The vertices of the given hyperbola are (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)), the foci are (-1, -1 ± sqrt(66)), the center is (-1, -1), and the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
To find the vertices, foci, center, and asymptotes of the given hyperbola, we need to use the standard form of a hyperbola equation:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1
First, we need to find the center (h, k) of the hyperbola. From the given equation, we can see that h = -1 and k = -1, so the center of the hyperbola is (-1, -1).
Next, we need to find the values of a and b. From the given equation, we can see that a^2 = 21 and b^2 = 45, so a = sqrt(21) and b = sqrt(45).
Now, we can find the vertices of the hyperbola. The vertices are located at (h, k ± a), so the vertices are (-1, -1 ± sqrt(21)). This gives us the vertices (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)).
Next, we need to find the foci of the hyperbola. The foci are located at (h, k ± c), where c = sqrt(a^2 + b^2). So, c = sqrt(21 + 45) = sqrt(66), and the foci are (-1, -1 ± sqrt(66)).
Finally, we need to find the asymptotes of the hyperbola. The equations of the asymptotes are y = k ± (a/b)(x - h). So, the equations of the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
So, the vertices of the given hyperbola are (-1, -1 + sqrt(21)) and (-1, -1 - sqrt(21)), the foci are (-1, -1 ± sqrt(66)), the center is (-1, -1), and the asymptotes are y = -1 ± (sqrt(21)/sqrt(45))(x + 1).
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PLSSS HELP IF YOU TURLY KNOW THISSSS
Answer: 2
5x - 7 = 3x + 3
2x - 7 = 3
2 is the new value of the coefficent
how much percent is 2882.45 if 3374.60 is 100%
( include working out and show answer as percentage )
Answer:
97271.1577
multiply 3374.60 by 2882.45%
In Exercises 3-6, find the indicated measure. Explain your reasoning. Show your work
The measured unknown sides are -
GH = 4.6
QR = 1.3
AB = 15
UW = 41
What are similar triangles?Two triangles are similar if the ratio of their corresponding sides is same and their corresponding angles are equal.
Given are the triangles as shown in the image.
{ 1 } -
We can write -
GH/HJ = GK/KJ
GH/4.6 = 3.6/3.6
GH = 4.6
{ 2 } -
QT/TS = QR/RS
4.7/4.7 = QR/1.3
QR = 1.3
{ 3 } -
AD/DC = AB/BC
AD/AD = AB/BC
AB/BC = 1
5x/4x + 3 = 1
5x = 4x + 3
x = 3
AB = 15
{ 4 } -
UW/UV = DW/DV
7x + 13 = 9x + 1
3x = 12
x = 4
UW = 41
Therefore, the measured unknown sides are -
GH = 4.6
QR = 1.3
AB = 15
UW = 41
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A lithotripter of height 15 cm and diameter 18 cm is to be constructed from a semi-ellipse. High energy underwater shock waves will be emitted from the focus that is closest to the vertex V in order to break down the kidney stone. How far from V in the vertical direction should a kidney stone be located to receive the greatest effect of the shock waves?
Answer:
27 cm
Step-by-step explanation:
You want the distance from the vertex of an ellipse to the opposite focus if the semi-major axis is 15 cm, and the minor axis is 18 cm.
Center to focusThe square of the distance from the center to either focus is the difference of the squares of the semi-axes. The semi-minor axis is half the diameter of the ellipsoid.
c² = 15² -(18/2)² = 225 -81 = 144
c = √144 = 12 . . . . . cm
Vertex to focusThe distance from the vertex (V) to the opposite focus is the distance from vertex to center plus the distance from center to focus:
15 cm +12 cm = 27 cm
The kidney stone should be located 27 cm from the vertex for greatest effect.
HELP PLEASE !!!!!
2g) Letty is simplifying the square root of 48 using the Product Property of Square Roots. She wants to use the factors 4 and 12 to simplify the radical. Explain why these are not the best factors to use.
2h) What factors would be a better choice to use to simplify the square root of 48? Why should you choose those
factors and not any other pair?
