Daryl's weekly allowance, given the amount he ended with and the amount spent on clothes, is $ 30
How to find the weekly allowance ?To find the weekly allowance, first find the amount that Daryl had after he had spend half his allowance on clothes. This amount can be found by the formula:
= Amount Daryl ended with - Amount gotten from cleaning oven
= 25 - 10
= $ 15
The amount of $ 15 is the amount out of Daryl's weekly allowance that he had after he had spent on clothes. The $15 is therefore half of the weekly allowance which means that the weekly allowance is:
= 15 x 2
= $ 30
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What do you know about the solution(s) to the system of linear equations
PLS I DONT UNDERSTAND
Answer:
see explanation
Step-by-step explanation:
given a system of equations graphically , then the solution is at the point of intersection of the 2 lines.
However, the 2 lines shown are parallel and never intersect and therefore have no solution.
Write an equation of the line passing through the point (4, 3) that is perpendicular to the line 4x - 7y=9.
An equation of the line is y=_x+_
Answer:
y = (7/4)x - 4
Step-by-step explanation:
Rewrite the equation in standard format of y=mx+b, where m is the slope and b is the y-intercept (the value of y when x=0).
4x - 7y=9
- 7y=9-4x
y=(-4x + 9)/(-7)
y = -(4/7)x - (9/7)
This line has a slope of -(4/7). A line perpendicular to this line will have a slope that is the negative inverse of the reference line. The prependicular line will have a slope of (7/4) [the negative inverse of -(4/7)].
We can now write the new line as
y = (7/4)x + b
We need a value of b that forces the line to go through point (4,3). This can be di=one by entering (4,3) into the perpendicular equation (y = (7/4)x + b) and solve for b:
y = (7/4)x + b
3 = (7/4)*4 + b for point (4,3)
3 = 7 + b
b = -4
The equation is y = (7/4)x - 4
See the attached graph.
The sum of the perimeters of 2 different regular hexagons is 66 and the difference between their perimeters is 30. If x represents the side length of the larger regular hexagon and y represents the side length of the smaller regular hexagon, which of the following systems of equations can be used to find the dimensions of the regular hexagons?
Perimeter of rectangular hexagon = 6 × length of side, length of side of given hexagon = 30/6 =5 cm and perimeter of new figure = Number of outer sides × length of each side =10×5=50 cm
What is rectangle?A rectangle in Euclidean plane geometry is a quadrilateral with four right angles. You might also describe it as follows: a quadrilateral that is equiangular, which indicates that all of its angles are equal. The parallelogram might also have a straight angle. Squares are rectangles with four equally sized sides. A quadrilateral of the shape of a rectangle has four 90-degree vertices and equal parallel sides. As a result, it is sometimes referred to as an equirectangular rectangle. Because its opposite sides are equal and parallel, a rectangle is also known as a parallelogram.
Perimeter of rectangular hexagon = 6 × length of side
length of side of given hexagon = 30/6 =5 cm
perimeter of new figure = Number of outer sides × length of each side
=10×5=50 cm
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Atiku is allowed a discount of 12.5 percent on a dress that cost$125. correct to the nearest dollar the amount paid for the dress.
The amount paid for the dress to the nearest dollar is $109
Calculating the amount paid for the dressFrom the question, we are to calculate the amount paid for the dress
From the given information,
Atiku is allowed a discount of 12.5% on a dress that cost $125
First, we will calculate the discount
Discount = 12.5% of $125
Discount = 0.125 × $125
Discount = $15.625
Amount paid = Cost - Discount
Thus,
Amount paid = $125 - $15.625
Amount paid = $109.375
Amount paid ≈ $109
Hence, the amount paid is $109
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What is the equation of this graphed line?
y=2x+2
y=-2x-2
y=6x+6
y=-6x-6
Use the drawing tools to form the correct answers on the provided dot plot.
Create a dot plot of the data on the average cruising speeds of Red Eagle's planes.
