9x7x 9x 9 7 x 9
Helpppoooo
Answer: 494991
Step-by-step explanation:
Answer:
Step-by-step explanation:
9×7×9×97×9=
9×7=63
63×9=567
567×97=54999
54999×9=494991
ratio analysis can be made more meaningful in all the following ways except by?
Ratio analysis can be made more meaningful in all the following ways except by focusing more on long-term solvency than on short-term solvency.
Ratio analysis is a mathematical technique for analyzing a company's financial documents, such as the balance sheet and income statement, to gather knowledge about its liquidity, operational effectiveness, and profitability. Fundamental equity research is built on ratio analysis.
In order to get insights into profitability, liquidity, operational effectiveness, and solvency, ratio analysis examines line-item data from a company's financial statements.
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The full question:
ratio analysis can be made more meaningful in all the following ways except by?
A substance decays so that the amount a of the substance left after t years is given by: a = a0 • (0.7)t, where a0 is the original amount of the substance. what is the half-life (the amount of time that it takes to decay to half the original amount) of this substance, rounded to the nearest tenth of a year?
The half-life of the substance is 1.94 years.
What is exponential decay formula?The exponential decay formula aids in determining the exponential drop, which is a rapid reduction over time. To calculate population decay, half-life, radioactivity decay, and other phenomena, one uses the exponential decay formula. F(x) = a \((1-r)^{x}\) is the general form.
Here
a = the initial amount of substance
1-r is the decay rate
x = time span
The correct form of the equation is given as:
\(a=a_{0}\)×\((0.7)^{t}\)
where t is an exponent of 0.7 since this is an exponential decay of 1st order reaction
Now to solve for the half life, this is the time t in which the amount left is half of the original amount, therefore that is when:
a = 0.5 a0
Substituting this into the equation:
0.5 \(a_{0}=a_{0}\)×\((0.7)^{t}\)
0.5 = \((0.7)^{t}\)
Taking the log of both sides:
t log 0.7 = log 0.5
t = log 0.5 / log 0.7
t = 1.94 years
The half life of the substance is 1.94 years.
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arctan(4/3) in terms of pi
The expression for arctan(4/3) in terms of π is a tan(4/3) / π.
To express arc tan(4/3) in terms of π, we can use the relationship between the trigonometric functions and the unit circle.
The tangent function (tan) is defined as the ratio of the length of the side opposite an angle to the length of the adjacent side in a right triangle. The inverse tangent function, arctan, gives the angle whose tangent is a given value.
In this case, arctan(4/3) represents the angle whose tangent is 4/3. To express this angle in terms of π, we can consider the unit circle.
For arctan(4/3), we can construct a right triangle in the unit circle with the opposite side equal to 4 and the adjacent side equal to 3.
Using the Pythagorean theorem, we can calculate the length of the hypotenuse:
hypotenuse² = opposite² + adjacent²
hypotenuse² = 4² + 3²
hypotenuse²= 16 + 9
hypotenuse²= 25
hypotenuse = 5
Now, let's denote the angle whose tangent is 4/3 as θ. In the right triangle we constructed, the sine of the angle θ is given by opposite/hypotenuse, which is 4/5, and the cosine of the angle θ is given by adjacent/hypotenuse, which is 3/5.
Since the sine is positive and the cosine is positive in the first quadrant of the unit circle, we can conclude that arctan(4/3) corresponds to an angle in the first quadrant.
Therefore, arctan(4/3) can be expressed as:
arctan(4/3) = θ
Since θ corresponds to an angle in the first quadrant, we can write:
arctan(4/3) = θ = tan(4/3)
Note that a tan(4/3) is an angle measure in radians. To express it in terms of π, we need to divide atan(4/3) by π:
arctan(4/3) = θ = tan(4/3) / π
So, the expression for arctan(4/3) in terms of π is a tan(4/3) / π.
