The height of the pile is increasing at a rate of approximately 0.056 feet per minute when the pile is 12 feet high.
We are given that sand is falling off at a rate of 14 cubic feet per minute, and the diameter of the base of the cone is approximately three times the altitude. Let h be the height of the pile at a given time t, then we can write the volume of the cone as V = (1/3)πr^2h, where r is the radius of the base of the cone. Since the diameter is three times the altitude, we have r = 3h/2.
Taking the derivative of the volume with respect to time, we get dV/dt = (1/3)π(9h^2/4)(dh/dt). Plugging in the given values, we get:
14 = (1/3)π(9h^2/4)(dh/dt)
Simplifying, we get:
dh/dt = 56/(3πh^2)
When the height of the pile is 12 feet, the rate of change of the height of the pile is:
dh/dt = 56/(3π(12)^2) ≈ 0.056 ft/min.
Therefore, the height of the pile is changing at a rate of approximately 0.056 ft/min when the pile is 12 feet high.
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A 90% confidence interval for the mean
of a population is computed from a random sample and is found to be 90 ±
30.
The 95 % confidence interval based on the same data is 90 ± 30. Option b is right choice.
To calculate the 95% confidence interval based on the same data, we need to use the formula:
CI = [tex]\bar x[/tex] ± zα/2 x (σ / √n)
where:
[tex]\bar x[/tex] is the sample mean
zα/2 is the z-score corresponding to the desired confidence level (in this case, 95% confidence level, so zα/2 = 1.96)
σ is the population standard deviation (which is unknown, so we'll use the sample standard deviation instead)
n is the sample size
Given that the 90% confidence interval is 90 ± 30, we can conclude that:
[tex]\bar x[/tex] = 90
the margin of error = 30 / 1.645 (corresponding to the z-score for 90% confidence level) = 18.22
the sample standard deviation is unknown, so we cannot determine the exact value of σ.
Therefore, the 95% confidence interval can be calculated as follows:
CI = 90 ± 1.96 x (18.22 / √n)
To find the answer choice that could be the 95% confidence interval, we need to check which one contains the range of values that can be obtained using this formula for different sample sizes.
a. 90 ± 21: This range is too narrow to be a 95% confidence interval. It corresponds to a margin of error of
21 / 1.96 = 10.71
which is smaller than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).
Therefore, this option is not correct.
b. 90 ± 30: This is the same range as the 90% confidence interval, which we already know is correct.
Therefore, this option is also correct.
c. 90 ± 39: This range is too wide to be a 95% confidence interval. It corresponds to a margin of error of
39 / 1.96 = 19.90
which is larger than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).
Therefore, this option is not correct.
d. 90 ± 70: This range is also too wide to be a 95% confidence interval. It corresponds to a margin of error of 70 / 1.96 = 35.71, which is much larger than the margin of error for the 90% confidence interval (which was 30 / 1.645 = 18.22).
Therefore, this option is not correct.
Therefore, the correct answers are b. 90 ± 30.
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Question:
A 90% confidence interval for the mean μ of a population is computed from a random sample and is found to be 90 ± 30. Which of the following could be the 95 % confidence interval based on the same data?
a. 90 ± 21
b. 90 ± 30
c. 90 ± 39
d. 90 ± 70
I NEED HELP ON THIS ASAP!!
THE GRAPH SHOWING (X=2,5) (Y=10,9) (Z=6,1) Thus, taking YZ as the base, the area of the triangle XYZ is 2√(98)square units.
DEFINE THE TRIANGLE'S AREA?The total area filled by a triangle's three sides in a two-dimensional plane is what is known as the triangle's area. The basic formula for calculating a triangle's area is A = 1/2× b× h, which equals half the product of the triangle's base and height.
We can use the following formula to determine a triangle's area given its coordinates:
Area is equal to half of |x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|.
where the triangle's vertices' coordinates are (x₁,y₁), (x₂,y₂), and (x₃,y₃).