Dataset
0.78
0.86
0.83
Drawing Tools
Select
Point
0.83
0.79
0.83
0.85
0.85
0.85
0.78
0.81
0.84
0.84
0.80
0.82
0.82
0.82
0.81
0.84
0.83
0.84
0.83
0.86
0.85
0.85
0.81
0.88
0.80
0.82
0.82
0.80
0.82
0.87
0.84
0.84
0.86
So on solving the provided question we can say we can plotting data by x = 0.83; y = 0.79; z = 0.83
what is plotting data?The most typical approach to display data using a chart is a graph that shows the relationship between two additional variables. Diagrams created by hand or on a computer are also acceptable. Move 2 units to the right after starting at the origin before going 3 units up. The coordinates for the points 2, 3, should be shown on the coordinate plane. Clearly state your points. The pink dot with the letter P thus stands for 2.3. Before creating a line chart, you need first generate a number line for each value in your data collection. Put an X (or dot) over each value of the data on the number line after that. For each instance when a value appears in a record, place an X over the corresponding number.
x = 0.83
y = 0.79
z = 0.83
xa = 0.85
ya = 0.85
za = 0.85
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An arcade sells game cards that customers can use to play the arcade games. A game card
comes stocked with credits, and each game costs 5 credits to play. Shivani was able to play
16 games before her card ran out of credits.
Graph the function that models the relationship between the number of games Shivani
played, n, and the number of credits remaining on her card, C(n).
Select points on the graph to plot them.
Answer: The function that models the relationship between the number of games played (n) and the number of credits remaining on the card (C(n)) is C(n) = 5n.
Step-by-step explanation: This function can be derived by realizing that each game costs 5 credits to play, so as the number of games played (n) increases, the number of credits remaining on the card (C(n)) decreases by 5 for each game played.
For example, if Shivani played 16 games before her card ran out of credits, we can substitute n = 16 into the function to find that C(16) = 5(16) = 80 credits. This means that her card had 80 credits on it before she started playing games.
To plot points on the graph, we can select a few different values of n and substitute them into the function to find the corresponding values of C(n). Here are a few examples:
When n = 0, C(n) = 5(0) = 0. This represents the case when Shivani has not played any games yet and has no remaining credits on her card.
When n = 8, C(n) = 5(8) = 40. This represents the case when Shivani has played 8 games and has 40 credits remaining on her card.
When n = 16, C(n) = 5(16) = 80. This represents the case when Shivani has played 16 games and her card has no remaining credits.
We can plot these points on the graph and label them (0,0), (8,40), (16,0) respectively.
Note that the y-intercept is (0,0) and the x-intercept is (16,0)
Graph:
n C(n)
0 0
8 40
16 0
The graph is a straight line starting from (0,0) and going down till (16,0) with a slope of -5.
In the figure below ABCDE A’B’C’D’E’
Which transformation maps figure ABCDE onto figure A’B’C’D’E’?
Answer: Rotation around the origin
Step-by-step explanation:
If you rotate ABCDE around the point (0,0), you will see that it maps onto A'B'C'D'E'.
Okay so basically help and say what to label with and whoever says it I’ll give brainly or wtv
Answer:
The area is 254.34 [tex]yds^{2}[/tex]
The circumference is 56.52 yards.
Step-by-step explanation:
a = [tex]\pi r^{2}[/tex]
a = (3.14) ([tex]9^{2}[/tex])
a = 3.14(81)
a = 254.34
The area is 254.34 [tex]yds^{2}[/tex]
c = 2[tex]\pi[/tex]r
c = 2(3.14)(9)
c = 56.52
The circumference is 56.52 yards.
How to write 12 and half kilometers in digits and decimal
Answer:
1000 mitre=1km
1.25 ×1000
Answer:
DIGIT FORM = 13km
DECIMAL FORM: 12.5km
Step-by-step explanation:
DIGIT Form: A digit can only be a whole number, so round 12 and a half (12/5 or 12.5) to the nearest whole number.
DECIMAL Form: It cannot be a whole number and must consist of a number followed by a decimal point (a period) and then another number.
DON'T FORGET TO ADD KILOMETERS (km) AFTER YOUR ANSWER.