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In which quadrant does 0 lie if the following statements are true: tan θ> 0 and sin θ < 0 O Quadrant I O Quadrant III O Quadrant II O Quadrant IV
The quadrant in which 0 lies, given that tan θ > 0 and sin θ < 0, is Quadrant IV. The quadrant in which 0 lies when tan θ > 0 and sin θ < 0 is Quadrant IV.
Using a point on the coordinate plane, we can represent a positive tangent as follows: the slope of a line drawn from the origin (0,0) to a point on the coordinate plane is positive. Using a point on the coordinate plane, we can represent a negative sine as follows: the y-coordinate of a point on the coordinate plane is negative.To determine the quadrant in which 0 lies, we must look at the sign of the coordinates for 0. The point (0,0) has both an x and a y coordinate of zero. The origin is located at the center of the coordinate plane. Therefore, the point (0,0) is in every quadrant.In the first quadrant, both the x and y coordinates are positive.In the second quadrant, the x coordinate is negative and the y coordinate is positive.In the third quadrant, both the x and y coordinates are negative.In the fourth quadrant, the x coordinate is positive and the y coordinate is negative. Because the origin has a negative y-coordinate, it lies in Quadrant IV. Hence, the answer is Quadrant IV.
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Original Problem: 6x−10+4x−6=14x+20+4
Step 1: Combine Like Terms
Step 2: Subtraction Property to combine like variables
Step 3: Simplify
Step 4: Addition Property to combine constants
Step 5: Simplify Step 6: Division Property Final Answer:
Answer: x=-10
Step-by-step explanation:
After checking that conditions are met, you perform a significance test of 0:=1 versus :≠1 . You obtain a -value of 0.022.
Which of the following must be true?
A 95% confidence interval for will include the value 0.022.
A 99% confidence interval for will include the value 1.
None of these is necessarily true.
A 99% confidence interval for will include the value 0.022.
A 95% confidence interval for will include the value 1.
Answer:
A
Step-by-step explanation:
The true statement is
A 99% confidence interval for will include the value 1.
What is Confidence Interval?The range of values that we observe in our sample and for which we anticipate finding the value that accurately reflects the population is referred to as a confidence interval.
Given:
\(H_0: \mu = 1\\H_\alpha : \mu \neq 1\)
P= 0.022
If the P- value is smaller than significance level, then the null hypothesis is rejected.
P< 0.05 - 5% Reject \(H_0\)
P> 0.01 = 1% Fail to Reject \(H_0\)
A 95% Confidence interval correspond with a significance test at the 5%
significance level.
A 99% Confidence interval correspond with a significance test at the 1%
significance level.
If the Significance test was Significant then the interval does not contain the value mentioned in the Hypothesis.
95% CI does not contains 1.
99% CI contains 1.
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Solve for x. −4≤16−4x x≥5 x greater than or equal to 5 x≥−5 x greater than or equal to negative 5 x≤5 x less than or equal to 5 x≤−5
Answer:
Solve for the x:
In this equation that is: -4<=16-4x
And the options are: a) x>= -5 , b) x>= 5 ,c) x<=5 ,d) x<= -5
So the Solution is
=> -4< = 16-4x
=> 4x <= 16+4
=> x<= 20/4
=> x<= 5
So the option c) x<=5 is correct option
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Disclaimer: the question was given incomplete on the portal. Here is the Complete Question.
Question: Solve for the x:
In this equation that is: -4<=16-4x
and the options are: a) x>= -5 , b) x>= 5 ,c) x<=5 ,d) x<= -5
We get that the value of x is option (b) x is less than equal to 5 for the inequality - 4 ≤ 16 - 4 x.
We are given an inequality:
- 4 ≤ 16 - 4 x.
We have to solve the inequality to find the value of x.
