Let's call the amount of money Rosa spent on art supplies "x". Since Rosa spent 1/4 of her money on art supplies, we can write the equation:
x = (1/4) * 300
x = 75
So Rosa spent $75 on art supplies.
Now we can find out how much money she had left after spending the $75 on art supplies.
remaining = 300 - 75
remaining = 225
And we can find out how much she spent on the boots.
boots = (2/5) * remaining
boots = (2/5) * 225
boots = 90
So Rosa spent $90 on the boots.
Let's call the amount of money Rosa spent on art supplies "A". Since Rosa spent 1/4 of her original $300 on art supplies, we can write the equation:
A = 1/4 * $300
A = $75
So, Rosa spent $75 on art supplies.
Next, let's find out how much money she had left after buying the art supplies. We can subtract the amount she spent on art supplies from her original $300:
$300 - $75 = $225
Now, let's find out how much she spent on the boots. We know that she spent 2/5 of the remainder on the boots, so we can write the equation:
B = 2/5 * $225
B = $90
So, Rosa spent $90 on the boots.
What are the features of the quadratic function graphed in the figure?
A) Vertex = (–3,4), y-intercept = (0,–5), x-intercepts = (–5,0) and (–1,0), axis of symmetry is x = –3
B) Vertex = (3,–4), y-intercepts = (–1,0) and (–5,0), x-intercept = (0,5), axis of symmetry is x = 3
C) Vertex = (–3,4), y-intercept = (0,–5), x-intercepts = (1,0) and (5,0), axis of symmetry is x = –3
D)Vertex = (–4,3), y-intercept = (5,0), x-intercepts = (0,1) and (0,5), axis of symmetry is x = –4
Answer:
A) Vertex = (-3, 4), y-intercept = (0, -5), x-intercepts = (-5, 0) and (-1, 0), axis of symmetry is x = -3
Step-by-step explanation:
The vertex of a quadratic function is the turning point. As this parabola opens downwards, the vertex is the maximum point of the graph. From inspection of the graph, the maximum point is at (-3, 4). Therefore:
The vertex of the quadratic function is (-3, 4).The y-intercept is the point at which the curve crosses the y-axis. From inspection of the graph, the curve crosses the y-axis at y = -5. Therefore:
The y-intercept of the quadratic function is (0, -5).The x-intercepts are the points at which the curve crosses the x-axis. From inspection of the graph, the curve crosses the x-axis at x = -5 and x = -1. Therefore:
The x-intercepts of the quadratic function are (-5, 0) and (-1, 0).The axis of symmetry is the vertical line that passes through the vertex of the parabola so that the left and right sides of the parabola are symmetrical. So the axis of symmetry is the x-value of the vertex. Therefore:
The axis of symmetry of the quadratic function is x = -3.Can anyone figure this out? Ive been stuck on it for a while and cant figure out the correct angle
The required scaled copy of polygon B using a scale factor of 0.75 as shown.
What is a scale image?Scale image is defined as a ratio that represents the relationship between the shape and size of a figure and the corresponding dimensions of the actual figure or object.
The polygon B is given as shown, which dimensions are below:
Length of polygon = 8 units
Height of polygon = 10 units
Here, the scale factor = 0.75
So, the dimensions of the scaled copy of polygon B are below:
Length of polygon B' = 8 × 0.75 = 6 units
Height of polygon B' = 10 × 0.75 = 7.5 units
Thus, the scaled copy of polygon B using a scale factor of 0.75 as shown.
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Solve the system of equations using the linear combination method.
-4x - 2y = 26
-
-5x – 2y = 35
-
Enter your answers in the boxes.
X =
y =
The value of x is 9.
The value of y is 5.
What is an equation?An equation contains one or more terms with variables connected by an equal sign.
Example:
2x + 4y = 9 is an equation.
2x = 8 is an equation.
We have,
4x - 2y = 26
This can be written as,
4x - 26 = 2y ______(1)
5x – 2y = 35
This can be written as,
5x - 35 = 2y _______(2)
From (1) and (2),
4x - 26 = 5x - 35
35 - 26 = 5x - 4x
9 = x
x = 9
And,
Substituting x = 9 in (1),
4x - 26 = 2y
4 x 9 - 26 = 2y
36 - 26 = 2y
2y = 10
y = 5
Thus,
The solution is (9, 5).
