[tex]5-3=2\\\\6-10=-4\\\\11.6-3=8.6\\\\63-4\times 5=63-20=43[/tex]
So, the first, second, and fourth equations are integers.
The first and last are positive integer and second is negative integer.
What are integers?Integers come in three types:
Zero (0)Positive Integers (Natural numbers)Negative Integers (Additive inverse of Natural Numbers)Given:
a) 5-3 =2
b) 6-10 = -4
c) 11.6 - 3 =8.6
d) 63 - 4 x 5 =63- 20 = 43
So, among the above first and last are positive integer and second is negative integer.
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HELP PLEASE I HAVE 15 MINUTES
1. Clark looks at the architectural plan of a four-walled room in which the walls meet each other at right angles. The length of one wall in the plan is 12 inches. The length of the diagonal of the floor
of the room in the plan is 20 Inches.
Answer:
Step-by-step explanation: Multiply 20 by 12 and divied 8 that should give u your answer
If A is a set with A = { 2, 5, 7, 11 } then what is |(A xA) U A)|
20
16
32
None of the above
Which one is the correct answer and why, please.
The given series consists of prime numbers starting from 2. So, the missing term is the prime number after 11, which is 13.
So, None of the above will be the correct answer.
A×(B∩C)=(A×B)∩(A×C)
We have B∩C={1,2,3,4}∩{5,6}=ϕ
∴ L.H.S = A×(B∩C)=A×ϕ=ϕ
A×B={(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4)}
A×C={(1,5),(1,6),(2,5),(2,6)}
∴R.H.S.=(A×B)∩(A×C)=ϕ
∴L.H.S=R.H.S
Hence A×(B∩C)=(A×B)∩(A×C)
To verify: A×C is a subset of B×D
A×C={(1,5),(1,6),(2,5),(2,6)}
B×D={(1,5),(1,6),(1,7),(1,8),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),
(3,8),(4,5),(4,6),(4,7),(4,8)}
which is 13.
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if the first quadrant starts on (12,11) and moves 8 units left and 7 units down where will it be
The first quadrant starts on (12,11)
If the quadrant moves 8 units left it will be located at (12-8, 11) = (4,11)
and if it moves 7 units down it will be located at (4,11-7) = (4,4)
So the new location of the quadrant is (4,4)
It's important to notice that the first quadrant starts on (12,11) and the coordinates are (x,y) where x is on the horizontal axis and y is on the vertical axis. So when it moves left it's decreasing the x value and when it moves down it's decreasing the y value
Share £32 in the ratio 1 : 5 : 2.
Step-by-step explanation:
In order to share the money equally
We take the variables as 'x'
This is because the given values are in the form of a ratio
So to do this we plug these values into a equation which is,
[tex]x + 5x + 2x = 32[/tex]
Adding the values we get,
[tex]8x = 32[/tex]
Dividing both sides by 8 we get,
[tex]\frac{8x}{8}= \frac{32}{8}[/tex]
[tex]x = 4[/tex]
Now we multiply the given values by x which is 4 in order to find how much money each ratio get
[tex]x = 4[/tex]
[tex]5x = 5X4 = 20[/tex]
[tex]2x = 2X4 = 8[/tex]
4 + 20 + 8 = £ 32 (Proving)
y=x^2+9x+8 in vertex form
The vertex form of parabola y = x² + 9 · x + 8 is the quadratic equation y + 49 / 4 = (x + 9 / 2)².