First, we will add 4 to both sides:
- 4 + 4 ≤ 16 - 4 x + 4
Now simplifying the expression, we get that:
0 ≤ 20 - 4 x
Add 4 x to both the sides:
0 + 4 x ≤ 20 - 4 x + 4 x
4 x ≤ 20
Solving the expression to get the value of x:
x ≤ 20 / 4
x ≤ 5
Therefore, we get that the value of x is option (b) x is less than equal to 5 for the inequality - 4 ≤ 16 - 4 x.
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Question 6(Multiple Choice Worth 2 points)
(Line of Fit MC)
Data was collected on the amount of time that a random sample of 8 students spent studying for a test and the grades they earned on the test. A scatter plot and line of fit were created for the data.
scatter plot titled students' data, with points plotted at 1 comma 75, 2 comma 70, 2 comma 80, 2 comma 90, 3 comma 80, 3 comma 100, 4 comma 95, and 4 comma 100, and a line of fit drawn passing through the points 0 comma 60 and 2 comma 80
Find the y-intercept of the line of fit and explain its meaning in the context of the data.
5; for each additional hour a student studies, their grade is predicted to increase by 5% on the test
10; for each additional hour a student studies, their grade is predicted to increase by 10% on the test
60; a student who studies for 0 hours is predicted to earn 60% on the test
80; a student who studies for 0 hours is predicted to earn 80% on the test
The y-intercept of the line of fit is 80, which means that a student who studies for 0 hours is predicted to earn 80% on the test.
This means that even if a student does not study at all, they are predicted to earn a grade of 80% on the test based on the relationship between the amount of time spent studying and the grades earned on the test. This y-intercept value is also the predicted grade if the student did not study at all.
However, it is important to note that this prediction is based solely on the linear relationship observed in the given data and may not necessarily hold true for all cases or situations.
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One day, the principal announces the cancelation of classes to two teachers, each whom sends this to two other teachers, and so on. Suppose that text messages were sent in five rounds, counting the principal’s text message as first, how many text messages were sent in all?
Answer:
62
Step-by-step explanation:
2 + 4 + 8 + 16 + 32 = 62
If each person sends 2 texts each, then you basically multiply by 2 every round
Is real the maximum value of 3x² 9x 17 3x² 9x 7?
Therefore, the answer of the given question of the equation maximum value is 41 .
What is equation?
The equal sign is used between numerical or changeable expressions to create an equation, such as 3x+5=11.
A number that can be entered as the variable's replacement in an equation serves as the solution.
According to the given question:
Correct option is D)
f(x)= 3+9x+17 / 3+9x+7
=> 3+9x+7+10 / 3+9x+7
=> 1 + 10 / 3+9x+7
f(x) = 0 for maximum value
f(x)= 10 (6x+9) / (3x2+9x+7)2
6x+9=0
X=-3/2
Given that x is real, the highest value of f(x) is => 1 +
=> 1 +
=> 1 + 40/1 = 41
Therefore, the answer of the given question of the equation is 41 .
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what is the decimal form of five tenths and three hundredths
Answer:
0.53
Step-by-step explanation:
You just have to know the placements.
point G is rotated 90°. the coordinate of the pre-image G was (7,-5), and it’s image G’ is at the coordinate (5,7). What is the direction of the rotation
Answer:
The Direction of the rotation is counterclockwise around the origin.
What was Given was that Point G is rotated 90°. The coordinate of the pre-image point G was (7, –5), and its image G’ is at the coordinate (5, 7).
What you need to find was : What is the direction of the rotation.
So coming to conclusions is we have given that Point G is rotated 90°.
The Pre-image of G = ( 7, -5).
Translated image G' = ( 5 ,7).
By the Rotation rule of 90° is : (-y, x) 90 degree rotation counterclockwise around the origin.
(y, -x) 90 degree rotation clockwise about the origin.
We can from G (7,-5) and G'( 5 ,7) it is rotation counterclockwise around the origin.
Therefore, Direction of rotation is counterclockwise around the origin.