What is the name of the segment inside the large triangle?
perpendicular bisector
altitude
median
angle bisector
Answer:
Perpendicular Bisector
In the given figure, △ADE ∼ △ABC. Which triangle similarity theorem justifies
this similarity? Show proof toyour answer.
a. Using the figure below, name the three similar triangles.
b. Write the proportions that exist among corresponding parts of similar
By similarity theorem The three similar triangles are △CLB~△LUB~△CUL
(B) The proportions in corresponding parts of similar are in △CLB~△LUB: CL/LU=LB/UB=CB/LB and in △CLB~△CUL:CL/CU=LB/UL=CB/CL
What is similarity theorem?Triangle similarity in Euclidean geometry. According to the fundamental theorem of similarity, a line segment can divide two triangle sides into proportionate segments if and only if it is parallel to the third side of the triangle.
In the given figure, △ADE~△ABC.
1. ∠D≅∠B
2. DE II BC converse of basic proportionality
3. ∠E ≅ ∠C corresponding angle are congruent
4. ∠DAE≅∠BAC reflexive property
5. △ADE ≅△ABC (AAA similarity theorem)
The three similar triangles are-
△CLB~△LUB~△CUL
The proportions that exist among corresponding parts of similar
→ In △CLB~△LUB
CL/LU=LB/UB=CB/LB
→ In △CLB~△CUL
CL/CU=LB/UL=CB/CL
→ In △LUB~△CUL
LU/CU=UB/UL=LB/CL
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Find the range for the possible measure of the third side of a triangle given two sides?
2.7 cm, 4.2 cm
range:
< n <
On solving the provided question, we can say that the range is between the two sides' difference and their sum.
What is range?The variable's range is obtained by finding its largest observed value (maximum) and subtracting its smallest observed value (minimum). Variational bounds or possible range: various steel prices; various styles; The extent or magnitude of a procedure or action: insight. the maximum or expected range of a weapon's projectile. The range of a list or set is the number between the minimum and maximum. Prior to identifying the region, align all the numbers. Remove (remove) the lowest number from the highest number next. The list's range is provided in the solution.The solution provides the list's range.
T is length of the third side
[tex]18-13 < T < 18+13[/tex]
[tex]5 yards < third side < 31 yards[/tex]
The range is between the two sides' difference and their sum.
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a lecture hall has 200 seats with folding arm tablets, 30 of which are designed for left- handers. the typical size of classes that meet there is 188, and we can assume that about 13% of students are left-handed. a) show that a binomial model is appropriate to use. then find the probability that a right-handed student in one of these classes is forced to use a lefty arm tablet. b) solve this problem again using the normal model. first show that a normal model is appropriate to use.
The probability of a right-handed student being forced to use a left-handed arm tablet is 0.085 or 8.5% using either the binomial or normal model.
A binomial model is appropriate to use for this problem since there are two possible outcomes (left or right-handed) and the probability of each outcome is independent of the other. The probability of a right-handed student being forced to use a left-handed arm tablet is calculated by subtracting the number of left-handed arm tablets (30) from the total number of students in the class (188) and dividing that by the total number of arm tablets (200). This yields a probability of 0.085 or 8.5%.
Using the normal model, the probability of a right-handed student being forced to use a left-handed arm tablet can also be calculated. This model is appropriate since there are a large number of students (188) and the probability of a student being right or left-handed is close to the population average (13%). The probability can be calculated by subtracting the number of left-handed arm tablets (30) from the total number of students in the class (188) and dividing that by the total number of arm tablets (200). This yields a probability of 0.085 or 8.5%.
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marcus is testing a hair sample from the telogen stage. what is likely to be true about this sample?
When Marcus tests a hair sample from the telogen stage. The result likely to be true about this sample is there will be a nuclear DNA present.
DNA (Deoxyribonucleic acid) is a polymer comprised of two polynucleotide chains that loop around each other to form a double helix having genetic instructions for the development, functioning, growth and reproduction of all known organisms and many viruses. DNA and ribonucleic acid both are nucleic acids.
Moreover, DNA is the information molecule. It holds the instructions for creating other large molecules, called proteins. These instructions are kept inside each of our cells, distributed among 46 long structures called chromosomes.
Hence, DNA is made of chemical structures called nucleotides. These structures are made of three parts: a sugar group, a phosphate group, and one of four types of nitrogen bases.
Hence the correct option is c.
--The given question is incomplete, the complete question is
''Marcus is testing a hair sample from the telogen stage. What is likely to be TRUE about this sample?