Answer:
2g) Letty cannot use the factors 4 and 12 to simplify the square root of 48 using the Product Property of Square Roots because 4 is a perfect square, but 12 is not. The Product Property of Square Roots only applies to factors that are both perfect squares.
2h) A better choice to simplify the square root of 48 would be to use the factors 16 and 3. This is because 16 is a perfect square and is a factor of 48, which means it can be taken out of the radical completely. The remaining factor is 3, which cannot be simplified any further since it is not a perfect square. Therefore, the square root of 48 can be simplified to 4 times the square root of 3. It is important to choose 16 and 3 as the factors and not any other pair because 16 is the largest perfect square factor of 48, and 3 is the remaining factor after taking out 16 that cannot be simplified any further.
2g) These factors (4 and 12) are not the best choice to simplify the square root of 48 because 4 is a perfect square, but 12 is not.
2h) a better choice to simplify the square root of 48 would be to use the factors 16 and 3. This is because 16 is the largest perfect square factor of 48, which simplifies the radical the most.
Now, Using the Product Property of Square Roots, we can simplify the square root of 48 as :
√48 = √(4 x 12)
However, these factors (4 and 12) are not the best choice to simplify the square root of 48 because 4 is a perfect square, but 12 is not.
Hence, We want to simplify the radical by finding the largest perfect square that is a factor of 48.
For this, we can break down 48 into its prime factors:
48 = 2 x 2 x 2 x 2 x 3
Then, we group the prime factors into pairs of the same number:
48 = (2 x 2) x (2 x 2) x 3
This gives us two perfect squares, 4 and 16.
Hence, We can simplify the square root of 48 by using the largest perfect square factor, which is 16:
√48 = √(16 x 3)
√48 = √16 x √3
√48 = 4√3
Therefore, a better choice to simplify the square root of 48 would be to use the factors 16 and 3.
This is because 16 is the largest perfect square factor of 48, which simplifies the radical the most.
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pls help me with this math assignment ! give you a like
pic down below ⬇️
The three points that are not collinear are J, K and L
The three points that are not collinearIn geometry, points that are not collinear are points that do not lie on the same straight line. i.e. points that are not collinear are points that are not on the same line
Using the above as a guide, the non-collinear points are points J, K and L
How to determine the opposite raysOpposite rays are two rays that have the same endpoint and extend indefinitely in opposite directions.
Using the above definition, we have
Opposite rays = HG and HI
Another name for line JHThe line JH can be named based on its endpoints
So, another name for JH is HJ
Planes that contain HLThree points are required to form a plane
This means that the planes that contain HL can be named as GHL and JHL
There are other planes that contained HL too
Sketch of planes that intersect in line pSee attachment for the sketch
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Gary earns $9.12 per hour at his job. He worked 24 hours last week. Calculate Gary's pay before taxes.
Gary's pay before taxes is $218.88.
Gary's pay before taxes can be calculated by multiplying his hourly rate by the number of hours he worked.
Taxes are compulsory payments made by a government organisation, whether local, regional, or federal, to people or businesses. Tax revenues are used to fund a variety of government initiatives, such as Social Security and Medicare as well as public infrastructure and services like roads and schools.
Step 1: Identify the hourly rate and number of hours worked.
Hourly rate: $9.12
Hours worked: 24
Step 2: Multiply the hourly rate by the number of hours worked.
$9.12 * 24 = $218.88
Step 3: The result is Gary's pay before taxes.
Gary's pay before taxes is $218.88.
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If f(x) = 3x² + 2x - 10, what is f(2)?
Answer:
f(2)=3x²+2x-10
f(2)=3(2)² +2(2)-10
=12+4-10
=6
Find the volume of each figure round to nearest hundredth if needed
For the given cuboid, it's volume value is deduced as 30 cm³.
What is volume?
Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity.
The figure is given as a cuboid.
The length of the cuboid is 5 cm.
The breadth of the cuboid is 6 cm.
The height of the cuboid is 1 cm.
The formula for volume of cuboid is given as -
Volume = length × breadth × height
Substitute the values into the equation -
Volume = 5 × 6 × 1
Volume = 30 cm³
Therefore, the volume value is obtained as 30 cm³.