As YZ serves as our base in this instance, we must first determine its length. When we apply the distance formula, we obtain:
√((1-9)2 + (6-1)2) Equals √YZ (98)
Finding the perpendicular distance from X to YZ will allow us to determine the height of the triangle now that we know YZ. This distance is equivalent to 2-6 = -4, which is the x-coordinate of X less the x-coordinate of Z.
As a result, the triangle's area is:
Area is equal to 1/2 × YZ × height
= 1/2×√(98)× |-4|
= 2 √ (98)
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a student has scores of 84%, 71%, 86%, and 73% on four exams. what grade does the student need on the next exam to have an overall mean of 80%?
Answer:
Step-by-step explanation:
So, first you add the four exam percentage which will be 314 then you multiply 80 and 5. We use 5 because there are total 5 exams which will give the mean of 80%. After you get the answer for 80x5 which will be 400. Then you do, 400 - 314 which will be 86. So, the answer is 86%. To figure out if that answer is right, You add all the scores percentage and divide that by 5 and you should get 80%.
To have an overall mean of 80% after five exams, the student needs to score 88% on the next exam.
To find the student's overall mean, we need to add up all the scores and divide by the number of exams.
We have 4 exams scores:84%, 71%, 86%, and 73%.
Therefore, the total of the four exam scores is:
84% + 71% + 86% + 73% = 314%
To find the overall mean after five exams, the total score after five exams should be 5 × 80% = 400%.
Therefore, to have an overall mean of 80%, the student must score (400 - 314)% on the fifth exam, which is equivalent to: 86%.
Hence, the student needs to score 86% on the next exam to have an overall mean of 80%.
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PLEASE HELP At a school picnic, 1 junior and 1 senior will be selected to lead the activities. If there are 125 juniors and 100 seniors at
the picnic, how many different 2-person combinations of 1 junior and 1 senior are possible?
As a result, there are 25,200 potential 2-person combinations that can be made up of a junior and a senior.
FACTORIAL: WHAT IS IT?
In mathematics, n is used to represent the factorial of a non-negative number, n! n less than or equal to and are said to be the product of all positive integers. For instance:
5! = 5 x 4 x 3 x 2 x 1 = 120
Several branches of mathematics, including algebra, mathematical analysis, and combinatorics, use the factorial function.
It is used to determine how many different ways there are to arrange given objects.
Using the formula for combinations, we can determine how many different 2-person combinations with a junior and a senior are conceivable.
nCr = n / r!(n-r)!
where r is the number of persons we wish to choose from (2), n is the total number of people (125 juniors and 100 seniors), and! stands for the factorial function.
Adding these values to the formula yields the following results:
(125 plus 100)C2
= (125 plus 100)! / 2!(125 plus 100-2)!
= (225)! / 2!(223)!
= (225 x 224 x 223!) / (2 x 1 x 223!)
= (225 x 224)! / 2
= 25,200
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C is the midpoint of BD. Are these triangles similar, congruent or neither? What theorem supports your answer?
options:
Similar: SSS
Similar: SAS
Similar: HL
Congruent: AAS
Congruent: SAS
Congruent: HL
Neither
The given triangles are congruent by AAS congruence. The solution has been obtained by using the concept of congruent triangles.
What are congruent triangles?
Triangles which have the same size and the same shape are said to be congruent. This implies that the respective sides and angles of the triangles are both equal.
We are given two triangles as ABC and CDE in which C is the midpoint of BD.
So, from this, we get
⇒ AC = EC (Given)
⇒ ∠ABC = ∠CDE (Right Angle)
⇒ ∠ACB = ∠ECD (Common Angle)
Hence, the triangles are congruent by the AAS congruence.
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what is the equivalent of 12 to the power of 4?
Answer:
20736
Step-by-step explanation:
12*12*12*12=20736
The diagram shows a triangular prism. What is the area of the prisms base?
What is the answer to the question?
You can follow the order of operations with all of the other operations in the equation and treat the operations in the expression separately. Option 2.