A. There will be several medullas present.
B. There will be mitochondrial DNA present.
C. There will be nuclear DNA present.
D. There will be very little DNA present."--
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You pick a card at random. 2 3 4 5 What is P(3)?
If you picked a card at random, then the probability of picking a three is 1 / 4 or 25 %.
How to find the probability ?The probability of an event occurring is defined by probability. There are many instances in real life where we may need to make predictions about how something will turn out. The outcome of an event may be known to us or unknown to us. When this happens, we say that there is a chance that the event will happen or not. In general, probability has many wonderful uses in games, in business to make forecasts based on likelihood, and in this emerging branch of artificial intelligence.
The probability that an event will occur is determined by the probability formula. The ratio of positive outcomes to all possible outcomes is used as the formula to determine an event's probability.
The probability that you can pick a three out of the cards given is:
= Number of cards of 3 / Number of total cards
= 1 / 4
= 25 %
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the final exam for miguel's math class is worth 120 points and will be comprised of an assortment of 50 multiple choice and free response question. each multiple choice is worth 2 points and each free response question is worth 3 points
The Linear Equations that shows the given problem statement will be the ones given in option D.
i.e.
x + y = 50
2x + 3y = 120
What are linear equations?
A linear equation is one in which the variable's maximum power is always 1. A one-degree equation is another name for it. A linear equation in one variable has the conventional form Ax + B = 0. In this equation, x is a variable, A is a coefficient, and B is constant. A linear equation in two variables has the conventional form Ax + By = C. The variables x and y are variables, the coefficients A and B are coefficients, and the C is a constant.
e.g.
y = 8x - 9 - Linear
y = x2 - 7 - Non-Linear, the power of the variable x is 2
Now,
Let's let x = the number of Multiple choice problems and
y = the number of free response questions.
Therefore, the equation will be
x + y = 50
Since the multiple choice problems are worth 2 points and the free response questions are worth 3 points and the total number of points is 120, then, the equation will be
2x + 3y = 120
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Answer: The Linear Equations that shows the given problem statement will be the ones given in option D.i.e.x + y = 502x + 3y = 120What are linear equations?A linear equation is one in which the variable's maximum power is always 1. A one-degree equation is another name for it. A linear equation in one variable has the conventional form Ax + B = 0. In this equation, x is a variable, A is a coefficient, and B is constant. A linear equation in two variables has the conventional form Ax + By = C. The variables x and y are variables, the coefficients A and B are coefficients, and the C is a constant.e.g.y = 8x - 9 - Lineary = x2 - 7 - Non-Linear, the power of the variable x is 2Now,Let's let x = the number of Multiple choice problems andy = the number of free response questions.Therefore, the equation will bex + y = 50Since the multiple choice problems are worth 2 points and the free response questions are worth 3 points and the total number of points is 120, then, the equation will be2x + 3y = 120
Step-by-step explanation:hope this helps 0^0
Is finding half of a number the same as dividing a number by ½ Why or why not?
No, finding half of a number is not the same as dividing a number by ½. When finding half of a number, the result is the number divided by 2. When dividing a number by ½, the result is the number multiplied by 2.
When finding half of a number, the result is the number divided by 2.
For example, if we want to find half of the number 8, we would divide 8 by 2 to get 4.
When dividing a number by ½, the result is the number multiplied by 2.
For example, if we want to divide the number 8 by ½, we would multiply 8 by 2 to get 16.
No, finding half of a number is not the same as dividing a number by ½. When finding half of a number, the result is the number divided by 2. When dividing a number by ½, the result is the number multiplied by 2.
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2. ALGEBRA On April 14, Mikos Souvakis borrowed $100,000 to remodel
his restaurant kitchen with a single-payment loan at 10. 5% ordinary
interest. If his loan's maturity value was $104,375, when does Mikos
have to pay it back?
and
We may apply the simple interest formula to find the due date for repayment from Mikos is 5 months.
I = P*r*t, where I represents interest, P the principal (initial sum), r the interest rate, and t the period of time in years.
The maturity value formula can be used to calculate interest given that the loan's principal is $100,000 and its maturity value is $104,375:
Where MV is the maturity value, P is the principal, and I is the interest, the formula is: MV = P + I.
