Answers
Answer:
-31/30
Step-by-step explanation:
-5/6 + 1/3 Times -3/5
-5/6 - 1/5
Related Questions
What is the formula for the volume of a rectangular prism?
Answers
Answer:
V = (l)(w)(h)
Step-by-step explanation:
This is essentially volume equals length times width times height.
V: Volume
L: Length
W: Width
H: Height
This means, that to find the volume of a rectangular prism, you multiply length times width times height. This also works if you have the values of volume and 2 values from length, width, or height, because you can rearrange the equation.
Please give me Brainliest :)Which graph best represents a function with a range of all real numbers greater than or equal to -3
Answers
Answer: Without the graphs posted in your question, unfortunately it is not possible for me to provide you an answer. Please include the graphs given in your question so that I can provide an accurate response.
evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) ∫ (3x^2 + 2x – 3) / (x^3 – x) dx
_____________
Answers
The integral ∫ (3x^2 + 2x – 3) / (x^3 – x) dx evaluates to:
-ln(|x|) + ln(|x - 1|) + ln(|x + 1|) + C, where C is the constant of integration.
To evaluate the integral ∫ (3x^2 + 2x – 3) / (x^3 – x) dx, follow these steps:
The denominator (x^3 - x) can be factored as x(x^2 - 1). Further, (x^2 - 1) can be factored as (x - 1)(x + 1). So, the denominator becomes x(x - 1)(x + 1).Perform partial fraction decomposition.
Write the integrand as a sum of partial fractions:
(3x^2 + 2x - 3) / (x(x - 1)(x + 1)) = A/x + B/(x - 1) + C/(x + 1) Clear the denominators.
Multiply both sides by x(x - 1)(x + 1) to get:
3x^2 + 2x - 3 = A(x - 1)(x + 1) + Bx(x + 1) + Cx(x - 1)Solve for A, B, and C.
By comparing coefficients or plugging in suitable x values, we find:
A = -1, B = 1, and C = 1 Rewrite the integral.
Now the integral becomes:
∫ (-1/x + 1/(x - 1) + 1/(x + 1)) dxEvaluate the integral.
The integral can be evaluated by finding the anti-derivative of each term:
∫ (-1/x) dx = -ln(|x|) + C1
∫ (1/(x - 1)) dx = ln(|x - 1|) + C2
∫ (1/(x + 1)) dx = ln(|x + 1|) + C3
Combine all three anti-derivatives and the constants:
- ln(|x|) + ln(|x - 1|) + ln(|x + 1|) + C
Thus, the integral ∫ (3x^2 + 2x – 3) / (x^3 – x) dx evaluates to:
-ln(|x|) + ln(|x - 1|) + ln(|x + 1|) + C, where C is the constant of integration.
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Below is a table giving the fox and rabbit populations in a national park
during a 12-month period. Note that each value of t corresponds to the beginning of
the month
According to the data in the table, is F a function of R? Is R a function of F? Explain.
Is either R or Fa function of t? Explain.
Answers
The table only gives us information about the values of R and F at certain points in time, but it does not give us a functional relationship between the two variables. To determine whether F is a function of R, we would need to know the number of foxes for each value of R.
What is function ?A function in mathematics is a rule that converts each element from one set, known as the domain, into a particular element in another set, known as the range. The domain of a function is the set of all potential inputs, whereas the range is the set of all potential outputs. A graph, a table of values, or an equation can all be used to describe a function. In science, engineering, economics, and other disciplines, functions are used to represent the connections between variables.
Neither R nor F is a function of t because there are multiple values of R and F for some values of t. For example, at t=2, the number of rabbits is 750, and at t=3, the number of rabbits is 567.
From the data in the table, we cannot determine whether F is a function of R or whether R is a function of F. To determine whether F is a function of R, we would need to know the number of foxes for each value of R. Similarly, to determine whether R is a function of F, we would need to know the number of rabbits for each value of F. The table only gives us information about the values of R and F at certain points in time, but it does not give us a functional relationship between the two variables.
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A player kicks a football off the ground so that if travels with a velocity of 32 miles per hour at an angle of 34° with the ground. Find the magnitude of the horizontal and vertical components.