Order of operationsThe order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed in an expression. These rules are designed to ensure that the expression is evaluated correctly and consistently.
When interpreting an expression as a given quantity, you are essentially substituting a value for a variable in the expression. This does not contradict the order of operations because the rules still apply, but the variables in the expression are treated as known values rather than unknown variables.
For example, if you have the expression 2 + 3 × 4, following the order of operations, you would first perform the multiplication (3 × 4 = 12) and then add the result to 2 (2 + 12 = 14).
If you are asked to interpret this expression as a given quantity when x = 3, you would substitute 3 for x and get 2 + 3 × 4 = 14, which is the same result obtained by following the order of operations.
In short, interpreting an expression as a given quantity does not contradict the order of operations because the rules still apply, but the variables in the expression are treated as known values.
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Help what’s 5/8 divided by 6
Answer:
5/48
Step-by-step explanation:
the first thing you would do is change 6 into the reciprocal, which is 1/6. After, you multiply the fractions just like that.
hope this helps!! :3
What is the value of 1/4+1/8+1/2?
A. 3/8
B. 4/8
C. 6/8
D. 7/8
Answer:
D
Step-by-step explanation:
Convert all of the fractions' denominators so that they are 8. This is because 4,2, and 8's least common multiple is 8.
Remember that when multiplying the denominator, we must also multiply the numerator. So:
[tex]\frac{1}{2} * \frac{4}{4} = \frac{4}{8} \\[/tex]
[tex]\frac{1}{4} *\frac{2}{2} =\frac{2}{8}[/tex]
Since we are asked to add all of them, we can do that easily since all of the fractions now share a common denominator.
[tex]\frac{2}{8} +\frac{4}{8} + \frac{1}{8} = \frac{7}{8}[/tex]
Therefore, the answer is D.
can someone help me with this please..........
Answer:
m = [tex]\frac{y-b}{x}[/tex]
Hope this helps!
Step-by-step explanation:
y = mx + b
mx = y - b
[tex]\frac{mx}{x}[/tex] = [tex]\frac{y-b}{x}[/tex]
m = [tex]\frac{y - b}{x}[/tex]
consider the toss of four coins. there are 24 possible outcomes of a single toss. develop a histogram of the number of heads (one side of the coin) that can appear on any toss. does it look like a normal distribution? should this be expected? what is the probability that three heads will appear on any toss?
From the histogram, we can see that there are 4 outcomes in which three heads appear. Therefore, the probability that three heads will appear on any toss is 4/16 = 0.25.
Consider the toss of four coins. There are 24 possible outcomes of a single toss. Develop a histogram of the number of heads (one side of the coin) that can appear on any toss. Does it look like a normal distribution? Should this be expected? What is the probability that three heads will appear on any toss?Solution:When four coins are tossed, there are 16 possible outcomes.
Each outcome has an equal probability of 1/16 of occurring. A histogram of the number of heads that can appear on any toss is shown below:Based on the histogram, the probability of getting 0 heads or 4 heads is 0.375, while the probability of getting 1 head or 3 heads is 0.25, and the probability of getting 2 heads is 0.125.
The histogram does not look like a normal distribution. This is because a normal distribution has a bell-shaped curve, which is not the case with this histogram. This is because there are a finite number of possible outcomes, and the probabilities of these outcomes are not normally distributed.
Thus, the histogram is not expected to look like a normal distribution .The probability that three heads will appear on any toss is the sum of the probabilities of the outcomes in which three heads appear.
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can we predict the heights of school-aged children from foot length? below is computer output from a regression analysis of this relationship for 15 randomly-selected canadian children from 8 to 15 years old, along with a residual plot. the explanatory variable is each child's foot length (in centimeters), and the response variable is the child's height (in centimeters).
The equation of the least-squares regression line based on these data is, y= 106.92 +2.044x where y =predicted height of child and x = child’s foot length.