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How do you find the vertex and y-intercept of a quadratic function
Answer:
The graph is a parabola that opens downward. Generally, a parabola has the following equation: y=ax2+bx+c If a is positive it opens upward
Question 4. 1. Find the number of non-negative integer solutions ofx1+x2+x3+x4=10. 2. (True/False) : The answer to the above question is same as the number of non-negative integer solutions ofx1+x2+x3≤10. Justify. Is it the same as the number of non-negative integer solutions ofx1+x2+x3+x4+x5=10? Justify. 3. Find the number of positive integer solutions ofx1+x2+x3+x4=10
1. 286
2. False
3. 84
1. The number of non-negative integer solutions of x1+x2+x3+x4=10 is 286.
2. The answer to the above question is not the same as the number of non-negative integer solutions of x1+x2+x3≤10. The reason is that the first equation has 4 variables, while the second equation has only 3 variables.
Therefore, the number of solutions will be different.
3. The number of positive integer solutions of x1+x2+x3+x4=10 is 84.
1. To find the number of non-negative integer solutions of x1+x2+x3+x4=10, we can use the formula: C(n+k-1, k-1) = C(10+4-1, 4-1) = C(13, 3) = 286
Therefore, there are 286 non-negative integer solutions of x1+x2+x3+x4=10.
2. The answer to the above question is not the same as the number of non-negative integer solutions of x1+x2+x3≤10 because the first equation has 4 variables, while the second equation has only 3 variables.
Therefore, the number of solutions will be different.
3. To find the number of positive integer solutions of x1+x2+x3+x4=10, we can use the formula: C(n-1, k-1) = C(10-1, 4-1) = C(9, 3) = 84
Therefore, there are 84 positive integer solutions of x1+x2+x3+x4=10.
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Segment in a circle: XY=
PLEASE HELP THIS IS MY LAST QUESTION
If the correlation coefficient for the data shown in the table is -1, A should be what value?
Time (hours) (x)
4 1
3 6 5 7
Distance from destination (miles) (y) 1,000 A 1,060 940 760 820 690
920
880
800
none of these
O 1030
2
Answer:
Step-by-step explanation:
Here is the answer
The correlation coefficient measures the strength of the linear relationship between two variables, and it ranges from -1 to 1. A correlation coefficient of -1 indicates a perfect negative linear relationship between the variables, which means that as one variable increases, the other decreases at a constant rate.
To find the value of A in this scenario, we need to look for a perfect negative linear relationship between the two variables, time (x) and distance from destination (y). The table shows that as time increases, the distance from the destination decreases, but we need to find the exact rate of change.
We can calculate the rate of change by finding the slope of the line that represents the relationship between time and distance. We can use the formula for the slope of a line, which is:
slope = (change in y) / (change in x)
If we choose the first and last points in the table, we get:
slope = (690 - 1060) / (7 - 1) = -70
This means that for every hour of time that passes, the distance from the destination decreases by 70 miles. Therefore, if the correlation coefficient is -1, we should see a perfect negative linear relationship between time and distance, with a slope of -70.
To check if A is the correct value, we can use the formula for the equation of a line in slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept. We can plug in the values of m and b and solve for A:
y = -70x + b
If we use the first point in the table, where x = 4 and y = 1000, we get:
1000 = -70(4) + b
b = 1220
So the equation of the line is:
y = -70x + 1220
If we plug in the values of x for the remaining points in the table, we get:
y = 1030 when x = 0
y = 880 when x = 2
y = 800 when x = 3
y = A when x = 5
y = 760 when x = 6
To find the value of A, we can plug in the corresponding value of y and solve for A:
1030 = -70(0) + 1220
880 = -70(2) + 1220
800 = -70(3) + 1220
760 = -70(6) + 1220
A = -70(5) + 1220 = 850
Therefore, the value of A should be 850 if the correlation coefficient for the data shown in the table is -1.