With the above data, we obtain: 104,375 = 100,000 + I
I = 4,375.
Now we can use this value of I to find the time t in years:
I = P*r*t
4,375 = 100,000 (0.105) t
t = 4.375 / (100,000 x 0.105)
t = 4.375 / 10,500
t = 0.417 years or approximately 5 months
So, it takes approximately 5 months to repay the pay back amount .
Therefore, Mike must repay the amount by April 14, which is around five months after the day he took out the loan.
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brainly The linear functions f(x) and g(x) are represented on the graph, where g(x) is a transformation of f(x): Part A: Describe two types of transformations that can be used to transform f(x) to g(x). (2 points) Part B: Solve for k in each type of transformation. (4 points) Part C: Write an equation for each type of transformation that can be used to transform f(x) to g(x). (4 points) Source StylesFormatFontSize
Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.
What is transformation ?
A transformation is a function that changes the position, size, or shape of an object or a graph. In mathematics, transformations are often used to map a function from one coordinate space to another. There are several types of transformations, including translations, rotations, reflections, and scalings.
Part A: Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.
Part B:
Horizontal stretching/shrinking: k is the horizontal scale factor.
Vertical stretching/shrinking: k is the vertical scale factor.
Part C:
Horizontal stretching/shrinking: g(x) = f(kx)
Vertical stretching/shrinking: g(x) = kf(x)
Two types of transformations that can be used to transform f(x) to g(x) are horizontal and vertical stretching/shrinking.
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how to show that a function does not have a local minimum from r^2 to r when the second derivative test fails
Answer:
derivative on r^2 with respect to r
one time derivative is 2r
second time derivative is 2
third time derivative is 0
a piece of clothes was 5 m long,after washing it srank to 4.2m long find the percentage decrease in length
The length of the piece of clothing decreased by 16% after washing.
What is the percentage decrease?
To compute a percentage drop, first determine the difference (decrease) between the two values being compared. Then, divide the reduction by the original number and multiply the result by 100. The percentage change is expressed as a result of the alteration.
The percentage decrease in length of the piece of clothing can be calculated as follows:
Percentage decrease = (original length - new length) / original length * 100
Percentage decrease = (5 - 4.2) / 5 * 100
Percentage decrease = 0.8 / 5 * 100
Percentage decrease = 16%
Hence, the length of the piece of clothing decreased by 16% after washing.
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consider the following binomial experiment: a company owns 5 copiers. the probability that on a given day any one copier will break down is 0.66. what is the probability that at least 3 copiers will break down on a given day?
The probability that at least 3 copiers will break down on a given day is 0.0909.
1. Calculate the probability that 0 copiers will break down:
P(0) = (1-0.66)^5
= 0.0729
2. Calculate the probability that 1 copier will break down:
P(1) = 5 * 0.66 * (1-0.66)^4
= 0.2221
3. Calculate the probability that 2 copiers will break down:
P(2) = 5C2 * 0.66^2 * (1-0.66)^3
= 0.3088
4. Calculate the probability that 3 copiers will break down:
P(3) = 5C3 * 0.66^3 * (1-0.66)^2
= 0.2197
5. Calculate the probability that at least 3 copiers will break down:
P(at least 3) = 1 - (P(0) + P(1) + P(2))
= 0.0909
The probability that at least 3 copiers will break down on a given day is 0.0909. This was calculated by first calculating the probability of 0 copiers breaking down (P(0)) which is (1-0.66)^5, then the probability of 1 copier breaking down (P(1)) which is 5 * 0.66 * (1-0.66)^4, and then the probability of 2 copiers breaking down (P(2)) which is 5C2 * 0.66^2 * (1-0.66)^3. Finally, the probability that at least 3 copiers will break down (P(at least 3)) was calculated by subtracting the sum of P(0), P(1), and P(2) from 1. This gives us the final result of 0.0909.
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a researcher is 95% confident that the interval from 2.9 minutes to 6.83 minutes captures mu the true mean amount of time mice take to complete a maze for a piece of cheese. is it plausible that the true mean number of times for all mice to complete this maze may be less than 7 minutes?