Answers
As a result, the horizontal component of velocity has a magnitude of 11.8 m/s\ while the vertical component of velocity has a value of 8.0 m/s.
What is velocity?The pace at which an item changes its position is described by a vector quantity called velocity. It is described as the rate at which displacement changes in relation to time The amount and direction of velocity are both present. Speed refers to the velocity's magnitude.
The football's velocity in this scenario is 32 miles per hour, or around 14.3 meters per second.
We can use trigonometry to determine the sizes of a velocity vector's horizontal and vertical components.
Vₓ = V cos (theta), where V is the magnitude of the velocity vector and theta is the angle that the velocity vector makes with the horizontal axis, gives the horizontal component of velocity.
Vy = V sin (theta), where V is the magnitude of the velocity vector and theta is the angle that the velocity vector makes with the horizontal axis, gives the vertical component of velocity.
In this instance, we are aware of the football's velocity, which is 32 miles per hour, or around 14.3 meters per second6. It makes a 34 degree 6 angle with the ground.
Therefore,
Vₓ= V cos (theta) * cos (34 degrees) = 14.3 m/s * 11.8 m/s
Vy = V sin (theta) * sin (34 degrees) = 14.3 m/s * 8.0 m/s
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fix her error, please help!!!
Answers
The trigonometric ratios of sine, cosine, and tangent can be used to find the lengths of the sides of the triangle.
How do you solve the right triangles?A right triangle has one angle that measures exactly 90 degrees. Identify this angle in the triangle.
Label the sides of the triangle as the hypotenuse, adjacent, and opposite sides, relative to one of the non-right angles. The hypotenuse is always the side opposite the right angle.
We know that;
Tan 45 = 5.9/w
w = 5.9/Tan45
w = 5.9
Then;
Tan 35 = 5.9/w + x
0.7 = 5.9/5.9 + x
0.7(5.9 + x) = 5.9
4.13 + 0.7x = 5.9
x = 2.5
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or differentiable parameterization of arc-length 2 in space ( not in a plane) how would you set up the function?
Answers
The function of differentiable parameterization of arc length 2 in space is ∫√(1)² + (2t)² + (cos(t))² dt.
To set up a differentiable parameterization of arc-length 2 in space, we can use the arc-length formula:
s = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
where s is the arc length, t is the parameter, and (x, y, z) is the position vector.
We want the arc length to be 2, so we can set up the following equation:
2 = ∫√(dx/dt)² + (dy/dt)² + (dz/dt)² dt
We can then choose a function for each component of the position vector, such as:
x = t
y = t²
z = sin(t)
We can now find the derivatives with respect to t:
dx/dt = 1
dy/dt = 2t
dz/dt = cos(t)
We can substitute these into the arc-length formula:
2 = ∫√(1)² + (2t)² + (cos(t))² dt
Solving this integral for t will give us the desired parameterization of arc-length 2 in space. However, this integral may not have a closed-form solution, so numerical methods may need to be used to approximate the solution.
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The question is -
Differentiable parameterization of arc-length 2 in space ( not in a plane) how would you set up the function?
the diameters of cylinders form a normal distribution with the mean of 5 cm and the standard deviation of 0.2 cm. katie ordered a cylinder from this population and the store manager told her that the cylinder she ordered is bigger than average in size. the manager specifically mentioned that her cylinder would be in the upper 10% in its size (diameter). what is the possible lowest diameter of katie's cylinder?
Answers
The possible lowest diameter of Katie's cylinder is approximately 5.26 cm.
To find the possible lowest diameter of Katie's cylinder, we'll use the concepts of normal distribution, mean, and standard deviation.
1. Recall the given information:
- Mean diameter (µ) = 5 cm
- Standard deviation (σ) = 0.2 cm
- Katie's cylinder is in the upper 10% in size (diameter).
2. Determine the z-score that corresponds to the upper 10% of the distribution:
To find the z-score for the upper 10%, you can use a z-table or a calculator with a built-in function to find the inverse of the cumulative distribution function (commonly known as the "inverse CDF" or "quantile" function). The z-score corresponding to the upper 10% is approximately 1.28.