Select the correct answer. A company sells bikes that are light and fast. The revenue earned over a period of x years can be represented by the function, . The cost of manufacturing the bikes can be represented by the function, . Which function, , describes the profit earned from the sales of the bikes over a period of x years? A. B. C. D.
The function that describes the profit earned from the sales of the bikes over a period of x years is: [tex]P(x) = 100x^3 + 6x^2 + 9x[/tex]. The Option A is correct.
Which function describes the profit earned from the sales of the bikes?The profit earned from the sales of the bikes can be calculated by subtracting the cost of manufacturing from the revenue earned. Therefore, the function that describes the profit earned can be represented as:
[tex]P(x) = S(x) - C(x)[/tex]
Substituting the given functions for S(x) and C(x), we get:
[tex]P(x) = (100x^3 + 6x^2 + 97x + 215) - (88x + 215)[/tex]
Simplifying this expression, we get:
[tex]P(x) = 100x^3 + 6x^2 + 9x.[/tex]
Full question "A company sells bikes that are light and fast. The revenue earned over a period of x years can be represented by the function, S(x) = 100x3 + 6x2 + 97x + 215. The cost of manufacturing the bikes can be represented by the function, C(x) = 88x + 215. Which function, P(x), describes the profit earned from the sales of the bikes over a period of x years?"
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Use PEMDAS to evaluate the expression:
8+(36 x 8-204) ÷ 6
Answer: 22
Step-by-step explanation:
Answer:
22
Step-by-step explanation:
(36 x8-204)
multiply first
36(8)=288
then subtract
288-204=84
8+84 ÷6
divide first
84 ÷6=14
8+14=22
Solve using the quadratic f -9d^(2)+6=-6d Write your answers as intec form, or decimals rounded
The answer to the quadratic equation -9d^(2)+6=-6d is 11/20 and -11/20. To solve the given quadratic equation f -9d^(2)+6=-6d, we need to rearrange the terms and use the quadratic formula.
Step 1: Rearrange the terms to get the equation in the standard form of ax^(2) + bx + c = 0
-9d^(2) + 6d + 6 = 0
Step 2: Identify the values of a, b, and c.
a = -9
b = 6
c = 6
Step 3: Use the quadratic formula to find the solutions for d.
The quadratic formula is given by:
d = (-b ± √(b^(2) - 4ac))/(2a)
Substitute the values of a, b, and c into the formula:
d = (-(6) ± √((6)^(2) - 4(-9)(6)))/(2(-9))
Step 4: Simplify the expression inside the square root:
d = (-(6) ± √(36 + 216))/(-18)
d = (-(6) ± √(252))/(-18)
Step 5: Simplify the expression further:
d = (-(6) ± 15.87)/(-18)
Step 6: Solve for d using both the positive and negative values inside the square root:
d = (-(6) + 15.87)/(-18) = 0.55
d = (-(6) - 15.87)/(-18) = -0.55
So, the solutions for d are 0.55 and -0.55 or 11/20 and -11/20 using the quadratic formula to solve the given quadratic equation.
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2. Write an equation with the given characteristics: a. Goes through: \( (2,13) \) and \( (2,-11) \)
An equation passing through the points (2, 13) and (2, -11) is x = 2.
The equation of a line can be written in the form of y = mx + b, where m is the slope and b is the y-intercept.
To find the equation of the line that goes through the given points, we first need to find the slope. The slope can be calculated using the formula:
[tex]\[m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}\][/tex]
Plugging in the given points, we get:
[tex]\[m = \frac{-11 - 13}{2 - 2} = \frac{-24}{0}\][/tex]
Since the denominator is 0, the slope is undefined. This means that the line is vertical. A vertical line has the equation x = a, where a is the x-coordinate of any point on the line. Since both of the given points have the same x-coordinate of 2, the equation of the line is x = 2.
Therefore, the equation with the given characteristics is x = 2.
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Julia placed the number cards 1, 2, 3, 5, 8, and 13 in a bag. A card is drawn at random. Determine the theoretical
probability of drawing an even number. Express your answer as a fraction in simplest form.