To determine the equation of the least-squares regression line, we need to fit a linear model to the data using the method of least squares. The equation of the line can be written as:
height = intercept + slope * foot length
where the intercept and slope are the parameters we need to estimate from the data. The intercept represents the predicted height when foot length is 0, and the slope represents the change in height for each unit increase in foot length.
Substituting the values form the graph,
y= 106.92 +2.044x where y =predicted height of child and x = child’s foot length.
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--The complete question is, Can we predict the heights of school-aged children from foot length? below is computer output from a regression analysis of this relationship for 15 randomly-selected canadian children from 8 to 15 years old, along with a residual plot. the explanatory variable is each child's foot length (in centimeters), and the response variable is the child's height (in centimeters).
What is the equation of the least-squares regression line based on these data? Define any parameters used.--
What set of reflections and rotations would carry rectangle ABCD onto itself? Parallelogram formed by ordered pairs A at negative 4, 1, B at negative 3, 2, C at 0, 2, D at negative 1, 1. Rotate 180°, reflect over the x-axis, reflect over the line y = x Reflect over the x-axis, rotate 180°, reflect over the x-axis Rotate 180°, reflect over the y-axis, reflect over the line y = x Reflect over the y-axis, reflect over the x-axis, rotate 180°
The only transformation that would carry rectangle ABCD onto itself is a single rotation of 180 degrees in a parallelogram.
Rectangle ABCD's characteristics and how each transformation affects its orientation and location must be examined in order to identify the collection of reflections and rotations that would carry the rectangle onto itself of a parallelogram.
First, using the provided coordinates, let's determine the sides of the rectangle: AB is parallel to DC, and AD is parallel to BC. Also, we can observe that the lengths of AB and AD are equal to DC and BC, respectively, indicating that the opposite sides are congruent.
The rectangle is first suggested to be rotated 180 degrees. This transformation simply maps each point to its corresponding point on the rectangle; the rectangle's orientation is left unchanged. Hence, rectangle ABCD would be carried onto itself by this transformation alone.
The rectangle is first reflected over the x-axis, rotated 180 degrees, and then reflected once more over the x-axis is the second transformation suggested. This series of transformations flips the rectangle's orientation while mapping each point to its equivalent point on the other side of the rectangle. Hence, the rectangle ABCD would not be carried onto itself by this transformation sequence.
The third suggested transformation involves reflecting the rectangle first over the y-axis, then over the line y = x, and then back over the y-axis. The rectangle's orientation is maintained but its position is altered by this series of modifications. Hence, the rectangle ABCD would not be carried onto itself by this transformation sequence.
The rectangle is to be rotated 180 degrees, reflected over the line y = x, and then reflected over the x-axis as the fourth transformation suggested. Each point is moved and mapped to a point on the other side of the rectangle via this series of transformations. The rectangle's orientation is also altered, though. Hence, the rectangle ABCD would not be carried onto itself by this transformation sequence.
Hence, a single 180° rotation is the only transformation that could turn rectangle ABCD upon itself.
In conclusion, we studied the features of the rectangle and the consequences of each transformation on its orientation and location to find the set of reflections and rotations that would transport rectangle ABCD onto itself. In contrast to the other suggested transformations, we discovered that only a single rotation of 180 degrees would keep the rectangle's orientation and location intact.
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4. A person has a beginning balance of $3,320. She charges $1,260 on the 10th day, and
she charges $270 on the 22nd day. What is her account's average daily balance?
a.
b.
d.
e.
$4,283
$4,367
$4,231
$4,345
$4,226
०२८
Ook d
Answer: (D) $4,345.
Step-by-step explanation:
In rectangle EFGH, diagonals EG = 8x - 1
and FH = 3x +9
Answer:
The length of diagonal EG and FH in rectangle EFGH can be expressed as 8x - 1 and 3x + 9, respectively.
Step-by-step explanation:
In a rectangle, the diagonals are equal in length. Therefore, we can set 8x - 1 equal to 3x + 9 and solve for x:
8x - 1 = 3x + 9
5x = 10
x = 2
Now that we know x = 2, we can find the length of both diagonals:
EG = 8x - 1 = 8(2) - 1 = 15
FH = 3x + 9 = 3(2) + 9 = 15
Therefore, the length of both diagonals in rectangle EFGH is 15.