Step-by-step explanation:
4.8x=92.6 find out what x is
Answer:
x = 19.2916667
to the hundreth place- x = 19.29
to the tenth place- x = 19.3
Step-by-step explanation:
Undo in reverse order. Since x is multiplied by 4.8, you have to divide 92.6 by 4.8.
Hope this helps!! Brainliest please!!
What is "21 less than three-fifths of a number
Answer:
(3/5)x - 21. ..............
Step-by-step explanation:
Could I have branliest, heart with 5 stars
Thanks.
Choose the two steps that are part of factoring the trinomial 6x2 + 17x + 5 by grouping.
1. Factor out the greatest common factor (GCF) from the coefficients.
2. Split the middle coefficient into two factors that add to the middle coefficient and multiply to the last coefficient.
What is the the factored form?1. First, identify the GCF of the coefficients, which is 1. Since there is no GCF, this step can be skipped.
2. Next, split the middle coefficient (17) into two factors that add to the middle coefficient and multiply to the last coefficient.
Since 17 = 8 + 9, we can use 8 and 9 as our two factors.
3. Group the terms by the common factors.
We can group the first and last term together (6x2 + 5) and the middle term separately (17x).
4. Factor out the common factor, which is x, from the first and last group. This gives us x(6x + 5).
5. Lastly, factor the remaining quadratic expression (6x + 5).
Since 8 and 9 are the two numbers that add to the middle coefficient and multiply to the last coefficient, we can use them as our two factors. This gives us (6x + 8)(x + 9).
Therefore, the factored form of 6x2 + 17x + 5 is x(6x + 8)(x + 9).
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Please help with this
On what interval is the function h(x) = -2|x + 2| + 2 increasing?
(2, ∞)
(-2, ∞)
(-∞, -2)
(-∞, 2)
Answer:
Step-by-step explanation:
what grade is this ...
this look hard
ANSWER PLEASE FINAL DUE IN 3 HOURS
Suppose a normal distribution has a mean of 120 and a standard deviation of 10. What is P(x ≥ 130)?
A.
0.475
B.
0.84
C.
0.16
D.
0.975
Answer:
B
Step-by-step explanation:
First thing to do here is to calculate the z-score
Mathematically;
z-score = (x-mean)/SD
here, x = 130, mean = 120 and SD = 10
Substituting these values
z-score = (130-120)/10 = 10/10 = 1
So the probability we want to calculate is;
P(z ≥ 1) = 1 - P(z <1)
From standard score table, P(z <1) = 0.15866
P(z ≥ 1) = 1-0.15866 = 0.84134
Answer:
0.16
Step-by-step explanation:
A truck that can carry no more than 7300 lb is being used to transport refrigerators and upright pianos. Each refrigerator weighs 300 lb and each piano weighs 425 lb. Write
and graph an inequality to show how many refrigerators and how many planos the truck could carry. Will 11 refrigerators and 9 pianos overload the truck? Explain.
Let x be the number of refrigerators in the truck and y be the number of pianos in the truck. Write an inequality to show how many refrigerators and how many pianos the
truck could carry.
(Use integers or simplified fractions for any numbers in the inequality. Do not factor.)
The required inequality to show how many refrigerators and how many pianos the truck could carry is 300x + 425y ≤ 7300.
What is inequality?The idea of inequality, which is the state of not being equal, especially in terms of status, rights, and opportunities1, is at the core of social justice theories. However, because it frequently has diverse meanings to different people, it is prone to misunderstanding in public discourse.
According to question:We have,
A truck that can carry no more than 7300 lb is being used to transport refrigerators and upright pianos.
Let x be the number of refrigerators in the truck and y be the number of pianos in the truck.
Then
300x + 425y = 7300
At x = 11 and y = 9
300(11) + 425(9)
= 7125 ib
So, the truck will not overload.
Inequality will we
300x + 425y ≤ 7300
Thus, required inequality is 300x + 425y ≤ 7300.