How to determine the equation of a parabola in vertex form
Herein we find the equation of a parabola in standard form, that is, a quadratic equation of the form:
y = a · x² + b · x + c
Where:
a, b, c - Real coefficientsx - Independent variable.y - Dependent variable.And we are asked to transform this expression into vertex form, that is, a quadratic equation of the form:
y - k = C · (x - h)²
Where:
(h, k) - Coordinates of the vertex.C - Vertex constant.This can be done by means of algebra properties. First, write the quadratic equation:
y = x² + 9 · x + 8
Second, use algebra properties until a perfect square trinomial is found:
y = x² + 9 · x + 8
y + 49 / 4 = x² + 9 · x + 81 / 4
Third, factor the perfect square trinomial:
y + 49 / 4 = (x + 9 / 2)²
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Question 7(Multiple Choice Worth 2 points)
(Linear Relationships MC)
Which linear equation shows a proportional relationship?
y equals two thirds times x
y equals negative 3 times x minus one seventh
y equals three fourths times x minus 5
y equals 3 times x plus 7
The correct linear equation which shows a proportional relationship is,
⇒ y equals two thirds times x.
What is linear expression?A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.
Now,
We know that;
The proportion relationship is,
⇒ y = kx
Where, k is constant of proportional.
By option 1;
The expression is,
⇒ y = 2/3 × x
⇒ y = 2/3x
Hence, It shows the linear equation.
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Blood alcohol concentrations of drivers involved in fatal crashes and then given jail sentences are shown below. Find mean, median, and mode. Fill in blanks.
0.26, 0.18, 0.18, 0.16 0.13, 0.24
0.29, 0.24, 0.14, 0.16, 0.10, 0.16
The mean is:________________. The median is: _____________. The mode(s) is (are): _______________
I need help
Find the length of the segment with the given endpoints.
(-3, -12.5), (-3, 16.5)
Answer:
The length would be 29.
Step-by-step explanation:
Follow the distance formula in the image.
-Hope this Helped
write -40.425 as a mixed number
Answer:
-40 17/40
Step-by-step explanation:
Answer: -40 7/40
Step-by-step explanation:
Rewrite the decimal number as a fraction with 1 in the denominator
40.425=40.4251/1
Multiply to remove 3 decimal places. Here, you multiply top and bottom by 103 = 1000
40.4251×1000/1000=40425/1000
Find the Greatest Common Factor (GCF) of 40425 and 1000 and reduce the fraction by dividing both numerator and denominator by GCF = 25
40425÷25
1000÷25
=1617/40
Simplify the improper fraction
in conclusion
-40 17/40
you must not forget the minus/negative sign :)
The two concentric circles shown have diameters a mm and b mm, where a and b are integers with a < b. The gray region between the two circles has area 48π mm2 . What is the sum of all possible values of b?
The sum of all possible values of b is 124 in the given question.
What is concentric circles?If two or more objects have a common centre, they are said to be concentric in geometry.
Concentric shapes include spheres, circles, regular polyhedral, and regular polygons because they all share a central axis. In Euclidean geometry, concentric circles always have different radii but the same centre.
The difference between the areas is [tex]$ \frac{\pi(b^2 - a^2)}{4} = 48 \pi[/tex]
Therefore b² = a² +1 92
We can write the equation as (b−a)(b+a)=192
So b−a and b+a are factors of 192
Here are the possibilities:
b−a=1, b+a=192 ;
b−a=2, b+a=96 ;
b−a=3, b+a=64 ;
b−a=4, b+a=48 ;
b−a=6, b+a=32 ;
b−a=8, b+a=24 ;
and
b−a=12, b+a=16
Add the pairs of equations and divide by 2 (as long as the sum is even): b=49; b=26; b=19; b=16; and b=14.
49 + 26 + 19 + 16 + 14 = 124
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PLEASE HELP MEEEEEE IM STRUGGLING
Answer:
Step-by-step explanation: i can't see the lab top
Are there times when you use or calculate percentages in your personal life or at work?
Provide a detailed description of ONE (1) real-life application of percentages. (Note: Please do not use Blood Alcohol Content as your topic - I will explain why when going over this assignment with the class)
Explain the benefits and/or drawbacks that result from using percentages in this application, i.e. explain your reasons for why using percentages for this application is a good thing and/or reasons why using percentages for this application is NOT a good thing.
Create an original word problem that illustrates the application that you have described. This word problem should be written in a question-and-answer format and should include BOTH a clearly stated math problem AND a detailed solution to your math problem (show ALL of your work).