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The surface of a cylinder is represented , where r is the radius of the cylinder and h is its height. Factor the right side of the formula.
The formula for the surface area of a cylinder is:
S = 2πrh + 2πr²
To factor the right side of the formula, we can first take out the common factor of 2πr:
S = 2πr(h + r)
Therefore, the right side of the formula can be factored as 2πr(h + r).
I need Helpppp my teacher sucks at teaching
If the length breadth and height of 3x+y cm 2x+2y cm and 3y cm respectively and find its volume
Answer: The length, breadth, and height of the given object are 3x+y cm, 2x+2y cm, and 3y cm, respectively. To find the volume of this object, we need to multiply these three dimensions:
Volume = Length x Breadth x Height
= (3x+y) x (2x+2y) x (3y)
= 18x^2y + 27xy^2 + 6xy^2 + 6y^3
= 18x^2y + 33xy^2 + 6y^3
Therefore, the volume of the object is 18x^2y + 33xy^2 + 6y^3 cubic cm.
Step-by-step explanation:
pls answer need help
The equation of the graph are
y = -0.5x + 1How to find the equation of the graphsFor the straight line graph, the equation of a straight line is of the formula
y = mx + b
where m is the slope and b is the y intercept
solving for the slope, using (2, 0) and (6, -2)
m = (0 - -2) / (2 - 6)
m = 2 / -4
m = -0.5
Equation of a line of slope -0.5 passing through point (2, 0)
y - 0 = -0.5(x - 2)
y = -0.5x + 1
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can yall help me this please?
Answer:
-31/30
Step-by-step explanation:
-5/6 + 1/3 Times -3/5
-5/6 - 1/5
Find the measure of angleq, the smallest angle in a triangle whose sides have lengths 4, 5, and 6. round the measure to the nearest whole degree. 34° 41° 51° 56°
The measure of the smallest angle, angle Q, is approximately 41°.
To find the smallest angle in the triangle with side lengths 4, 5, and 6, we can use the cosine rule.
The cosine rule states that for any triangle with sides a, b, and c and corresponding angles A, B, and C:
[tex]cos(A) = (b^2 + c^2 - a^2) / (2 * b * c)[/tex]
In our case, since we want to find the smallest angle, we need to find the angle opposite the smallest side.
The smallest side is 4, so we are looking for angle A.
The other sides, b and c, are 5 and 6 respectively.
Step 1: Plug in the side lengths into the cosine rule formula.
[tex]cos(A) = (5^2 + 6^2 - 4^2) / (2 * 5 * 6)[/tex]
Step 2: Calculate the values inside the parentheses.
cos(A) = (25 + 36 - 16) / (60)
Step 3: Simplify the equation.
cos(A) = (45) / (60)
Step 4: Divide the numbers.
cos(A) = 0.75
Step 5: Now, to find angle A, take the inverse cosine [tex](cos^(-1))[/tex] of the result.
A = cos^(-1)(0.75)
Step 6: Calculate the angle in degrees.
A ≈ 41.41°
Step 7: Round the angle to the nearest whole degree.
A ≈ 41°.
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Please help! Urgent!
Answer:
x = 10
Step-by-step explanation:
using the cosine ratio in the right triangle and the exact value
cos60° = [tex]\frac{1}{2}[/tex] , then
cos60° = [tex]\frac{adjacent}{hypotenuse}[/tex] = [tex]\frac{5}{x}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
x = 5 × 2 = 10
Rita sells boxes of cookies for $10 each. John sells boxes of cookies for $8 each. Each of them sold the same dollar amount. What was the dollar amount each of them sold?
This result doesn't make sense in the context of the problem, so it's likely that there was a mistake in the problem setup or in the information given.
What is Algebraic expression ?