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Set up the integral that would give the volume V generated by rotating the region bounded by the given curves about the y-axis. y = x3, y = 0, x = 5 Disk/Washer Method v= V = --Select-- 4 ---Select--- Cylindrical Shells Method V= V = ---Select---
The integral that would give the volume V generated by rotating the region bounded by the given curves about the y-axis is V = ∫[from x=0 to x=6] π[(13 - x²/3)² - 13²]dx
To find the volume of a rotational solid, we can use the method of disks/washers, which involves slicing the solid into thin disks or washers, calculating the volume of each slice, and then adding them up using integration.
To use the method of disks/washers, we need to first determine the radius of each disk or washer. Since we're rotating the region around a horizontal line, the radius will be the distance from each point on the curve to the line of rotation, which in this case is y = 16. To find this distance, we subtract 16 from the y-coordinate of each point on the curve.
The outer radius is the distance from the point on the curve y = x^2/3 + 3 to the line y = 16, which is
=> r = 16 - (x²/3 + 3) = 13 - x²/3.
The inner radius is the distance from the point on the curve y = 3 to the line y = 16, which is
=> r = 16 - 3 = 13.
Next, we need to express the volume of each disk or washer in terms of these radii. This gives us the following formula for the volume of each slice:
dV = π[(13 - x²/3)² - 13²]dx
Finally, we can find the total volume of the solid by integrating over the range of x values that define the region we're rotating:
V = ∫[from x=0 to x=6] π[(13 - x²/3)² - 13²]dx
Evaluating this integral will give us the volume of the solid created by rotating the region between y = x²/3 + 3 and y = 3 about the line y = 16.
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Complete Question:
Set up the integral that uses the method of disks/washers to find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified lines. y = x²/3 + 3 , y = 3 , x = 6 About the line y = 16.
Give your overall description of a box plot
Box plot is a chart that shows data from a five-number summary including one of the measures of central tendency.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
A box plot, also known as a box and whisker plot, is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.
The box plot consists of a rectangular box with whiskers that extend from the edges of the box.
The box represents the middle 50% of the data, and the vertical line inside the box represents the median (50th percentile) of the data.
The whiskers extend to the smallest and largest observations that are within 1.5 times the interquartile range (IQR) of the lower and upper quartiles, respectively.
Any observations that fall outside the whiskers are considered outliers and are plotted as individual points.
Box plots are useful for comparing the distribution of data between different groups of data.
Hence, Boxplot is a graphical representation of a set of numerical data that displays its distribution, central tendency, and variability.
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HELP RN PLSPLS!!!! I CANT FIGURE THIS OUTT!!
Answer:m=2/5
Step-by-step explanation:
m[tex]\neq[/tex]0
5/6=1/3m
15m=6
m=2/5, m[tex]\neq[/tex]0
a certain town of population size 100,000 has three newspapers: i, ii, and iii. the proportions of townspeople that read these papers are: i: 10%, i and ii: 8%, i and ii and iii: 1%, ii: 30%, i and iii: 2%, iii: 5%, ii and iii: 4%. (note that, for example, the 10% of people who read newspaper i might read only i or might read i and some other paper(s) ).
Out of a population of 100,000, the number of people who read at least two newspapers is = 33,000.
Let's approach this problem using the inclusion-exclusion principle.
First, we can add up the proportions of people who read each paper to get:
P(I) + P(II) + P(III) = 10% + 30% + 5% = 45%
However, this includes the people who read two or more papers multiple times, so we need to subtract those out. We can calculate these as follows:
P(I&II) + P(I&III) + P(II&III) = 8% + 2% + 4% = 14%
2P(I&II&III) = 2%
Using the inclusion-exclusion principle, we can now find the proportion of people who read at least two papers:
P(at least 2 papers) = P(I) + P(II) + P(III) - (P(I&II) + P(I&III) + P(II&III)) + 2P(I&II&III)
Plugging in the values, we get:
P(at least 2 papers) = 45% - 14% + 2% = 33%
So, the number of people who read at least two newspapers is:
0.33 * 100,000 = 33,000
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Complete question is:
A certain town of population size 100,000 has three newspapers: I , II and III the proportions of townspeople that read these papers are:
I= 10 percent
II= 30% percent
II=5 percent
I&II=8 percent
I&III=2 percent
II&III=4 percent
I&II&III=1 percent
How many people read at least two newspapers?
solve for y
2y - 3(2y-3)+2=31
Answer:
y = -5
Step-by-step explanation:
You want to solve for y in 2y -3(2y -3) +2 = 31.