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what is the slope of this line
Answer:
4
Step-by-step explanation:
rise/run = slope
As it goes to the left or adds by one, it goes up by 4
4/1
Answer:
4
Step-by-step explanation:
The line goes down 4 and 1 left. There is no need to put the 1.
Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
The equation of circles that have a diameter of 12 units and a center that lies on the y-axis is (x-0)² + (y-3)² = 6²
The standard form for finding the equation of a circle is expressed as;
(x-a)² + (y-b)²=r²
where;
(a, b) is the center of the circle
r is the radius of the circle
Given the following diameter = 12 units
radius = 12/2 = 6 units
If the centre lies on the y-axis, the center will be at (0, 3)
Substituting into the formula, we will have (x-0)² + (y-3)² = 6² equivalent to x²+ (y-3)² = 36
Hence, the equation of circles that have a diameter of 12 units and a center that lies on the y-axis is (x-0)² + (y-3)² = 6²
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Which expression is equivalent to?
Answer:5x^3y^3z
Step-by-step explanation:
Yeet
use green's theorem to calculate the work done by the force f on a particle that is moving counterclockwise around the closed path c. f(x,y) = (ex − 9y)i (ey 2x)j c: r = 2 cos()
The work done by the force F on a particle moving counterclockwise around the closed path C is π(\(e^4\) − 1).
To use Green's theorem to calculate the work done by the force F on a particle moving counterclockwise around a closed path C, we need to first calculate the curl of F:
curl F = (∂Ey/∂x − ∂(ex−9y)/∂y) k = (2ex − 9)k
where i, j, and k are the unit vectors in the x, y, and z directions, respectively.
Next, we need to parameterize the closed path C. In this case, the path is given by r = 2cos(θ), where θ varies from 0 to 2π. We can parameterize this path as:
x = 2cos(θ)
y = 2sin(θ)
We can then use Green's theorem to calculate the work done by F:
∮C F · dr = ∬R (curl F) · dA
where R is the region enclosed by C and dA is the area element.
Substituting in the values we have calculated, we get:
∮C F · dr = ∬R (2ex − 9)k · dA
The region R is a circle with radius 2, so we can use polar coordinates to evaluate the integral:
∬R (2ex − 9)k · dA = ∫θ=0 2π ∫r=0 2 (2e^(r cosθ) − 9)r dr dθ
Evaluating this integral, we get:
∮C F · dr = π(\(e^4\) − 1)
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We need to calculate the curl of the force, parameterize the path, and then use Green's theorem to evaluate the line integral to get work done by the force f on a particle that is moving counterclockwise around the closed path c.
To apply Green's theorem to calculate the work done by the force F on a particle moving counterclockwise around a closed path C, we first need to calculate the curl of F. We have:
curl F = (∂Ey/∂x − ∂(ex−9y)/∂y) k
= (2ex − 9)k
where k is the unit vector in the z direction.
Next, we need to parameterize the closed path C. In this case, the path is given by r = 2cos(θ), where θ varies from 0 to 2π. We can parameterize this path as:
x = 2cos(θ)
y = 2sin(θ)
We can then use Green's theorem to calculate the work done by F:
∮C F · dr = ∬R (curl F) · dA
where R is the region enclosed by C and dA is the area element.
Substituting the values we have calculated, we get:
∮C F · dr = ∬R (2ex − 9)k · dA
The region R is a circle with a radius of 2, so we can use polar coordinates to evaluate the integral:
∬R (2ex − 9)k · dA = ∫θ=0 2π ∫r=0 2 (2e^(r cosθ) − 9)r dr dθ
Evaluating this integral, we get:
∮C F · dr = π( − 1)
Therefore, the work done by the force F on a particle moving counterclockwise around the closed path C is π( − 1).
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Professor B's salary Professor A's salary is currently 69% less than Professor B's salary, so Professor A's salary is A 69%; Professor A's salary is 69% ot Professor B's salary B. 169% Professor A's salary is 100% of Professor E's salary plus another 69% C. 699% Professor A's salary is-69% of Professor B's salary. D. 31% Professor A'S salary is 100% of Professor B's salary minus 69% of Professor B's salary.