3. Calculate the possible lowest diameter of Katie's cylinder using the z-score formula:
z = (X - µ) / σ
where:
- z is the z-score
- X is the diameter we want to find
- µ is the mean diameter
- σ is the standard deviation
4. Rearrange the formula to solve for X:
X = z * σ + µ
5. Plug in the values and solve for X:
X = (1.28 * 0.2) + 5
X = 0.256 + 5
X ≈ 5.26 cm
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the function f(x)=-5x^2 + 17x + 21 models the predicated sales of a pen after x price increase. use the table showing actual and predicted sales to find the residual for the models
Answers
To find the residual for each data point, we subtract the predicted sales from the actual sales. The residuals for the model are 1, 0, 1, and 1 for x=0, 1, 2, and 3 respectively.
To find the residual for each data point, we need to subtract the predicted sales from the actual sales:
For x=0: Residual = Actual sales - Predicted sales = 22 - 21 = 1
For x=1: Residual = Actual sales - Predicted sales = 33 - 33 = 0
For x=2: Residual = Actual sales - Predicted sales = 36 - 35 = 1
For x=3: Residual = Actual sales - Predicted sales = 28 - 27 = 1
Therefore, the residuals for the model are 1, 0, 1, and 1 for x=0, 1, 2, and 3 respectively.
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I have a question,what is 3x + 2 = 17
Answers
Answer: x=5
Step-by-step explanation:
First you gotta box the variable so you don't get confused. The opposite of adding 2 is subtracting so you subtract 2-2 which gives you 0 and subtract 17-2 which gives 15. THEN, you will have left 3x=15 in order to get rid of the three you have to divide by the 3 on both sides. Lastly, you get your answer! Hope this helps :D
How many square centimeters are in the total surface area
of this cylinder? Express your answer to 2 decimal places?
4.0 cm
2.5 cm
Answers
Total surface area of this cylinder, the answer is (C) 163.36 cm².
What is an area ?
The total surface area of a cylinder can be found using the formula:
surface area = 2πr² + 2πrh
where r is the radius and h is the height.
Substituting the given values, we get:
surface area = 2π(4.0)² + 2π(4.0)(2.5)
= 2π(16.0) + 2π(10.0)
= 32π + 20π
= 52π
Using the approximation 3.14 for π and rounding to 2 decimal places, we get:
surface area ≈ 52(3.14) ≈ 163.36 cm²
Therefore, the answer is (C) 163.36 cm².
What is cylinder?
A cylinder is a three-dimensional shape that has two circular bases that are parallel and congruent to each other. The sides of a cylinder are curved, and they connect the bases. A cylinder has a constant cross-sectional area throughout its height, and it is a type of prism. The height of a cylinder is the perpendicular distance between the bases.
The volume of a cylinder can be calculated using the formula V = πr²h, where r is the radius of the base and h is the height of the cylinder. The surface area of a cylinder can be calculated using the formula A = 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder. Cylinders are commonly found in everyday objects such as cans, pipes, and containers.
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Emily got her monthly credit card statement, which showed a bill of $156. She paid the minimum payment due, which was $30. If the interest rate was 15.5%, then Emily had to pay interest on the amount of $ _________ and the amount of interest to be paid would be __________.
Answers
Emily had to pay interest on the amount of $126 and the amount of interest to be paid would be $16.41.
What is interest rate?
Interest rate is the percentage charged by a lender to a borrower for the use of money, usually expressed as an annual percentage of the principal borrowed.
Emily had to pay interest on the remaining balance after making the minimum payment of $30.
Remaining balance = $156 - $30 = $126
The amount of interest to be paid would be calculated as:
interest = (interest rate/12) * remaining balance
Where 12 is the number of months in a year.
So,
interest = (15.5/12) * $126 = $16.41
Therefore, Emily had to pay interest on the amount of $126 and the amount of interest to be paid would be $16.41.
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Max went to the store and bought a new iPad that was 45% of the original price he saved $135 because of the sale how much was the original price of the iPad
Answers
Answer:
$245.45
Step-by-step explanation:
.45c = $135, so c = $245.45
.45 × $245.45 = $110.45, a $135 savings.