SimplifyParentheses can be eliminated using the distributive property.
2y -6y +9 +2 = 31
Like terms can be combined.
-4y +11 = 31
SolveWe can separate the constant and variable terms by subtracting 11 from both sides.
-4y = 20
The value of y is now found by dividing by -4.
y = 20/(-4) = -5
y = -5
The area of a rectangular room is 750 square feet. The width of the room is 5 feet less than the length of the room. Which equations can be used to solve for y, the length of the room? Select three options. y(y + 5) = 750 y2 – 5y = 750 750 – y(y – 5) = 0 y(y – 5) + 750 = 0 (y + 25)(y – 30) = 0
The three equations that can be used to solve for y, the length of the room, are:
1. y(y + 5) = 750
2. y^2 – 5y = 750
3. (y + 25)(y – 30) = 0
Explanation:
Let's assume that the length of the room is y and the width of the room is y - 5.
We know that the area of the room is the product of its length and width, so we can write an equation:
y(y - 5) = 750
Simplifying this equation, we get:
y^2 - 5y - 750 = 0
Now we can solve this quadratic equation using the quadratic formula or factoring method. By factoring, we can get equation 3. By using the quadratic formula, we can get equation 2. Equation 1 is just another form of equation 2. Therefore, options 1, 2, and 3 can be used to solve for y. Option 4 is not a valid equation as it doesn't represent the area of the room.
In triangle ABC, B = 40, a = 7, c = 10. Find the measure of angle C.
We can use the Law of Cosines to solve for angle C:
cos C = (a^2 + b^2 - c^2) / (2ab)
where b is the length of side BC.
We know that B = 40 degrees, so we can find the measure of angle A:
A = 180 - B - C
A = 180 - 40 - C
A = 140 - C
We also know that the sum of the angles in a triangle is 180 degrees, so:
A + B + C = 180
140 - C + 40 + C = 180
180 = 180
This checks out, so we can continue with solving for angle C:
cos C = (a^2 + b^2 - c^2) / (2ab)
cos C = (7^2 + b^2 - 10^2) / (2*7b)
cos C = (49 + b^2 - 100) / (14b)
cos C = (b^2 - 51) / (14b)
We can use the Law of Sines to solve for b:
sin B / b = sin A / a
sin 40 / b = sin (140 - C) / 7
b = sin 40 * 7 / sin (140 - C)
Now we can substitute this value of b into the equation for cos C:
cos C = (b^2 - 51) / (14b)
cos C = [(sin 40 * 7 / sin (140 - C))^2 - 51] / (14 * sin 40 * 7 / sin (140 - C))
Simplifying this equation and solving for cos C, we get:
cos C = 11/14
Therefore, angle C is:
C = cos^-1(11/14)
C ≈ 38.8 degrees
$300,9%,3 years ??????????????
Answer:
$381
Step-by-step explanation:
9% = 1 year
27% = 3 years
$300 = 100%
After 3 years, we have
100% + 27% = 127%
127% = 1.27
300 tines 1.27 = $381
So, after 3 years has $381
Please Help me!
thank you for the help!
The expression representing the perimeter of the rectangle is given as follows:
P = 6x + 4.
The perimeter of the rectangle when x = 7 is given as follows:
46 feet.
How to obtain the perimeter of a rectangle?The perimeter of a rectangle of length l and width w is given by the expression presented as follows:
P = 2(l + w).
The dimensions for this problem are given as follows:
x + 4.2x - 2.Hence the expression for the perimeter of the rectangle is given as follows:
P = 2(x + 4 + 2x - 2)
P = 2(3x + 2)
P = 6x + 4.
When x = 7, the perimeter of the rectangle is given as follows:
P = 6(7) + 4
P = 46 feet.
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The United States form of government is a...
O Democratic Parliament
O Republic
O Democratic Republic
O Republican Congress
Answer:
The United States form of government is a...
O Republic
can I please get the five points:)
Answer:
Republic
Step-by-step explanation:
Option B
I hope this helps :) if not let me know
Find the inequality represented by the graph.
A snack mix recipe calls for 1 1 2 cups of chips and 1 5 cup of dip. Luke wants to make the same recipe using 1 cup of dip. How many cups of chips will Luke need?