There are times when I use or calculate percentages in my personal life or at work. This is true.
The benefits that result from using percentages in this application is that it gives one an idea of the value to be paid as tax. The drawback is that it can be complicated for large number.
A word problem that can Illustrate percentage is that "An income is $1000 and there's a tax of 10%, how much will be paid?"
How to illustrate the percentage?A number or ratio expressed as a fraction of 100 is called a percentage. Divide the value by the total value to get the percentage, then multiply that number by 100.
For instance, many statistics in the media, bank interest rates, retail discounts, and inflation rates are all expressed as percentages. For understanding the financial aspects of daily life, percentages are crucial.
For example, an income is $1000 and there's a tax of 10%, the tax will be:
= 10% × $1000
= $100
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Bart found 20 quadrilaterals in his classroom. He made a Venn diagram using the properties of the quadrilaterals, comparing those with four equal side lengths. (E) and those with four right angles (R). Given that a randomly chosen quadrilateral has four right angles, what is the probability that the quadrilateral also has four equal side lengths? Express your answer in percent form, rounded to the nearest whole percent.
The probability that a randomly chosen quadrilateral with four right angles will also have four equal side lengths is 25%.
What is probability?Probability is a branch of mathematics that deals with the likelihood of an event occurring. It is the measure of how likely it is that something will happen or be the case. Probability is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability can be used to calculate the chances of a certain outcome based on a set of conditions or data. Probability can also be used to make decisions and predictions about the future.
This is because of the Venn diagram that Bart created. If a quadrilateral has four right angles, then it can either have four equal side lengths (which makes it a square) or it can have four unequal side lengths. Since there are 8 quadrilaterals in the set with four equal side lengths (E) and 16 quadrilaterals in the set with four right angles (R), the probability that a randomly chosen quadrilateral with four right angles will also have four equal side lengths is 8/16, which is equivalent to 25%. Therefore, the probability that a randomly chosen quadrilateral with four right angles will also have four equal side lengths is 25%, rounded to the nearest whole percent.
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A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x, is found to be 113, and the sample standard deviation, s, is found to be 10.
(a) Construct a 95% confidence interval about μ if the sample size, n, is 26.
(b) Construct a 95% confidence interval about μ if the sample size, n, is 15.
(c) Construct a 90% confidence interval about μ if the sample size, n, is 26.
(d) Could we have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed?
a) The 95% confidence interval about μ with n = 26 is given as follows: (109, 117).
b) The 95% confidence interval about μ with n = 15 is given as follows: (107.5, 118.5).
c) The 90% confidence interval about μ with n = 26 is given as follows: (109.7, 116.3).
d) The intervals could not have been computed if the population was not normally distributed, as the sample size is less than 30.
What is a t-distribution confidence interval?The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the following rule:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
In which the variables of the equation are presented as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The sample mean and standard deviation for this problem are given as follows:
[tex]\overline{x} = 113, s = 10[/tex]
The critical values are given as follows:
95% confidence, 25 df: 2.0595.95% confidence, 14 df: 2.1448.90% confidence, 25 df: 1.7081.The bounds of the interval for item a are given as follows:
[tex]113 - 2.0595\frac{10}{\sqrt{26}} = 107.5[/tex][tex]113 + 2.0595\frac{10}{\sqrt{26}} = 118.5[/tex]The bounds of the interval for item b are given as follows:
[tex]113 - 2.1448\frac{10}{\sqrt{15}} = 109[/tex][tex]113 + 2.1448\frac{10}{\sqrt{15}} = 117[/tex]The bounds of the interval for item c are given as follows:
[tex]113 - 1.7081\frac{10}{\sqrt{26}} = 109.7[/tex][tex]113 + 1.7081\frac{10}{\sqrt{26}} = 116.3[/tex]The Central Limit Theorem states that the intervals can only be calculated for non-normal populations if the sample size is greater than 30.