Razak's mother weighs, 64.8kg. If she weighs 4 times the weight of Razak, What is Razak's weight?
Answers
Razak weighs 16.2 kg. To find Razak's weight, we need to use the information given in the problem that his mother weighs 4 times his weight.
Let's use the variable "R" to represent Razak's weight in kilograms. According to the problem, Razak's mother weighs 4 times Razak's weight, so we can write an equation:
Mother's weight = 4 * Razak's weight
64.8 = 4R
To solve for R, we can divide both sides by 4:
64.8 / 4 = R
16.2 = R
Therefore, Razak weighs 16.2 kg.
It's important to note that in solving this problem, we assumed that the weight of Razak's mother was accurately given in the problem. Additionally, we assumed that the units of measurement were in kilograms. If the problem used different units, we would need to convert them to kilograms before solving the equation.
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M and N are the mid point of opposite sides of a square ABCD.a point is selected randomly in the square find the probability that it lies
1. in ∆ADM
2.in ∆ADM but not ∆ADN
3neither in ∆ADM nor in ∆ADN
Answers
The probability of selecting a point
1) in ΔADN is also 1/8.
2) in ΔADM but not in ΔADN is 0.
3) lies neither in ΔADM nor in ΔADN is 3/4.
What is probability?
Probability is a measure of the likelihood or chance of an event occurring. It is a number between 0 and 1, with 0 representing an impossible event and 1 representing a certain event. The probability of an event is calculated by dividing the number of ways the event can occur by the total number of possible outcomes.
To solve this problem, we need to use the fact that the diagonals of a square bisect each other and are perpendicular. Let's denote the midpoint of AB as P and the midpoint of BC as Q.
To find the probability that a point selected randomly in the square lies in ΔADM, we need to find the ratio of the area of ΔADM to the area of the square ABCD.
Since M is the midpoint of AD, we have DM = AM = 1/2 * AB. Similarly, since N is the midpoint of CD, we have DN = CN = 1/2 * BC.
Thus, the area of ΔADM is (1/2 * DM * AM) = (1/2 * 1/2 * AB * 1/2 * AB) = 1/8 * AB². The area of the square is AB². Therefore, the probability of selecting a point in ΔADM is (1/8 * AB²) / AB² = 1/8.
To find the probability that a point selected randomly in the square lies in ΔADM but not in ΔADN, we need to subtract the probability of selecting a point in ΔADN from the probability of selecting a point in ΔADM. Since N is the midpoint of CD, we have AN = 1/2 * AB, and therefore DN = 1/2 * BC = 1/2 * AB.
Thus, the area of ΔADN is (1/2 * DN * AN) = (1/2 * 1/2 * AB * 1/2 * AB) = 1/8 * AB², which is the same as the area of ΔADM.
Therefore, the probability of selecting a point in ΔADN is also 1/8.
The probability of selecting a point in ΔADM but not in ΔADN is (1/8 * AB²) - (1/8 * AB²) = 0.
To find the probability that a point selected randomly in the square lies neither in ΔADM nor in ΔADN, we need to subtract the probability of selecting a point in ΔADM from the probability of selecting a point in the square and then subtract the probability of selecting a point in ΔADN from that result.
The probability of selecting a point in the square is 1.
The probability of selecting a point in ΔADM is 1/8, and the probability of selecting a point in ΔADN is also 1/8.
Therefore, the probability of selecting a point that lies neither in ΔADM nor in ΔADN is 1 - (1/8 + 1/8) = 3/4.
Hence, the probability of selecting a point
1) in ΔADN is also 1/8.
2) in ΔADM but not in ΔADN is 0.
3) lies neither in ΔADM nor in ΔADN is 3/4.
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What is 2^2000
2 to the power of 2000
Answer in number form. Not scientific notation or e notation.