Answer:8
Step-by-step explanation:
15/15=1
112/15=7.466...
you can't buy 7.466... cups, so you round up to 8.
hope I'm right ;)
help please. I dont know what a im doing on this problem
The domain of the function is [0, 3) and the range of the function is [-4, 5).
What is Domain and Range of a Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Here the input values in set A is called the domain and the output values in set B is called the range.
Given is a graph of a function.
The input values or domain includes the value of x which forms the curve in the graph.
The output values or range consists of the values of y formed as a result of the input of x.
Look at the x values corresponding to the end points of curve.
The curve extends from x = 0 to x = 3.
But at x = 3, it is an open circle. So x = 3 is not included.
So the input values are from x = 0 to x = 3, where 3 is not included.
Domain in interval notation = [0, 3).
Similarly the values of y extends from y = -4 to y = 5, where 5 is not included, since there is an open circle there.
Range in interval notation = [-4, 5).
Hence the domain and range are [0, 3) and [-4, 5) respectively.
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3. The polynomial function H(x)=x³-4x + 5x-2 has a known linear factor
of (x-1). Use long division and factoring of quadratics to rewrite H(x) as a product of
linear factors.
The polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).
What is a polynomialA polynomial is a mathematical expression which have a sum of powers in one or more variables with coefficients. The highest power of the variable in a polynomial is called its degree.
We shall divide the polynomial x³ - 4x³ + 5x² - 2 by x - 1 as follows;
x³ divided by x equals x²
x - 1 multiplied by x² equals x³ - x²
subtract x³ - x² from x³ - 4x³ + 5x² - 2 will result to -3x² + 5x - 2
-3x² divided by x equals -3x
x - 1 multiplied by -3x equals -3x² + 3x
subtract -3x² + 3x from x³ - 4x³ + 5x² - 2 will result to 2x - 2
2 divided by x equals 2
x - 1 multiplied by 2 equals 2x - 2
subtract 2x - 2 from 2x - 2 will result to a remainder 0
Therefore, the polynomial H(x) = x³ - 4x³ + 5x² - 2 divided by (x - 1), will give a quotient of x² - 3x + 2 and a remainder of 0 using the long division, we can write H(x) = (x -1)(x² - 3x + 2).
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True or False: The equation x = x2u + x3v, with x2 and x3 free (and neither u or v a multiple of the other), describes a plane through the origin
False. The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane.
The equation [tex]x = x2u + x3v[/tex] describes a line through the origin, not a plane. This equation can be written as [tex]x = x2u + x3v[/tex], which is the same as [tex]ux2 + vx3 - x = 0[/tex]. This equation is in the form of [tex]ax + by + cz = d[/tex], with a = u, b = v, c = -1, and d = 0. Since the coefficients of [tex]x2, x3[/tex], and x are all non-zero, we can conclude that this equation describes a line, not a plane. To confirm, we can calculate the direction vector for the line. The direction vector is (u, v, -1), which is a single vector and not two or more vectors that would be necessary to describe a plane.
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find invertible matrices such that is non-invertible. choose so that (1) neither is a diagonal matrix and (2) are not scalar multiples of each other.
Invertible matrices P = [1 -2; 0 1] and Q = [1 0; 2 1] such that
A = PQ = [1 2; 2 -3] is non-invertible.
To find invertible matrices such that a given matrix is non-invertible, we can use the fact that if A is non-invertible, then the system of linear equations Ax = 0 has a non-trivial solution. This means that there exists a non-zero vector x such that Ax = 0.
Let's start with a non-invertible matrix A, for example:
A = [1 2; 2 4]
The determinant of A is 0, which means that A is non-invertible.
To find a non-zero vector x such that Ax = 0,
We can solve the system of linear equations:
x + 2y = 0
2x + 4y = 0
This system is equivalent to the single equation:
x + 2y = 0
If we choose y = 1, then x = -2, and we get the non-zero vector:
x = [-2; 1]
Now we can use x to construct invertible matrices P and Q such that
PQ = A, as follows:
P = [1 -2; 0 1]
Q = [1 0; 2 1]
The inverse of P is:
P^-1 = [1 2; 0 1]
And the inverse of Q is:
Q^-1 = [1 0; -2 1]
We can verify that P and Q are invertible and that PQ = A:
PQ = [1 -2; 0 1][1 0; 2 1]
PQ = [1 -2; 2 -4 + 1]
PQ = [1 2; 2 -3] = A
Therefore, we have found invertible matrices P and Q such that A = PQ is non-invertible.