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What is the solution to the equation below?
12+v1 – 5x = 18
A. x = -7
B. x = 7
C. x = 1
D. x = -1
Solving given algebra problem, value of x is equal to -1. It is the area of mathematics that uses arithmetic to manipulate or operate abstract symbols rather than actual numbers.
In algebra, how do you solve square roots?Isolate the squared term and the constant term on the opposite sides of the equation to solve quadratic equations using the square root method. After that, take the square root of both sides, plus or minus the side with the constant term.
Given equation,
[tex]12+\sqrt{1-5x} =18\\\sqrt{1-5x}=18-12\\ \sqrt{1-5x}=6\\\\square\\ case1\\1-5x=64x=-1\\case2\\1-5x=-6\\x=7/5[/tex]
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кя
KH
⠀
Problem 5.1
A publisher wants to know the thickness of a new book.
The book has a front cover and a back cover, each with a
thickness of of an inch.
The paper has a thickness of
inch per 100 pages.
Write an equation that represents the total width of the
book, y, for every 100 pages of paper, x.
Given that for every hundred pages a thickness of 1/4 inch is added to the book, then the equation is: y = 1/2 + 1/4*x
How to find the Solution?Considering that the book's front and back covers each have a thickness of 1/4 inch, adding them together results in a thickness of 1/2 inch for the book, regardless of how many pages it has.
Given that a book's thickness increases by a quarter of an inch for every hundred pages, the equation is:
y = 1/2 + 1/4*x
where y denotes the book's width and x is the number of bond paper pages printed every 100 pages. Y has a length of inches.
Equations that have the same roots or solutions are said to be equivalent.
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From midnight until 8:00 AM, the temperature tripled and then rose 5 degrees. The temperature at 8:00 AM was –7 degrees Celsius. Write and solve an equation to find the temperature at midnight.
The equation for the temperature at midnight and the solution are Tm = 3T + 5 and -16 degrees Celsius respectively
How to write and solve an equation?
An equation is a mathematical statement that is made up of two expressions joined by an equal sign.
Let T represent the temperature
The temperature tripled and then rose 5 degrees can be written as:
Tm = 3T + 5
where Tm is the temperature at midnight.
If the temperature at 8:00 AM was –7 degrees Celsius, we have:
Tm = 3(-7) + 5
Tm = -21 + 5
Tm = -16 degrees Celsius
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What is -2 2/5 times -1/3?
(With work)
The product of -22/5 × - 1/3 is 12/15
What is product of a number?A product in math is defined as the result of two or more numbers when multiplied together. For example if 5 is multiplied by 2 , the product is 10 i.e 5 × 2 = 10 , 10 is the product where 5 and 2 are the factors. Also 5×4 = 20, 20 is the product and the 5 and 4 are the products.
The division of this of the product with any of the factors will give the other factors.
To solve -2 2/5 × -1/3
we first change the mixed fraction to improper fraction.
We then multiply -12/5 with -1/3
-2 2/5 = -12/5
-12/5 × -1/3
= 12/15
therefore the value of -2 2/5 × -1/3 is 12/15
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The histogram shows the distribution of the annual hours of commuting delay per traveler for 46 small and medium urban areas, fewer than one million in population. Kebay 5 10 15 20 hours Which of the following must be true?
a. The mean is greater than the median.
b. The mean is less than the median.
c. The mean is the same as the median.
In this problem the statement that the mean is greater than the median is true. So, the correct answer is option(a).
We have a histogram which shows the distribution of the annual hours of commuting delay per traveler for 46 small, medium and urban areas, fewer than one million in population. See the diagram carefully and try to draw the conclusion. The shape of the histogram showing that positive skewed or right skewed distribution since Curve increases fastly and decreases slowly. A positive skewed distribution is a type of distribution where most of the values are concentrated in the left tail of the distribution and the right tail of the distribution is longer. In positive skewed distribution,
i) Mean > Median
ii) (Q₃ - Q₂) > (Q₂ - Q₁)
iii) Sqrt(β₁) >0
Hence, The mean is greater than the median is true.