Answers
Step-by-step explanation:
Not sure why you would want this...here is the result using a big number calculator on the internet:
2^2000 =
114,813,069,527,425,452,423,283,320,117,768,198,402,231,770,208,869,520,047,764,273,682,576,626,139,237,031,385,665,948,631,650,626,991,844,596,463,898,746,277,344,711,896,086,305,533,142,593,135,616,665,318,539,129,989,145,312,280,000,688,779,148,240,044,871,428,926,990,063,486,244,781,615,463,646,388,363,947,317,026,040,466,353,970,904,996,558,162,398,808,944,629,605,623,311,649,536,164,221,970,332,681,344,168,908,984,458,505,602,379,484,807,914,058,900,934,776,500,429,002,716,706,625,830,522,008,132,236,281,291,761,267,883,317,206,598,995,396,418,127,021,779,858,404,042,159,853,183,251,540,889,433,902,091,920,554,957,783,589,672,039,160,081,957,216,630,582,755,380,425,583,726,015,528,348,786,419,432,054,508,915,275,783,882,625,175,435,528,800,822,842,770,817,965,453,762,184,851,149,029,376
The time taken T (in seconds) for the length of rope R (in metres) to be rotated about a central point is given by the formula T = √8R\2.5
a) Make R the subject of the formula.
b) what is the length of rope if it takes 16 seconds for a single revolution?
c) what is the time taken for a rope of length 18 metres to complete a revolution? Answer correct to one decimal place.
Full working please <3
Answers
we have derived the formula T = √(8R/2.5) for the time taken for a rope of length R to be rotated about a central point, made R the subject of the formula, and used the formula to calculate the length of a rope that takes 16 seconds for a single revolution and the time taken for a rope of length 18 metres to complete a revolution.
How to solve questions?
a) To make R the subject of the formula, we need to isolate it on one side of the equation. We start by rearranging the formula:
T = √(8R/2.5)
Squaring both sides of the equation, we get:
T²= 8R/2.5
Multiplying both sides by 2.5, we get:
2.5T²= 8R
Finally, dividing both sides by 8, we get:
R = 2.5T²/8
Therefore, R is given by the formula R = 2.5T²/8.
b) To find the length of rope if it takes 16 seconds for a single revolution, we can substitute T = 16 into the formula we derived in part (a):
R = 2.5T²/8 = 2.5(16)²/8 = 100
Therefore, the length of the rope is 100 metres.
c) To find the time taken for a rope of length 18 metres to complete a revolution, we can again use the formula we derived in part (a), but this time we substitute R = 18:
T = √(8R/2.5) = √(8*18/2.5) = √(115.2) ≈ 10.7 seconds
Therefore, the time taken for a rope of length 18 metres to complete a revolution is approximately 10.7 seconds, rounded to one decimal place.
In summary, we have derived the formula T = √(8R/2.5) for the time taken for a rope of length R to be rotated about a central point, made R the subject of the formula, and used the formula to calculate the length of a rope that takes 16 seconds for a single revolution and the time taken for a rope of length 18 metres to complete a revolution.
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P ( rolling a composite number and the coin landing on heads and tails )
Answers
Assuming a standard six-sided die and a fair coin, the probability of rolling a composite number (any number that is not a prime number) on the die is 3/6 or 1/2, since there are three composite numbers (4, 6, 8) out of six possible outcomes (1, 2, 3, 4, 5, 6).
The probability of the coin landing on heads is 1/2, and the probability of it landing on tails is also 1/2. Therefore, the probability of both events happening together is:
P(composite number and heads and tails) = P(composite number) x P(heads) x P(tails)
P(composite number and heads and tails) = (1/2) x (1/2) x (1/2)
P(composite number and heads and tails) = 1/8
So the probability of rolling a composite number and the coin landing on heads and tails is 1/8.
Find the least-squares regression line y^=b0+b1x through the point
(−3,0),(2,9),(5,13),(9,19),(10,23).
For what value of x is =0?
Answers
Answer:
3
Step-by-step explanation:
On your own sheet of paper, make a stem-and-leaf plot of the following set of data and then find the range of the data.
83, 71, 62, 86, 90, 95, 61, 60, 87, 72, 95, 74, 82, 54, 99, 62, 78, 76, 84, 92.
can I get the range in a graph form?