Quick pleaseee
Answer: 1/3
Step-by-step explanation: so there is six cards total and only two of them are even so 2/6 and that can go to 1/3
1/3
There are 2 even numbers, and the total is 6. So 2/6 in the simplest form is 1/3.
Help pls due today
I’m stressing out
Answer:
If the relationship is proportional, we can use the proportionality constant, k, to relate the cost and number of climbs. The equation would be:
t = kc
We can find the value of k by using the given information that 5 climbs cost $52.50:
52.50 = k(5)
Solving for k, we get:
k = 10.50
Therefore, the equation that represents the total cost, t, of c climbs is:
t = 10.50c
PLEASE HELP
Write the equation of the line with a y-intercept of (4,0) and an x-intercept of (0,6)
Answer:
y = -3/2x + 6
Step-by-step explanation:
Slope m = (y2 - y1)/(x2 - x1)
m = (6 - 0)/(0 - 4) = 6/-4 = -3/2
Slope-intercept form is y = mx + b
Substitue m = -3/2, y = 0, x = 4
0 = -3/2(4) + b
b = 6
y = -3/2x + 6
Read and interpret the following conditions imposed on the variables \( a, b, c, d \), and \( x \). Determine and state whether the statements in Exercises 1 - 12 are true or false. If they are false,
The given conditions imposed on the variables \( a, b, c, d \) and \( x \) are:
\( a+b = c \) \( d = a^2 + b^2 \) \( x = a^3 + b^3 \)
To determine if the statements in Exercises 1-12 are true or false, use the given conditions to evaluate the expressions in the statement. If the statement matches the conditions, it is true; if it does not, it is false. For example, if the statement is: " \( a + b = d \) ", then this is false, as \( a + b \neq d \).
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Which equation best represents the relationship between x and y in the graph?
Ay = 3x - 2
- ¹x +3
-8-7-6-5
9
8
-7
6
S
-1
-2
-3
4
-5
-6
-7
-8
-9
4 5 6 7 8
X
An equation that best represents the relationship between x and y in the graph is: A. y = -3x/4 - 3.
What is the point-slope form?Mathematically, the point-slope form of a straight line can be calculated by using this mathematical expression:
y - y₁ = m(x - x₁) or y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
Where:
m represents the slope.x and y are the points.At point (-4, 0), an equation of this line can be calculated by using the point-slope form:
y - y₁ = (y₂ - y₁)/(x₂ - x₁)(x - x₁)
y - 0 = (-3 - 0)/(0 + 4)(x + 4)
y - 0 = -3/4(x + 4)
y = -3x/4 - 3.
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Triangle XYZ has vertices X(−2, −1), Y(0, 2), and Z(2, −1)
The image after the rotation would be X(2, 1) Y(0,-2) and Z(-2, 1) according to the graph below.
What is a Graph?An ordered graphical depiction of numbers or variables is how this is described.
180 degrees in each direction results in (x,y) becoming (-x,-y). The coordinates are thus inverted to either positive or negative values depending on their respective signs.
A graph is a mathematical structure made up of a set of lines linking some pairs of VERTICES that may or may not be empty and a collection of points called VERTICES. Graphs are a common method for displaying the relationships between data. Although taking up less space, a graph accomplishes the goal of presenting data that are either too many or complex to be adequately described in the text.
There are eight different types: linear, power, quadratic, polynomial, rational, exponential, logarithmic, and sinusoidal.
Hence, it means X(−2, −1), Y(0, 2), and Z(2, −1) = X(2, 1) Y(0,-2) and Z(-2, 1).
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Complete question -
Members of a softball team raised $1353 to go to a tournament. They rented a bus for $943.50 and budgeted $31.50 per player for meals. Write and solve an equation which can be used to determine xx, the number of players the team can bring to the tournament.
We round up to the nearest whole number because we are unable to have half a player. As a result, the softball team is allowed to send 7 players to the competition.
What are an example and an equation?The equal sign joins two expressions to create a mathematical formula called an equation. An illustration formula could be 3x - 5 = 16. By resolving this equation, we find that the value of the variable x is 7.