Note:- that neither P nor Q is a diagonal matrix, and they are not scalar multiples of each other.
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The complete question may be:
Find non-invertible matrices A, B such that A+B is invertible. Choose
A, B, so that (1) neither is a diagonal matrix and (2) A, B are not scalar multiples of each other.
Please help
At the beginning of spring, Savannah planted a small sunflower in her backyard. When it was first planted, the sunflower was 5 inches tall. The sunflower then began to grow at a rate of 0.5 inches per week. How tall would the sunflower be after 7 weeks? How tall would the sunflower be after w weeks?
Answer:
24.5in
Step-by-step explanation:
7 days in a week. 7 weeks. 7 x 7=49
49 x 0.5 + 24.5in
The sum of digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. Find the number.
(a) 18
(b) 27
(C) 36
(d) 72
With the sum of digits of a two-digit number is 9. If the digits are reversed, the number is increased by 63. The number is 18. So, the correct option is A.
Let the two-digit number be represented by 10x + y, where x is the tens digit and y is the ones digit. We are told that x + y = 9 and that if the digits are reversed, the number is increased by 63.
If we reverse the digits of the original number, we get 10y + x. The difference between this number and the original number is 63, so we can set up the equation:
10y + x - (10x + y) = 63
Simplifying this equation, we get:
9y - 9x = 63
Dividing both sides by 9, we get:
y - x = 7
Now we have two equations: x + y = 9 and y - x = 7. We can solve this system of equations by adding the two equations:
2y = 16
y = 8
Substituting y = 8 into x + y = 9, we get:
x + 8 = 9
x = 1
Therefore, the original number is 10x + y = 10(1) + 8 = 18. So the answer is (a) 18.
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At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 30 km/h. How fast (in km/hr) is the distance between the ships changing at 4:00 p.m.? (Round your answer to three decimal places.)
The distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.
How fast is the distance between the ships changing at 4:00 PM?From the question, we have the following parameters that can be used in our computation:
Distance, D = 150 km
Rates = 35 km/h and 30 km/h
Let t be the time elapsed from noon to 4:00 PM
So, we have
t = 4
The distance between the ships to their distances is represented as
d^2 = (D + rate 1 * t)^2 + (rate 2 * t)^2
So, we have
d^2 = (150 + 35t)^2 + (30t)^2
d^2 = (150)^2 + 10500t + 1225t^2 + 900t^2
Differentiate with respect to time (t)
2D d' = 10500 + 2450t + 1800t
So, we have
D d' = 5250 + 1225t + 900t
Substitute 4 for t
D d' = 13750
So, we have
d' = 13750/D
This gives
d' = 13750/150
d' = 91.6
So the distance between the ships is increasing at a rate of approximately 91.6 km/h at 4:00 PM.
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Which statements about this situation are true?
Select all the correct answers.
The maximum value in the range is $200.
The maximum value in the range is $320.
The maximum value in the domain is 200.
The minimum value in the domain is 0.
The statements about the given situation that are true about domain and range are;
B: The maximum value in the range is $320.
D: The minimum value in the domain is 0.
How to find the domain and range?The domain is defined by b, the number of bracelets sold.
The minimum value in the domain is 0, which represents no bracelets sold.
The maximum value in the domain is 260, which represents the largest number of bracelets the group can make, and the largest number they could sell.
The range is defined by f(b), the amount of profit on the bracelets.
To find the maximum value in the range, we find f(260), the profit on selling the maximum in the domain.
Substitute 260 for b in f(b) = 2b – 200 to get:
f(260) = 2(260) – 200
f(260) = 320
The maximum value in the range is $320.
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Complete question is;
The high school jazz band is selling homemade leather bracelets at a local craft fair to raise money for a trip. The group has a $200 budget to spend on supplies, which is enough to make 260 bracelets. The group is charging $2 per bracelet at the craft fair.
Which statements about this situation are true?
Select all the correct answers.
The maximum value in the range is $200.
The maximum value in the range is $320.
The maximum value in the domain is 200.