Answers
The stem-and-leaf plot reveals that 54 and 99 are the smallest and biggest values, respectively. Hence, the range is: range = 99 - 54 = 45
what is range ?The range of a number collection or data points in mathematics is the difference here between biggest and smallest values. It is a measurement of the data's dispersion or variability. The range is frequently employed in descriptive statistics and can serve to provide information about the data's distribution.
given
Certainly! The data are shown in a stem-and-leaf plot as follows:
5 | 4
6 | 0 1 2 2
7 | 1 2 4 6 8
8 | 2 3 4 4 6 7
9 | 0 5 5 9
The difference between the largest and smallest values is the data's range.
The stem-and-leaf plot reveals that 54 and 99 are the smallest and biggest values, respectively. Hence, the range is: range = 99 - 54 = 45
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Graph triangle DEF with vertices D (6,2), E (6,6), and F (10,4) and its image after a dilation centered at the origin with a scale factor of 1/2. Then find the difference between the area of the image and the area of the preimage.
Answers
Answer:
The graph is below.
The area of the pre-images is 8 and the area of the image is 2. The difference is 6 [tex]units^{2}[/tex]
Step-by-step explanation:
Area of the large triangle
a = [tex]\frac{bxh}{2}[/tex] I am going to choose the vertical line to be the base because I can count the base to be 4 and the height straight up from the base to be 4.
a = [tex]\frac{4x4}{2}[/tex] = 8
I found the base for the image in the same way
a = [tex]\frac{2x2}{2}[/tex] 2
Helping in the name of Jesus.
dominique is building a flower box for her yard. the outline of the box is shown in the diagram. if she wants to cover the box with burlap to protect it, how much burlap does she need to buy?
Answers
Once we have the total surface area, we will know how much burlap Dominique needs to buy to cover the flower box.
To determine how much burlap Dominique needs to buy to cover the flower box, we need to calculate the surface area of the box. Unfortunately, without the specific dimensions or a provided diagram,
I cannot give you an exact number.
However, here are the general steps to calculate the surface area of a rectangular flower box:
1. Identify the dimensions: length (L), width (W), and height (H).
2. Calculate the surface area of each face:
- Top/Bottom: 2 x (L x W)
- Sides: 2 x (L x H)
- Front/Back: 2 x (W x H)
3. Add the surface areas of all faces together to get the total surface area.
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explain what function (in mosaic ) and what parameters you should you use to find the value of z* for a specified confidence level c
Answers
This would give you a z* value of approximately 1.96.
To find the value of z* for a specified confidence level c in a mosaic function, you should use the inverse cumulative
distribution function (also called the quantile function or the z-score function). The parameters you need are the
desired confidence level (c) and the standard normal distribution.
Here are the steps to find the value of z*
1. Determine the desired confidence level (c). This is usually given as a percentage, like 95% or 99%.
2. Calculate the area under the curve corresponding to the confidence level. This is equal to (1 - c)/2.
3. Use the inverse cumulative distribution function (quantile function) with the standard normal distribution to find the z
score corresponding to the calculated area.
4. The resulting value is the z* value for the specified confidence level c.
For example, if you need to find the z* value for a 95% confidence level, you would calculate (1 - 0.95)/2 = 0.025, and
then use the inverse cumulative distribution function with the standard normal distribution to find the z-score
corresponding to 0.025. This would give you a z* value of approximately 1.96.
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8. A square and its dimensions are shown below. What is the perimeter of the square?
Answers
Step-by-step explanation:
Perimeter is the sum of all of the sides
add all of the 3x + 3 's
3x+3 + 3x +3 + 3x+3 + 3x + 3 = 12x + 12 units
the study shows that tayugineans drink an average of 1.64 cups of coffee per day with a variance of 0.0576. if five hundred tayugenians are selected, approximately how many of them will drink less than 1 cup of coffee per day?
Answers
If Tayugineans drink an average of 1.64 cups of coffee per day, then 0.38 percent will drink less than 1 cup-of-coffee per day.
The average number of cups of coffee drink is = 1.64 cups,
The variance is = 0.0576, So, the standard-deviation is = √0.0576 = 0.24,
Let us assume that the variable is normally-distributed with a standard deviation of 0.24 cup.
We have to find the persons who drink less than 1 cup of coffee,
It is calculated as : P(x < 1) ,
The corresponding z-value is = z = (1 - 1.64)/0.24 = -2.6667 ,
So, P(z < -2.67) = 0.0038 = 0.38%.