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HEL Two students created paintings for their art class. The canvas size used for each is shown.
2 rectangles: Rectangle 1 has a length of 6 feet and width of 4 feet. Rectangle 2 has a length of 3 feet and width of 2 feet.
Which statement is true about the ratios comparing the width and length of each painting?
The paintings' width to length ratios are equal because the ratio 4 to 6 is equal to the ratio 2 to 3.
The paintings' width to length ratios are equal because the ratio 6 to 4 is equal to the ratio 2 to 3.
The paintings' width to length ratios are different because the ratio 3 to 2 is not equal to 6 to 4.
The paintings' width to length ratios are different because the ratio 4 to 6 is not equal to 3 to 2.P ASAP
The statement that is true about the ratios comparing the width and length of each painting is:
The paintings' width to length ratios are equal because the ratio 4 to 6 is equal to the ratio 2 to 3.
Ratio of numbers.A ratio is a fraction which compares the values of two numbers. Such that they can be expressed as a ratio of one to the other. Example is the ratio of a to b which can be expressed as; a:b or a/ b.
From the first dimension of the canvas size, we have;
length = 6 feet
width = 4 feet
So that,
the ratio comparing the width and length = 4/ 6
= 0.6667
From the second dimensions of the canvas size, we have;
length = 3 feet
width = 2 feet
the ratio comparing the width and length = 2/ 3
= 0.6667
Therefore, the statement that is true about the ratios comparing the width and length of each panting is; The paintings' width to length ratios are equal because the ratio 4 to 6 is equal to the ratio 2 to 3.
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Determine the ratio adjacent over hypotenuse using
8. The ratio of the adjacent over the hypotenuse is 8:17
10. The ratio of the adjacent over the hypotenuse is 1:√2
Calculating the ratio of adjacent over hypotenuseFrom the question, we are to calculate the ratio of the adjacent over the hypotenuse in the given triangles
8.
Using angle A as the reference angle
Adjacent = 16
Hypotenuse = 34
Thus,
The ratio of the adjacent over the hypotenuse = 16/34
The ratio of the adjacent over the hypotenuse = 8/17
Hence, the ratio is 8:17
10.
First,
We will calculate the hypotenuse, H
From the Pythagorean theorem, we can write that
H² = 4² + 4²
H² = 16 + 16
H² = 32
H = √32
H = 4√2
Now,
Using angle A as the reference angle
Adjacent = 4
Hypotenuse = 4√2
Thus,
The ratio of the adjacent over the hypotenuse = 4/4√2
The ratio of the adjacent over the hypotenuse = 1/√2
Hence, the ratio is 1:√2
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determine the constant rate of change (slope) of the linear function. a high school basketball team notices that attendance at its games changes at a constant rate based on the number of losses the team has suffered. when the team had lost seven games, 295 people attended the next game. when the team had lost 13 games, 199 people attended the next game.
The team loses 20 people/audience with each loss.
The slope formula defines to the formula used to calculate the steepness of a line and determines how much it's inclined. To calculate the slope of the lines, the x and y coordinates of the points lying on the line can be used.
The formula to calculate slope is:
m = (y2 - y1)/(x2 - x1) = Δy/Δx
When the team lost eight games, 295 people attended the next game.
When the team lost 13 games, 199 people attended the next game.
Slope = 295-199/8-13
slope=100/-5
slope=-20
Therefore, the team loses 20 people/audience with each loss.
And the attendance loss is 20 times as great as the number of games lost/losses
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A ball is rolled at a speed of 12m/sec after 40 seconds it comes to a stop . What is the acceleration of the ball ? Answer to nearest hundredth
Therefore , the solution of the given problem of speed comes out to be
the ball accelerates at 0.33 m/s2.