The minimum value in the domain is 0.
find the lateral surface area of the prism
The lateral surface area of a prism is the sum of the areas of all its rectangular faces.
What is surface area?Surface area is a two-dimensional measure that refers to the total area of a surface, such as the area of a two-dimensional shape, a three-dimensional solid, or a combination of both. It is the sum of the areas of all the faces of a solid object. It is also referred to as the area of the boundary of a three-dimensional object. It can be used to calculate the volume of an object, and is also used in other calculations like area of the base of a triangle, area of a circle, and more.
It is calculated by multiplying the perimeter of the base by the height of the prism. For example, if the base of a prism is a square with side length s and the height is h, then the lateral surface area of the prism can be calculated as 4sh. If the base of the prism is a rectangle with length l and width w, then the lateral surface area of the prism can be calculated as 2lw + 2wh.
The lateral surface area of a prism can also be calculated by adding the areas of the triangular faces. If the base of the prism is a triangle with side lengths a, b, and c, then the lateral surface area of the prism can be calculated as a + b + c.
It is important to note that the lateral surface area of a prism is different from its surface area. The surface area of a prism is the sum of the areas of its lateral faces and its two bases.
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Complete question:
How to find the lateral surface area of the prism?
Communicate and Justify
A store made $650 on Monday. It made $233 on Tuesday
morning and $378 on Tuesday afternoon.
Leah says the store made more money on Tuesday.
Her work is shown at the right.
1. What is Leah's argument? How does she support it?
2. Tell how you can analyze Leah's reasoning.
3. Does Leah's reasoning make sense?
Leah's argument is that the rounded up figures for Tuesday sales are greater than the sales for Monday. She supports it by summing up the sales figures.
Leah's reasoning is wrong because she rounded up $ 233 to $ 300 instead of to $ 200.
Leah's reasoning therefore does not make sense.
What should Leah have done ?Leah attempts to round the Tuesday sales figures to the nearest 100. In doing so, she rounded $ 233 to $ 300 instead of $ 200 which was the closest.
If she had done so, the result would be :
= 200 + 400
= $ 600
This would then show that Monday's figures were higher than Tuesday.
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show that f(z) = z is nowhere differentiable ) i.e. there is no point z0 e c such that f1(z0) exists)
The limit of the difference quotient must not exist at any location z0 in the complex plane in order to demonstrate that f(z) = z is nowhere differentiable.
For f(z), the difference ratio is as follows:
[f(z Plus h) - f(z)] / h = [(z + h) - z] / h = h / h = 1
As h gets closer to 0, we take the maximum and obtain:
lim h0 [z + h - z] / = lim h 0 h / h = 1
This limit is constant at 1 and is unaffected by the number of z. The limit of the difference quotient must not exist at any location z0 in the complex plane in order to demonstrate that f(z) = z is nowhere differentiable. As a result, f(z) = z is never differentiable and the limit of the difference quotient is not present at any position z0 in the complex plane.
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Find the distance from the point P(2, 1, 4) to the plane through the points Q(1, 0, 0), R(0, 2, 0), and S(0, 0, 3).
The Distance from the point P(2, 1, 4) to the plane is approximately -0.9487.
The equation of a plane can be found using the normal vector and a point on the plane.
First, we can find the normal vector of the plane by finding the cross product of two vectors connecting the points on the plane.
The vector from point Q to R is (0, 2, 0) - (1, 0, 0) = (-1, 2, 0)
The vector from point Q to S is (0, 0, 3) - (1, 0, 0) = (-1, 0, 3)
The normal vector is the cross product of these two vectors:
n = cross(-1, 2, 0), (-1, 0, 3)) = (3, 3, -2)
Next, we can find the equation of the plane using the normal vector and a point on the plane, say Q(1, 0, 0):
ax + by + cz = d
3x + 3y - 2z = d
3x + 3y - 2z = 3(1) + 3(0) - 2(0) = 3
Finally, we can find the distance from the point P(2, 1, 4) to the plane by finding the perpendicular distance between the point and the plane. This can be done using the formula:
[tex]d = (Ax + By + Cz + D)/\sqrt{ {A^2 + B^2 + C^2}[/tex]
[tex]d = (3x + 3y - 2z + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]
[tex]d = (3(2) + 3(1) - 2(4) + 3)/\sqrt{ (3^2 + 3^2 + (-2)^2)[/tex]
[tex]d = (-3)/\sqrt{(18)[/tex]
d = -3/3.162 = -0.9487
So the distance from the point P(2, 1, 4) to the plane is approximately -0.9487.