Therefore, 0.38% drink less than 1 cup of coffee per day.
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Pls im begging you do my math hw I just got back from a track meet
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Answer:
x = 71
Step-by-step explanation:
180 degrees is how many degrees were in a triangle.
Add up the angles then subtract by 180 and there is your answer.
Faith kicks a football. The graph below shows the height of the football ℎ h in feet after � t seconds.
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Answer:
faith
Step-by-step explanation:
you are a Beautiful human being
Pls help! This is due today!
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Answer:
3(a - 1) / (a + 1)
Step-by-step explanation:
I double checked, its right
I need to find the general equation for the circle that passes through the given points
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The general equation for a circle with center (a,b) and radius r is:
[tex](x-a)^2 + (y-b)^2 = r^2[/tex]. By substituting the values and simplifying the equation, we will get 0.
What is circle ?
In math, a circle is a closed, two-dimensional shape made up of all elements that are spaced uniformly apart from the centre. The radius of something like the circle refers to this distance. Often, the letter "O" or "" is used to represent a circle. The circle is really a fundamental geometric shape and includes a number of significant characteristics, including its radius, which is the distance it around circle, or its area, which is the volume of the circle's interior. A circle's area is equal to 2 times its radius square, and its circumference is equal to 2 times its radius.
To find the equation for the circle that passes through the points (-5,0), (0,4), and (2,4), we need to find the center (a,b) and radius r.
First, we find the midpoint of the line segment between (-5,0) and (0,4), which is:
[tex]((-5+0)/2, (0+4)/2) = (-2.5,2)[/tex]
Next, we find the midpoint of the line segment between (0,4) and (2,4), which is:
[tex]((0+2)/2, 4/2) = (1,2)[/tex]
Since the circle passes through all three points, its center must be equidistant from all three points. Using the distance formula, we can set up three equations:
[tex](-2.5 - a)^2 + (2 - b)^2 = r^2[/tex]
[tex]a^2 + (4 - b)^2 = r^2[/tex]
[tex](1 - a)^2 + (2 - b)^2 = r^2[/tex]
Simplifying the equations and setting them equal to each other, we get:
[tex](-2.5 - a)^2 + (2 - b)^2 = a^2 + (4 - b)^2 = (1 - a)^2 + (2 - b)^2[/tex]
Expanding the squares and simplifying, we get:
[tex]6a - 10 = 0[/tex]
[tex]2b - 12 = 0[/tex]
Solving for a and b, we get:
a = 5/3
b = 6
Now we can find the radius of the circle by plugging in the values of a and b into one of the equations we set up earlier:
[tex]a^2 + (4 - b)^2 = r^2[/tex]
[tex](5/3)^2 + (4 - 6)^2 = r^2[/tex]
[tex]25/9 + 4 = r^2[/tex]
[tex]r^2 = 61/9[/tex]
Therefore, the equation of the circle is:
[tex](x - 5/3)^2 + (y - 6)^2 = 61/9[/tex]
Simplifying, we get:
[tex]4x^2 + 4y^2 - 40x + 48y + 167 = 0.[/tex]
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A rectangle garden has one side that is 6 feet longer than the other. A lawn care company installs synthetic grass that costs $8 per square foot in the entire.
If x represents the length, in feet, of the shorter side of the garden, and y represents the total cost, in dollars, to install the synthetic grass, which equation correctly shows the relationship between x and y?
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Answer:
Step-by-step explanation:
The area of the rectangle garden can be expressed as:
Area = Length × Width
Since one side is 6 feet longer than the other, we can express the length as:
Length = Width + 6
Substituting this expression for length into the area formula, we get:
Area = (Width + 6) × Width
Simplifying this expression, we get:
Area = Width² + 6Width
The cost to install synthetic grass is given as $8 per square foot, so the total cost y can be expressed as:
y = 8(Area)
Substituting the expression for area that we found earlier, we get:
y = 8(Width² + 6Width)
Simplifying this expression, we get:
y = 8Width² + 48Width
Therefore, the equation that correctly shows the relationship between x (the length of the shorter side of the garden) and y (the total cost to install synthetic grass) is:
y = 8x² + 48x