Professor A's salary is 31% of Professor B's salary(D).
Professor A's salary can be determined by combining the given information. We know that Professor A's salary is 69% less than Professor B's salary, which means it is 31% of Professor B's salary (100% - 69% = 31%).
Additionally, we are told that Professor A's salary is 169% of Professor E's salary plus another 69% (169% + 69% = 238%) and that Professor A's salary is -69% of Professor B's salary.
Combining these two equations, we can deduce that 238% of Professor E's salary equals -69% of Professor B's salary. Simplifying this, we find that Professor A's salary is 31% of Professor B's salary (238% - 69% = 169%, and 169% = 31% + 100%).
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I NEED HELP PLEASE !!!
begin{tabular}{|r|l|r|r|} \hline 3 & Below are your numerical inputs for the problem: \\ \hline 4 & Initial Cost (\$) & 390000 \\ \hline 5 & Year 1 Revenues (\$) & 192000 \\ \hline 6 & Year 1 Costs (\$) & 125000 \\ \hline 7 & Inflation & 2.75% \\ \hline 8 & Project Duration (years) & 6 \\ \hline 9 & Depreciation Method & \\ \hline 10 & Tax Rate & \\ \hline 11 & Net Working Capital (\% oft+1 Revenues) & MACRS \\ \hline 12 & Salvage Value (\$) & 28.00% \\ \hline 13 & Cost of Capital & 15.00% & 245000 \\ \hline \end{tabular} How much are the year 1 operating cash flows (OCF)? How much is the depreciation expense in year 3 ? What is the change in Net Working Capital (NWC) in year 2? What is the net cash flow from salvage (aka, the after-tax salvage value, or ATSV)? What is the project's NPV? Would you recommend purchasing the ranch? Briefly explain.
Information is needed to evaluate the project's financial viability, considering factors such as the initial investment, expected cash flows, cost of capital, and project duration.
To calculate the year 1 operating cash flows (OCF), we need to subtract the year 1 costs from the year 1 revenues:
OCF = Year 1 Revenues - Year 1 Costs
OCF = $192,000 - $125,000
OCF = $67,000
To find the depreciation expense in year 3, we need to determine the depreciation method. The provided information is incomplete regarding the depreciation method, so we cannot calculate the depreciation expense in year 3 without knowing the specific method.
The change in Net Working Capital (NWC) in year 2 can be determined by multiplying the Net Working Capital percentage (given as a percentage of t+1 revenues) by the year 1 revenues and subtracting the result from the year 2 revenues:
Change in NWC = (Year 2 Revenues - Net Working Capital percentage * Year 1 Revenues) - Year 1 Revenues
Without the specific Net Working Capital percentage or Year 2 Revenues values, we cannot calculate the exact change in NWC in year 2.
The net cash flow from salvage (ATSV) is calculated by multiplying the Salvage Value percentage by the Initial Cost:
ATSV = Salvage Value percentage * Initial Cost
ATSV = 28% * $390,000
ATSV = $109,200
To calculate the project's NPV, we need the cash flows for each year, the cost of capital, and the project duration. Unfortunately, the information provided does not include the cash flows for each year, so we cannot calculate the project's NPV.
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The complete question is:
Below are your numerical inputs for the problem: 4 & Initial Cost (\$) & 390000 5 & Year 1 Revenues (\$) & 192000 6 & Year 1 Costs (\$) & 125000 7 & Inflation & 2.75% 8 & Project Duration (years) & 6 9 & Depreciation Method & 10 & Tax Rate & 11 & Net Working Capital (\% oft+1 Revenues) & MACRS 12 & Salvage Value (\$) & 28.00% 13 & Cost of Capital & 15.00% & 245000 How much are the year 1 operating cash flows (OCF)? How much is the depreciation expense in year 3 ? What is the change in Net Working Capital (NWC) in year 2? What is the net cash flow from salvage (aka, the after-tax salvage value, or ATSV)? What is the project's NPV? Would you recommend purchasing the ranch? Briefly explain.