The correct option is option a) Yes, this is plausible for the population mean because the upper boundary of the 95% confidence interval is below 7 minutes.
A confidence interval is a range of values, derived from a sample of data, that is used to estimate an unknown population parameter. The interval has an associated confidence level, such as 90%, 95%, or 99%, which indicates the level of confidence that the interval contains the true population parameter.
The most commonly used method to calculate a confidence interval is the method of maximum likelihood estimation. The wider the interval, the less certain the estimate, and the narrower the interval, the more certain the estimate.
It is plausible that the true mean number of times for all mice to complete the maze may be less than 7 minutes, as the interval provided by the researcher (2.9 minutes to 6.83 minutes) does not include 7 minutes as an upper bound.
A 95% confidence interval indicates that if the study were to be repeated multiple times, 95% of the intervals created would contain the true mean, so there is a 5% chance that the true mean falls outside of the interval provided.
Therefore, The correct option is option a) Yes, this is plausible for the population mean because the upper boundary of the 95% confidence interval is below 7 minutes.
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Answer:Yes, this is plausible for the population mean because the upper boundary of the 95% confidence interval is below 7 minutes
Step-by-step explanation:
i took the quiz
what number represents an area receiving several weeks (to months) of constant daylight during their summer?
The number that represents an area receiving several weeks (to months) of constant daylight during their summer is 10.
The number 10 is typically found at latitudes above the Arctic Circle (about 66.5 degrees North) during the summer solstice, the longest day of the year.
During the summer solstice, the tilt of the Earth's axis is such that the North Pole is tilted towards the sun, providing more direct sunlight to areas within the Arctic Circle. This results in several weeks to months of constant daylight, known as the "midnight sun."
On the other hand, areas south of the Antarctic Circle, which is located at about 66.5 degrees South latitude, experience "polar night" or 24-hour darkness during the winter solstice.
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Which option distinguishes the element that needs to be adjusted on the chart in the following scenario?
Donna just completed her demographic chart for her final social studies project on population growth. Her
teacher returns it and says her independent variable is incorrect and needs to be changed.
title
O y-axis value
O legend
Ox-axis value
833
Sign out
Jan 16
An answer option which distinguishes the element that needs to be adjusted on the chart in the following scenario is: C. x-axis value.
What are the types of variables?In an experiment, there are two (2) main types of variables and these include the following:
Dependent variable.Independent variable.What is an independent variable?In Mathematics, an independent variable can be defined as the variable that is being manipulated by an experimenter or researcher. This ultimately implies that, an independent variable is typically considered to be the cause in an experiment and it is represented by the x-axis value on a graph.
In this scenario, we can reasonably infer and logically deduce that the independent variable (x-axis value) represents the element that should be adjusted on Donna's demographic chart on population growth.
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can someone help me with this please
2.3 + r = 7.9
r = ?
Answer:
r = 5.6
Step-by-step explanation:
To solve this problem, we need to isolate r on one side of the equation. To do this, we need to subtract 2.3 from both sides, giving us:
2.3 + r = 7.9
r = 5.6
So, r = 5.6
Hope this helped!
Answer:
5.6
Step-by-step explanation:
2.3+r=7.9
r=7.9-2.3
r=5.6
The function ƒ(x) = x−−√3 is translated 3 units in the negative y-direction and 8 units in the negative x- direction. Select the correct equation for the resulting function.
The final function after transformations, is [tex]f(x +8) = \sqrt[3]{x+8} - 3[/tex]
The technique of altering an existing graph or graphed equation to create a different version of the following graph is known as graph transformation. The modification of algebraic equations is a typical form of algebraic problems. Sometimes graphs are translated, or moved about the xy-plane; sometimes they are stretched, rotated, inverted, or a combination of these transformations.
Now, there are 2 different transformations applied to the function here.
First Transformation:
[tex]f(x) = \sqrt[3]{x}[/tex] is translated 3 units in the negative y direction.
After transformation, the function is [tex]f(x) = \sqrt[3]{x} - 3[/tex].
Second Transformation:
The function [tex]f(x) = \sqrt[3]{x} - 3[/tex] is translated 8 units in the negative x direction.
After transformation, the function is [tex]f(x +8) = \sqrt[3]{x+8} - 3[/tex].