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Grace had a savings of 9000 part of which was invested at 7% and the rest at 9%. How much has she invested at each rate if her annual income of from the investments was 741.60
The amount grace invested $3420 at 7% and the rest of her savings, $5580, at 9%.
What is simple interest?
Simple interest is a method of calculating the interest charge. Simple interest can be calculated as the product of principal amount, rate and time period.
Simple Interest = (Principal × Rate × Time) / 100
We are given that;
Amount grace invested= 9000 at 9%
Rate for rest of it= 9%
Annual income=741.6
Now,
Let's call the amount Grace invested at 7% "x".
Then the amount she invested at 9% would be "9000 - x", since she invested the rest at the higher rate.
We know that her annual income from the investments was $741.60.
The amount of money she made from the 7% investment would be 0.07x (7% expressed as a decimal multiplied by the amount invested), and the amount of money she made from the 9% investment would be 0.09(9000 - x) (9% expressed as a decimal multiplied by the amount invested).
So we can set up the equation:
0.07x + 0.09(9000 - x) = 741.60
Simplifying and solving for x:
0.07x + 810 - 0.09x = 741.60
-0.02x = -68.4
x = 3420
Therefore, by the given interest rate answer will be $5580, at 9%.
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susan currently walks to school from her apartment, which is 1.3 miles away from her first class. she typically walks at a speed of 3 miles per hour. she is considering buying a used bicycle from deseret industries to ride to campus. susan assumes that if she were riding a bike, she could go about 5 miles per hour.How many minutes could susan save getting to class each morning if she were to ride the bike?
Susan could save 10.4 minutes getting to class each morning if she were to ride the bike.
As per the data given:
The distance is given between the school and the apartment = 1.3 miles
Susan's walking speed = 3 miles/hr
Now we know that speed = distance ÷ time
Putting values in the above formulae, we get the time for walking situation
3 miles/hr = 1.3 ÷ time
Time = 1.3 ÷ 3 miles
Time = (13 ÷ 30 )hr
= (13 ÷ 30) × 60 minutes
= 13 × 2
= 26 minutes
The time taken when she is walking is 26 minutes.
Here we have to determine how many minutes could Susan save getting to class each morning if she were to ride the bike.
Now if she uses a bike, the speed is 5 miles/hr
Again applying same formulae speed = distance ÷ time
5 miles/hr= 1.3 ÷ time
Time= 1.3 ÷ 5 miles
= (13 ÷ 50)hr
= (13 ÷ 50) × 60 minutes
= 15.6 minutes
Time taken by bike is 15.6 minutes
Total time saved = time taken when walking - time taken using the bike
= 26 - 15.6
= 10.4 minutes
Hence, Susan could save 10.4 minutes if she uses a bike.
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Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
y=ln(x), y=0,x=5; about the y -axis___ dx
The integral for the volume of the solid is ∫2π(eᵇ)(dy)
In calculus, finding the volume of a solid is an important concept. One method to do this is by using integration.
To find the volume of the solid obtained by rotating the region bounded by y=ln(x), y=0, and x=5 about the y-axis, we can use the method of cylindrical shells. This involves dividing the region into infinitely thin vertical strips and rotating each strip around the y-axis to create cylindrical shells.
To set up the integral, we can use the formula for the volume of a cylindrical shell, which is 2πrhΔx, where r is the distance from the y-axis to the edge of the strip, h is the height of the strip, and Δx is the thickness of the strip.
Since we are rotating around the y-axis, we need to express the curves in terms of y instead of x. Using the inverse of the natural logarithm function, we can rewrite
=> y=ln(x) as x=eᵇ.
Thus, the region is bounded by x=5, y=0, and
=> x=eᵇ.
Next, we need to find the distance from the y-axis to the edge of the strip, which is simply the x-value at a given y. This is given by r=e^y.
The height of the strip is simply Δy, the thickness of the strip. To find this, we can take the difference between two consecutive y-values. Thus, Δy=dy.
Putting it all together, the integral for the volume of the solid is:
=> ∫2π(eᵇ)(dy)
This integral represents the sum of the volumes of all the cylindrical shells that make up the solid. However, we still need to evaluate the integral to get the actual volume of the solid.
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