An algebraic expression is a mathematical phrase that can include numbers, variables, and operators (such as addition, subtraction, multiplication, and division), as well as grouping symbols like parentheses.
Let's assume that they both sold a total of $x.
Since Rita sold each box of cookies for $10, she must have sold x/10 boxes.
Likewise, since John sold each box for $8, he must have sold x/8 boxes.
We know that they both sold the same dollar amount, so we can set their two sales expressions equal to each other and solve for x:
x/10 = x/8
Multiplying both sides by the least common multiple of 10 and 8, which is 40:
4x = 5x
Subtracting 4x from both sides:
x = 0
Therefore, This result doesn't make sense in the context of the problem, so it's likely that there was a mistake in the problem setup or in the information given.
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problem 1 cars arrive at a drive-through pharmacy at the rate of 4 every 10 minutes. the average service time at the pharmacy window is 2 minutes. the poisson distribution is appropriate for the arrivals, and service times are exponentially distributed. use the formulae for the m/m/1 queueing model to answer the following: (a) what is the average time a car is in the system (from the time it enters the pharmacy queue till it finishes service and exists)? (b) what is the average number of cars in the system? (c) what is the average number of cars waiting (in the queue) to receive service? (d) what is the average time a car is in the queue? (e) what is the probability that there are no cars at the window? (f) what percentage of the time is the serving pharmacist busy? (g) what is the probability that there are exactly 2 cars in the system?
The required probability and average for the given arrival rate and service rate are,
Average time a car in the given system is 2.13 minutes.
Average number of cars in the given system is 0.852 cars.
Average number of cars waiting is 0.052 cars.
Average time a car spends waiting is 0.13 minutes.
Probability of no cars is 0.2.
Percentage of time the serving pharmacist is 80%.
Probability of exactly 2 cars is 0.64.
Arrival rate per minute λ = 4/10 = 0.4
And service rate per minute µ = 1/2
= 0.5 .
Average time a car is in the system is given by,
W = Wq + 1/µ
where Wq is the average time a car spends waiting in the queue.
1/µ is the average service time.
Using Little's Law, we have,
L = λW
where L is the average number of cars in the system.
Using the given values,
Wq
= (0.4^2)/(2×(1-0.4))
= 0.13 minutes
W
= Wq + 1/µ
= 0.13 + 2
= 2.13 minutes
Average time a car is in the system is 2.13 minutes.
Average number of cars in the system is,
L = λW
= λWq + λ/µ
L = λWq + λ/µ
= 0.4×0.13 + 0.4/0.5
= 0.052 + 0.8
= 0.852 cars
Average number of cars in the system is 0.852 cars.
Average number of cars waiting in the queue is,
Lq = λWq
= 0.4×0.13
= 0.052 cars
Average number of cars waiting in the queue is 0.052 cars.
Average time a car spends waiting in the queue is,
Wq = Lq/λ
= 0.052/0.4
= 0.13 minutes
Average time a car spends waiting in the queue is 0.13 minutes.
Probability that there are no cars at the window is,
P(0) = (1 - λ/µ)
= (1 - 0.4/0.5)
= 0.2
Probability that there are no cars at the window is 0.2.
Percentage of time the serving pharmacist is busy is ,
ρ = λ/µ
= 0.4/0.5
= 0.8
Percentage of time the serving pharmacist is busy is 80%.
Probability that there are exactly 2 cars in the system is,
P(2) = ((λ/µ)^2/(1-ρ)) × P(0)
where P(0) is the probability that there are no cars at the window.
P(2)
= ((0.4/0.5)^2/(1-0.8))×0.2
= 0.64
Probability that there are exactly 2 cars in the system is 0.64.
Therefore, the probability for the given situations are,
Average time is 2.13 minutes.
Average number of cars is 0.852 cars.
Average number of cars waiting in the queue is 0.052 cars.
Average time a car spends waiting in the queue is 0.13 minutes.
Probability of no cars at the window is 0.2.
Percentage of time the serving pharmacist is busy is 80%.
