(x - 2) is (-1, -18)
Examining the function for the vertexFrom the question, we have the following parameters that can be used in our computation:
y = 2(x + 4)(x - 2)
Expand the equation
So, we have
y = 2x² + 4x - 16
Differentiate the function and set to 0
So, we have
4x + 4 = 0
So, we have
4x = -4
Evaluate
x = -1
Next, we have
y = 2(-1 + 4)(-1 - 2)
Evaluate
y = -18
This means that the vertex is (-1, -18)
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(|7 − 3 x 5) | ÷ 4)³ + √64
Answer:
16
Step-by-step explanation:
(|7 − 3 x 5| ÷ 4)³ + √64
(|7 − 3 x 5| ÷ 4)³ + √64
(|7 − 15| ÷ 4)³ + √64
(|7 − 15| ÷ 4)³ + √64
(8 ÷ 4)³ + √64
(8 ÷ 4)³ + √64
2³ + √64
8 + 8
16
Answer:
16
Step-by-step explanation:
To solve, use PEMDAS as it applies to the expression.
First, you must carry out what is in the parenthesis.
Multiply (3*5 = 15), then subtract (7 - 15 = -8). The two little vertical lines surrounding (7 - 3 x 5) mean absolute value. This means that you will write -8 as +8.
Continuing on in the parenthesis, carry out (8 ÷ 4 = 2).
Next, we must do exponents.
We see that everything in the parenthesis is being raised to the power of three (cubed). Since we've solved what was in the parenthesis, we simply need to carry out ([tex]2^{3}[/tex] = 8).
Now, we need to quickly carry out [tex]\sqrt{64}[/tex]. Square roots are just whatever number can be multiplied by itself to get, in this case, 64.
[tex]\sqrt{64} = 8[/tex].
Finally, we must add what remains.
8 + 8 = 16.
So, (|7 - 3 x 5) | ÷ [tex]4)^{3}[/tex] + [tex]\sqrt{64}[/tex] = 16.
If f(x) = 16x2, what is the value of f(2.5)?
Answer:
f(2.5) = 100
Step-by-step explanation:
to evaluate f(2.5) substitute x = 2.5 into f(x)
f(2.5) = 16(2.5)² = 16 × 6.25 = 100
A building contractor is building a backyard playground and wishes to put down rubber mulch to provide safety from falls. The contractor wishes to put the mulch in a pit in the shape of a rectangular solid 24 feet long, 13 feet wide, and 6 inches deep.
a) Determine the volume, in cubic feet, of mulch the contractor will need.
b) If mulch costs 9$ per cubic foot, what will the cost of mulch be?
Answer:
A= 220.5ft
B= $2425.5
Step-by-step explanation:
A= 220.5ft
B= $2425.5
This table shows the number of customers who have come in to Trent’s Hobby Shop each day since he opened the doors.
A 2-column table with 4 rows. Column 1 is labeled Number of Days Open with entries 1, 2, 3, 4. Column 2 is labeled Number of customers with entries 3, 11, 24, 30.
Is this function discrete or continuous?
Answer:
a
Step-by-step explanation:
a spinner with 10 equally sized slices 4 yellow, 4 red, 2 blue. probability that the dial stops on yellow?
The probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).
Therefore, there is a 40% chance that the spinner will stop on yellow.
To find the probability that the spinner stops on yellow, we need to determine the number of favorable outcomes (yellow) and the total number of possible outcomes.
The spinner has 10 equally sized slices, with 4 yellow, 4 red, and 2 blue.
The number of favorable outcomes (yellow) is 4 because there are 4 yellow slices.
The total number of possible outcomes is 10 because there are 10 slices in total.
Therefore, the probability of the spinner stopping on yellow can be calculated as:
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 4 / 10
Simplifying this fraction, we get:
Probability = 2 / 5
So, the probability that the dial stops on yellow is 2/5 or 0.4 (or 40%).
Therefore, there is a 40% chance that the spinner will stop on yellow.
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A company has recently been hiring new employees. Today the company has 32% more employees than it did a year ago. If there are currently 69,300 employees, how many employees did the company have a year ago?
Answer:
52500
Step-by-step explanation:
Let there be x employees in the previous year
Now, the company has 32% more employees whis is 69300
i.e.
[tex]x + \frac{32}{100} x = 69300\\\\ \implies\frac{132x}{100} = 69300 \\\\\implies 132x = 6930000\\\\\implies x = \frac{6930000}{132}\\[/tex]
⇒ x = 52500
There were 52500 employees in the previous year
X-2
5 = 8 using the change of base formula logby=
log y
log b
By using the change of base formula: The solution to the equation log(base y) (X-2) = 5 is [tex]X = y^5 + 2.[/tex]
To solve the equation log(base y) (X-2) = 5 using the change of base formula, we can rewrite the equation as log(base b) (X-2) / log(base b) y = 5.
Using the change of base formula, we can choose any base for b.
Let's choose base 10 for simplicity.
So the equation becomes log(base 10) (X-2) / log(base 10) y = 5.
We know that log(base 10) (X-2) represents the logarithm of (X-2) to the base 10, and log(base 10) y represents the logarithm of y to the base 10.
Now, to solve for X, we can isolate it by multiplying both sides of the equation by log(base 10) y:
log(base 10) (X-2) = 5 [tex]\times[/tex] log(base 10) y.
This simplifies to:
log(base 10) (X-2) [tex]= log(base 10) y^5.[/tex]
Since the logarithms on both sides have the same base, we can remove the logarithm and equate the arguments:
[tex]X - 2 = y^5.[/tex]
Now we can solve for X by adding 2 to both sides:
[tex]X = y^5 + 2.[/tex]
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Which of the following functions is graphed me below ?
Answer:
[tex]y = |x - 2| + 3[/tex]
The correct answer is C.
Prove the following?
The given statement proposition is a that suggests that if x is an inductive element, defined in terms of the set X as described, then it follows that every non-zero natural number can be expressed as the successor of some other natural number.
The given statement is a logical proposition involving the concept of induction. Let's break it down and analyze its components.
"If x is inductive..."
This implies that x is a concept or object that possesses the property of being inductive. In mathematics, the term "inductive" typically refers to the property of being a natural number or belonging to the set of natural numbers.
"...then so is (x ∈ X : x = Ф or x = y ∪ {y} for some y)"
Here, we have a set X that consists of elements satisfying a certain condition. The condition states that an element x in X can either be equal to the empty set (Ф) or the union of a set y with itself ({y}). This construction represents a recursive definition, where each element in X is defined in terms of a base case (Ф) and a recursive step ({y}).
"...hence each n ≠ 0 is m + 1 for some m."
This conclusion states that for every natural number n that is not equal to 0, there exists another natural number m such that n can be expressed as m + 1. In other words, any non-zero natural number is the successor of some other natural number.
Overall, the given statement is a proposition that suggests that if x is an inductive element, defined in terms of the set X as described, then it follows that every non-zero natural number can be expressed as the successor of some other natural number.
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