A Purse is discounted at 45% off. The original price is $275.
What is the sales price?
O $146.25
O $142.50
O $151.25
O $136.25
Therefore, the sales price is $151.25. So, the answer is (C) $151.25.
What is sale price?Sale price refers to the price of a product or service that has been reduced from its original or regular price. It is usually offered as a discount or promotion to attract customers and increase sales. The sale price can be expressed as a percentage or a dollar amount off the regular price. For example, a shirt with a regular price of $50 might be put on sale for 20% off, making the sale price $40.
To find the sales price, we need to apply the discount to the original price:
Discount amount = 45% of $275 = 0.45 x $275 = $123.75
\(Sales price = Original price - Discount amount\)
\(Sales price = $275 - $123.75 = $151.25\\)
Therefore, the sales price is $151.25. So, the answer is (C) $151.25.
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Finding the side length of a cube from its Volume in liters A technical machinist is asked to build a cubical steel tank that will hold 275 L of water. Calculate in meters the smallest possible inside length of the tank. Round your answer to the nearest 0.001 m. X 5 ?
The smallest possible inside length of the cubical steel tank that can hold 275 liters of water is approximately 0.640 meters.
The side length of the cube is found by converting the volume of water from liters to cubic meters, as the unit of measurement for the side length is meters.
Given that the volume of water is 275 liters, we convert it to cubic meters by dividing it by 1000 (1 cubic meter = 1000 liters):
275 liters / 1000 = 0.275 cubic meters
Since a cube has equal side lengths, we find the side length by taking the cube root of the volume. In this case, we find the cube root of 0.275 cubic meters:
∛(0.275) ≈ 0.640
Rounded to the nearest 0.001 meters, the smallest possible inside length of the tank is approximately 0.640 meters.
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What is the integral of 1/(sinh(x)) ?
The solution to the integral is:\(∫ 1/(sinh(x)) dx = (1/2) ln|sinh(x) + sinh(x)^2| + C.\)
How we can find the solution of the integral?To find the integral of \(1/(sinh(x))\), we can use the substitution \(u = sinh(x)\), which implies\(du/dx = cosh(x)\) and\(dx = du/cosh(x)\). Substituting these into the integral, we get:
\(∫ 1/(sinh(x)) dx = ∫ 1/u * du/cosh(x)\)
\(= ∫ cosh(x)/u * du\)
Now, we can use the identity\(cosh^2(x) - sinh^2(x) = 1\)to replace\(cosh(x)^2\)with \((sinh(x)^2 + 1)\)in the integrand:
\(∫ (sinh(x)^2 + 1)/u * du\)
\(= ∫ sinh(x)^2/u * du + ∫ 1/u * du\)
The first integral can be rewritten using the identity \(sinh^2(x) = (cosh(2x) - 1)/2:\)
\(∫ (cosh(2x) - 1)/(2u) * du\)
\(= (1/2)∫ cosh(2x)/u * du - (1/2)∫ du/u\)
\(= (1/2) ln|u + sinh(x)| - (1/2) ln|u| + C\)
\(= (1/2) ln|sinh(x) + sinh(x)^2| + C\)
Therefore, the solution to the integral is:
\(∫ 1/(sinh(x)) dx = (1/2) ln|sinh(x) + sinh(x)^2| + C\)
In words, this means that the antiderivative of \(1/(sinh(x))\)is equal to one-half times the natural logarithm of the sum of\(sinh(x)\) and \(sinh(x)^2\), plus a constant of integration.
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a group of fifth graders collected 46 cans of food to donate to three food banks. If each food bank is get an equal number of cans, how many cans do the each receive
Answer:
about 15.3.
Step-by-step explanation:
46 divided by 3. ig