What is speed?Speed is a pace that is measured in terms of how far an object travels in a given amount of time. Speed can be measured in a wide range of ways, including in miles an hour, laps every minute, and meters a second. There are connections between the units of time, distance, and speed. Three ways of expressing the relationship are
d = rt, r = dt, and t = dr.
Given -
The ball's initial speed is 0 meters per second.
v = 12 m/s is the ball's final speed.
36 seconds were used.
Determine the ball's acceleration.
Solution:
=> A = (v-u)/t
=>A = (12-0)/36
=> A = 12/36
=> A =1 /3
=> A = 0.33 m/s²
Consequently, the ball accelerates at 0.33 m/s2.
Therefore , the solution of the given problem of speed comes out to be
the ball accelerates at 0.33 m/s2.
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need a bit help, im stuck on these.
Answer:
7. VS = 17
8. QR = 6
9. UT = 17
Step-by-step explanation:
You have trapezoid QRTU with midsegment VS parallel to bases QR and TU. Given dimensions for two of these parallel lines, you want the measure of the third.
Segment relationshipsThe markings on the parts of segments QU and RT indicate that points V and S are their midpoints, respectively. This makes segment VS a "midsegment", one that joins the midpoints of opposite sides. As such, its length is the average of the lengths of QR and TU:
VS = (QR +TU)/2
This is the relation that is used for problem 7.
For problems 8 and 9, this is rearranged to give the values of QR and UT:
QR = 2VS -UT
UT = 2VS -QR
7. VSFrom above, ...
VS = (QR +UT)/2 = (12 +22)/2
VS = 17
8. QRFrom above, ...
QR = 2VS -UT = 2(9) -12
QR = 6
9. UTFrom above, ...
UT = 2VS -QR = 2(11) -5
UT = 17
__
Additional comment
You will note that the line segments form an arithmetic sequence: the difference in length between adjacent segments is the same.
7: 12, 17, 22
8: 6, 9, 12
9: 5, 11, 17
Simplify
8!/(8-2)
A. 720
b. 0.018
C. 8
d. 56
e. 40320
f. 1.333
Answer:
D) 56
Step-by-step explanation:
[tex]\displaystyle \frac{8!}{(8-2)!}=\frac{8!}{6!}=\frac{8*7*6*5*4*3*2*1}{6*5*4*3*2*1}=8*7=56[/tex]
Solve this please i will mark brainlest and give 5 hearts
Answer: 5.75pi(yards)
Step-by-step explanation: We know the radius is 3 so the area is 9pi. The shaded area has an angle of 230 and the total angle of a circle is 360. We multiply the area by the fraction of the angle which is 230/360(9pi).
please write an equation for each line.
By analytic geometry, the equations for lines represented by figure are:
Red line: y = 2 · x + 5 Yellow line: y = - 3.333 · x + 5 Black line: y = 2 · x - 6 Green line: y = - 2 Blue line: x = 5How to derive the equations of lines seen in a picture
In this problem we find the representation of five lines (red - yellow - green - blue - black). There is a horizontal line (green), a vertical line (blue) and three oblique lines (red - yellow - black). According to analytic geometry, we find the following definitions:
Horizontal lines: y = aVertical lines: x = bOblique lines: y = m · x + b (m - Slope, b - Intercept).The slope for oblique lines is determined by secant line formula:
m = Δy / Δx
Where:
Δx - Change in independent variable.Δy - Change in dependent variable.From picture we get the following hints:
Red line crosses the y-axis at (x, y) = (0, 5).Yellow line crosses the y-axis at (x, y) = (0, 5).Black line crosses the y-axis at (x, y) = (0, - 6).Green line crosses the y-axis at (x, y) = (0, - 2).Blue line crosses the x-axis at (x, y) = (5, 0).Finally, we proceed to find the equations for each line:
Red line
Slope
m = 5 / 2.5
m = 2
Intercept
b = y - m · x
b = 5 - 2 · 0
b = 5
The equation for the red line is y = 2 · x + 5.