Probability that there are exactly 2 cars is 0.64.
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Complete the following statement. Write your answer as a decimal or whole number.
__% of $3,000,000 = $2,250,000
Submit
Answer: 75
Step-by-step explanation:
[tex]x[/tex]% x 3000000 = 2250000
[tex]x[/tex]/100 x 3000000 = 2250000
[tex]x[/tex] x 30000 = 2250000
[tex]x[/tex] = 2250000/30000
[tex]x[/tex] = 75
emily is repaving her backyard patio space in the shape shown below. to order the correct number of concrete pavers, she needs to calculate the area of the patio. what is the area of emily’s patio?
The area οf Emily's patiο is 1,896 square feet.
What is square?A square is a geοmetric shape with fοur equal sides and fοur equal angles (90 degrees each). It is a special case οf a rectangle where all sides are οf equal length. The area οf a square is calculated by multiplying the length οf οne side by itself, which is expressed as side^2 οr s^2, where s is the length οf a side.
Tο calculate the area οf the patiο, we need tο first find the dimensiοns οf each rectangle that make up the patiο.
The first rectangle has a length οf 15 feet and a width οf 20 feet. Its area is:
Area οf the first rectangle = length x width = 15 ft x 20 ft = 300 square feet
The secοnd rectangle has a length οf 38 feet and a width οf 42 feet. Its area is:
Area οf the secοnd rectangle = length x width = 38 ft x 42 ft = 1,596 square feet
Tο find the tοtal area οf the patiο, we need tο add the areas οf the twο rectangles:
Tοtal area οf the patiο = Area οf the first rectangle + Area οf the secοnd rectangle
Tοtal area οf the patiο = 300 square feet + 1,596 square feet = 1,896 square feet
Therefοre, the area οf Emily's patiο is 1,896 square feet.
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Write equations of circles in standard form from graphs
The equation of the circle in standard form is (x - 3)²+ y² = 1.
What is equation of circle in standard form?The standard form of the equation of a circle is:
(x - h)² + (y - k)² = r²
where (h, k) is the center of the circle and r is its radius.
This equation represents all points in the xy-plane that are equidistant from the center (h, k) of the circle. The squared terms (x - h)² and (y - k)² represent the distances between the center and any point (x, y) on the circle, and r² is the squared length of the radius of the circle.
In the given question,
The center of the circle is (3, 0), and the radius extends to the point (4, 0), which means the radius has a length of 1 unit.
Substituting the values of the center and radius into the standard form equation of a circle, we get:
(x - 3)² + (y - 0)² = 1²
Simplifying, we get:
(x - 3)² + y² = 1
Therefore, the equation of the circle in standard form is:
(x - 3)² + y² = 1.
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Complete the following steps on the graph:
1) Draw the line x = -3 in black
2) Reflect ABC over the line x = -3 (draw on the graph in blue)
3) Translate A'B'C' by the directed line segment from (0,0) to (4,1) (draw on the graph in red)
Find the area between two curves: y=|x+3|-2 and y= 1-(1/3)|x-2|
The area between the two curves will be around 17/6 square units.
Let's start by setting the two equations equal to each other:
|+3|−2 = 1−(1/3)|−2|
Case 1: +3≥0
In this case, the equation simplifies to:
+3−2 = 1−(1/3)|−2|
Simplifying further, we get:
= 1
Case 2: +3<0
In this case, the equation simplifies to:
−(+3)−2 = 1−(1/3)|−2|
Simplifying further, we get:
= −2
two curves intersect at =−2 and =1.
we need to integrate the difference between the two curves with respect to , from =−2 to =1:
∫−2¹ [(1−(1/3)|−2|)−(|+3|−2)] d
Simplifying the absolute values,
∫−2¹ [(1−(1/3)(−2))−(|+3|−2)] d
Next, we can break up the integral into two parts:
∫−2¹ [1−(1/3)(−2)] d − ∫−2¹ (|+3|−2) d
Simplifying, (7/3) + (1/2) = 17/6
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