To begin, let's define a few variables:
x: The softball team's total roster size
y: the sum of the players' meal expenses
We can construct the equation shown below:
$1353 - $943.50 - $31.50x = y
Simplifying this equation, we get:
$409.50 - $31.50x = y
we can set up another equation:
y = $31.50x
Now we can substitute the second equation into the first equation to eliminate y:
$409.50 - $31.50x = $31.50x
Simplifying this equation, we get:
$409.50 = $63x
Dividing both sides by 63, we get:
x = 6.5
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Please help me I don’t want to fail
The measure of line KR is 8 inches
How to determine the measure of the length
It is important that we know the properties of an isosceles trapezoid.
Only one pair of sides are parallelNon-parallel sides are equal in measureThe diagonals are equal in measureThe opposite angles are supplementary, that is, their sum is equal to 180 degreesFrom the information given, we have that;
KR = 1/2x + 5
DH = 2x - 4
Since the non- parallel sides of an isosceles trapezoid are equal, then, we have;
KR = DH
1/2x + 5 = 2x - 4
collect the like terns, we have
1/2x - 2x = -4 - 5
x - 4x /2 = - 9
cross multiply
-3x = -18
x = 6
KR = 1/2(6) + 5 = 8 inches
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Are (9g – 11 + 10g) - (12 - 11g+ 13) and -3(-10g + 12) equivalent expressions? Explain.
Yes, these expressions are equivalent expressions.
What are expressions?A sentence qualifies as a mathematical expression if it comprises one or more mathematical operations, at least two numbers, or variables. Let's examine the writing style of expressions. A number is 6 times larger than the other number, x, by a factor of 2. The mathematical phrase x/2 + 6 represents this claim. Complex riddles are solved using mathematical expressions.
In the given question,
Expressions are as follows:
(9g-11+10g) - (12-11g+13)
Simplifying it, we get:
30g -36
In the 2nd expression given,
-3(-10g+12) and after simplifying:
30g-36
As the expressions after simplifying are equal to each other, we can say that these are equivalent expressions.
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A European company is selling a new brand of headphones. For h headphones sold, in thousands, a profit of E(h) = -4h^(4) + 7h^(3) - 7h + 20, in ten thousands of Euros, will be earned. How much will be earned in profit for selling 1,500 headphones?
The European company will earn 128,750 Euros in profit for selling 1,500 headphones.
How to find sale profitTo find the profit for selling 1,500 headphones, we need to plug in the value of h into the profit function E(h) and simplify.
Since h is measured in thousands of headphones, we need to divide 1,500 by 1,000 to get h = 1.5.
Now we can plug in h = 1.5 into the profit function and simplify:
E(1.5) = -4(1.5)^(4) + 7(1.5)^(3) - 7(1.5) + 20
E(1.5) = -4(5.0625) + 7(3.375) - 10.5 + 20
E(1.5) = -20.25 + 23.625 - 10.5 + 20
E(1.5) = 12.875
So the profit for selling 1,500 headphones is 12.875 ten thousands of Euros, or 128,750 Euros.
Therefore, the European company will earn 128,750 Euros in profit for selling 1,500 headphones.
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Kuro has a stack of 15 coins on a table with a volume of 8 cubic centimeters. He has stacked them perfectly straight creating a cylindrical shape. He accidentally bumps the table, causing the coins to shift to a leaning position; but are still stacked. What would be the best estimate of the volume of the stack once it has been bumped/shifted?
The best estimate of the volume of the stack once it has been bumped/shifted is 4πt³ cubic centimeters.
Describe Volume of cylinder?A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected by a curved surface. The volume of a cylinder is the amount of space occupied by the shape and is given by the formula:
Volume = πr²h
where π (pi) is a mathematical constant approximately equal to 3.14, r is the radius of the circular base, and h is the height of the cylinder.
Assuming that the coins have the same dimensions and are perfectly circular, the original stack of 15 coins formed a cylinder with a height of 15 times the thickness of a single coin, and a radius equal to the radius of a single coin.
The volume of this cylinder can be calculated using the formula V = πr²h, where r is the radius and h is the height.