Yellow line
Slope
m = - 5 / 1.5
m = - 3.333
Intercept
b = y - m · x
b = 5 - (- 3.333) · 0
b = 5
The equation for the yellow line is y = - 3.333 · x + 5.
Black line
Slope
m = 6 / 3
m = 2
Intercept
b = y - m · x
b = - 6 - 2 · 0
b = - 6
The equation for the black line is y = 2 · x - 6.
Green line
The equation for the green line is y = - 2.
Blue line
The equation for blue line is x = 5.
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Answer 2x4 pls ITS FOR A TEST PLS ANSWER ILL MARK U BRAINLYEST
Answer:
2x4 is 8
Step-by-step explanation:
4 to times so 4+4 =8 thx
Need this please !!!!!
Answer:
m∠H = 53°m∠I = 74°m∠J = 53°----------------------------------
Since the triangle is isosceles, the two angles opposite to equal sides are congruent:
m∠J = m∠H = (3x + 11)°The sum of the three angles is 180°:
2(3x + 11) + 4x + 18 = 1806x + 22 + 4x + 18 = 18010x + 40 = 18010x = 140x = 14The angle measures are:
m∠J = m∠H = (3*14 + 11)° = 53°m∠I = (4*14 + 18)° = 74°Computer the lower Riemann sum for the given function f(x) = 4 - x^3 over the interval x E [0,1] with respect to the partition P = [0, 1/2, 3/4, 1]
Answer:
[tex]\dfrac{917}{256}=3.58203125[/tex]
Step-by-step explanation:
The Riemann sum is a method by which we can approximate the area under a curve using a series of rectangles.
The Lower Riemann Sum uses the minimum height of the rectangle on each subinterval.
As the Lower Riemann Sum is entirely below the curve, it is an underestimation of the area under the curve.
The number of partitions is the number of rectangles used.
Partitions can be of equal length or not of equal length.
Given:
Function: f(x) = 4 - x³Interval: [0, 1]Partition, P = [0, 1/2, 3/4, 1]The given partition divides the interval [0, 1] into 3 subintervals:
[0, 1/2], [1/2, 3/4] and [3/4, 1]To calculate the areas of the rectangles, multiply the width of each rectangle by its height.
The width is the difference between the x-values of each subinterval.
The height is the minimum value of the function across the subinterval. For the given function, this is the value of the function for the right side of the subinterval.
First rectangle [0, 1/2]:
[tex]\begin{aligned}\implies \left(\dfrac{1}{2}-0\right) \cdot f\left(\dfrac{1}{2}\right)&=\dfrac{1}{2} \cdot \left(4-\left(\dfrac{1}{2}\right)^3\right)\\\\&=\dfrac{1}{2} \cdot \dfrac{31}{8}\\\\&=\dfrac{31}{16}\end{aligned}[/tex]
Second rectangle [1/2, 3/4]:
[tex]\begin{aligned}\implies \left(\dfrac{3}{4}-\dfrac{1}{2}\right) \cdot f\left(\dfrac{3}{4}\right)&=\dfrac{1}{4} \cdot \left(4-\left(\dfrac{3}{4}\right)^3\right)\\\\&=\dfrac{1}{4} \cdot \dfrac{229}{64}\\\\&=\dfrac{229}{256}\end{aligned}[/tex]
Third rectangle [3/4, 1]:
[tex]\begin{aligned}\implies \left(1-\dfrac{3}{4}\right) \cdot f\left(1\right)&=\dfrac{1}{4} \cdot \left(4-\left(1\right)^3\right)\\\\&=\dfrac{1}{4} \cdot 3\\\\&=\dfrac{3}{4}\end{aligned}[/tex]
Therefore, the Lower Reimann Sum for the given function over the given interval and partitions is the sum of the area of the rectangles:
[tex]\implies \dfrac{31}{16}+\dfrac{229}{256}+\dfrac{3}{4}=\dfrac{917}{256}=3.58203125[/tex]