Therefore, the final function, after transformations, is [tex]f(x +8) = \sqrt[3]{x+8} - 3[/tex].
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Pls help with theseeeeeeee
Yes, you can assume both statements. In the diagram, planes W and X intersect at the line KL.
Determine whether you can assume the statement?This means that the points K, L, M, and N lie on the same plane, making them coplanar.All four points are contained within the same plane and are connected by lines, so it is clear that they all lie on the same plane.This means that statement 11 is true and statement 12 is also true.Yes, you can assume the statement in Exercises 11 and 12.The statement in Exercise 11 is true because it is shown in the diagram that Planes W and X intersect at KL.The statement in Exercise 12 is true because the points K, L, M, and N all lie on the same plane, which is Plane W.This type of reasoning is called deductive reasoning.Deductive reasoning is a logical process in which a conclusion is drawn based on the evidence that is given.In this case, the evidence is the diagram and the conclusion is the statement in the exercise.
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find the exact length of the curve. y = √x − x2 + sin^−1 √x
The above statement suggests that perhaps the curve's approximate length is 2.
What in mathematics is a curve?Curve, An abstract phrase used only to describe the trajectory of a linearly stretching point in mathematics (see continuity). Usually, an equation will create such a route. The term may also be used to describe a linear model or a collection of connected line segments. The curve is referred to as a route, or a plot line (or just arc). If a curve is the continuous injective picture of either an interval or indeed a circle, it is considered simple.
Considering function is,
[tex]y=\sqrt{x-x^2}+\sin ^{-1} \sqrt{x}[/tex]
To determine the curve's actual length
First , Parametrise the Curve by x = sin²u
[tex]\begin{aligned}& y=\sqrt{\sin ^2 u-\sin ^4 u}+\sin ^{-1} \sqrt{\sin ^2 u} \\& y=\sin u \sqrt{1-\sin ^2 u}+\sin ^{-1} \sin u\end{aligned}[/tex]
y = sinucosu + u
[tex]\begin{aligned}& y=\frac{1}{2} \times 2 \sin u \cos u+u \\& y=\frac{\sin 2 u}{2}+u \\& \frac{\mathrm{d} y}{\mathrm{~d} u}=\frac{\mathrm{d}}{\mathrm{d} u}\left[\frac{\sin 2 u}{2}+u\right] \\& \frac{\mathrm{d} y}{\mathrm{~d} u}=\left[\frac{2 \cos 2 u}{2}+1\right]\end{aligned}[/tex]
[tex]\frac{\mathrm{d} y}{\mathrm{~d} u}=[\cos 2 u+1][/tex]
and, x = sin²u
[tex]\begin{aligned}& \frac{\mathrm{d} x}{\mathrm{~d} u}=\frac{\mathrm{d}}{\mathrm{d} u}\left[\sin ^2 u\right] \\& \frac{\mathrm{d} x}{\mathrm{~d} u}=[2 \sin u \cos u] \\& \frac{\mathrm{d} x}{\mathrm{~d} u}=[\sin 2 u] \\& \left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2=\left(\frac{\mathrm{d} x}{\mathrm{~d} u}\right)^2+\left(\frac{\mathrm{d} y}{\mathrm{~d} u}\right)^2 \\& \left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2=(\sin 2 u)^2+(1+\cos 2 u)^2\end{aligned}[/tex]
[tex]\begin{aligned}\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =\left[\sin ^2 2 u+1+\cos ^2 2 u+2 \cos 2 u\right] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =[1+1+2 \cos 2 u] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =[2+2 \cos 2 u] \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =4 \cos ^2 u \\\left(\frac{\mathrm{d} s}{\mathrm{~d} u}\right)^2 & =2 \cos u\end{aligned}[/tex]
Currently, the curve's exact length is,
[tex]\begin{aligned}& =\int_0^{\Pi / 2} 2 \cos u d u \\& =2 \int_0^{\Pi / 2} \cos u d u \\& =2[\sin u]_0^{\Pi / 2} \\& =2\left[\sin \frac{\Pi}{2}-\sin 0\right]\end{aligned}[/tex]
= 2
As a result, the curve's exact length is 2
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