Since the volume of the original stack is 8 cubic centimeters, we can set up the equation:
8 = πr²(15t)
where t is the thickness of a single coin.
Solving for r, we get:
r = √(8/15πt)
When the stack is bumped and shifts to a leaning position, the new shape will still be a cylinder, but the height will be shorter than before. Let's say that the new height is h2. We can calculate the new volume using the same formula:
V2 = πr²h2
To estimate the new height, we can use the fact that the coins are now leaning against each other, so the new height will be less than 15t. Let's say that the new height is approximately 12t.
Substituting in the values, we get:
V2 = π(√(8/15πt))²(12t)
Simplifying, we get:
V2 = 4πt³
Therefore, the best estimate of the volume of the stack once it has been bumped/shifted is 4π³ cubic centimeters.
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Mr Khumalo has a budget of R30000 to fence off the garden
Determine if he will have sufficient funds to fence off the garden if it is further giveb that a gate of 1. 2m width, thats cost R500, will be fitted on one side of the garden
Answer:
No R30000 will be enough as the gate of 1.2m width cost R500
If we know the length of the garden and the cost per meter of fencing, we can determine if Mr. Khumalo has sufficient funds to fence off the garden, taking into account the cost of the gate.
What is an expression?An expression contains one or more terms with addition, subtraction, multiplication, and division.
We always combine the like terms in an expression when we simplify.
We also keep all the like terms on one side of the expression if we are dealing with two sides of an expression.
Example:
1 + 3x + 4y = 7 is an expression.
3 + 4 is an expression.
2 x 4 + 6 x 7 – 9 is an expression.
33 + 77 – 88 is an expression.
We have,
To determine if Mr. Khumalo will have sufficient funds to fence off the garden,
We need to calculate the total cost of the fence excluding the cost of the gate.
Let's assume the length of the garden is L meters.
The perimeter of the garden is then 2L + 1.2 meters (due to the gate on one side).
If the cost of fencing per meter is R, then the total cost of the fence will be:
Cost of the fence.
= (2L + 1.2) × R
Since Mr. Khumalo has a budget of R30000, we can set up an inequality to determine if the cost of the fence is within his budget:
(2L + 1.2) × R + 500 ≤ 30000
Simplifying the inequality, we get:
(2L + 1.2) × R ≤ 29500
Therefore,
If we know the length of the garden and the cost per meter of fencing, we can determine if Mr. Khumalo has sufficient funds to fence off the garden, taking into account the cost of the gate.
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8.53 A random variable X has the normal distribution N(m, o2), where mo e R, if it is an absolutely continuous random variable with density (x - m) fx(x) = exp 202 Verify that fx is indeed a density.
∫fx(x)dx = 1 over the entire range of x, and fx is indeed a density function.
A random variable X has the normal distribution N(m, o^2), where m and o are both real numbers, if it is an absolutely continuous random variable with density fx(x) = (1/√(2πo^2)) * exp(-(x-m)^2/2o^2).
To verify that fx is indeed a density, we need to check that it satisfies the two properties of a density function:
1) fx(x) >= 0 for all x
2) ∫fx(x)dx = 1 over the entire range of x
First, let's check that fx(x) >= 0 for all x. Since the exponential function is always positive, we can see that fx(x) will always be positive as well. Therefore, fx(x) >= 0 for all x.
Next, let's check that ∫fx(x)dx = 1 over the entire range of x. To do this, we need to integrate fx(x) over the entire range of x, which is from -∞ to ∞:
∫fx(x)dx = ∫(1/√(2πo^2)) * exp(-(x-m)^2/2o^2)dx from -∞ to ∞
Using the substitution u = (x-m)/√(2o^2), we can rewrite the integral as:
∫(1/√(2πo^2)) * exp(-u^2/2) * √(2o^2)du from -∞ to ∞
Simplifying, we get:
∫(1/√(2π)) * exp(-u^2/2)du from -∞ to ∞
This integral is equal to 1, as it is the integral of the standard normal density function. Therefore, ∫fx(x)dx = 1 over the entire range of x, and fx is indeed a density function.
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