The measures of the other two angles in the parallelogram are both 143 degrees.
what is a parallelogram?A parallelogram is a quadrilateral with two sides that are parallel to one another. A parallelogram has equal-length opposite sides and equal-length opposite angles. The internal angles that are extra to the transversal on the same side. A parallelogram's total interior angles add up to 360 degrees.
A parallelogram is formed by the junction of two parallel lines, and its opposite angles are equal.
Then, there are two 35 degree angles. Two further angles that are equivalent are noted: A parallelogram's angles add up to 360 degrees. Following that, x+x+37+37=360
2x=360-74
2x=286
x=143.
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“Write a proof to show that the triangles are congruent.”
The given triangles ABD and CBD are congruent because their corresponding sides and angles are equal.
What are congruent triangles?Congruent triangles are two triangles that have exactly the same size and shape. In other words, all corresponding sides and angles of the two triangles are equal.
There are several ways to show that two triangles are congruent, including:
Side-Side-Side (SSS) Congruence: If the three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
In the given triangles ABD and CBD, each of their angles is 60⁰, since the triangles are equilateral, so the triangles are congruent.
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The rear windshield wiper blade on a car has a length of 8 inches. The blade is mounted on a 16 inch arm, 8
inches from the pivot point. If the wiper turns through an angle of 130 degrees, how much area is swept
clean?
blade
square inches.
The area swept clean by the wiper blade is approximately 239.3 square inches.
To find the area swept clean by the bladeWe need to find the area of the circular sector that is swept out by the blade as it rotates through an angle of 130 degrees. The radius of the circular sector is the length of the wiper arm, which is 16 inches.
The formula for the area of a circular sector is
A = (θ/360) × πr^2
Where
θ is the central angle of the sector (in degrees)r is the radius of the circle π is the constant pi (approximately 3.14)In this case, θ = 130 degrees and r = 16 inches, so we can plug these values into the formula to get:
A = (130/360) × 3.14 × (16)^2
Simplifying this expression, we get:
A ≈ 239.3 square inches
Therefore, the area swept clean by the wiper blade is approximately 239.3 square inches.
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how many square feet of carpet will we need for this hole
Answer: 44 ft²
Step-by-step explanation:
We will find the area of the large rectangle and subtract the area of the square. The area of a rectangle (and square) can be used with this formula:
A = LW
Large Rectangle:
A = LW
A = (12 ft)(4 ft)
A = 48 ft²
Square:
A = LW
A = (2 ft)(2 ft)
A = 4 ft²
Subtract:
48 ft² - 4 ft² = 44 ft²
It cost Chile 17. 70$ to send 177 text messages. How much would it cost to send 97 text message
The total cost of sending 97 text messages is $9.70
How to find the unit cost?A unit cost is defined as the total expenditure incurred by a specific company to produce, store, and sell one unit of a specific product or service. Unit costs are also synonymous with cost of goods sold (COGS). This accounting measure also involves all of the fixed and variable costs associated with the production of a good or service.
Now, we are told that It cost Chile 17.70$ to send 177 text messages. Thus:
Unit cost of message = 17.70/177 = $0.1 per message
Thus:
Cost of 97 text messages = 97 * 0.1 = $9.7
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Given a reduced echelon matrix in R3 onto iff for every vector b in R3, Ax = b has a solution iff every row of A has a pivot.
a. true b. false
Every vector b in R3, the equation Ax = b has a solution if every row of A has a pivot.
The statement is true.
True.
If a reduced echelon matrix in R3 is obtained by row operations from a matrix A, then the reduced echelon matrix is in row echelon form and is equivalent to the original matrix A.
If every row of A has a pivot, then every row has a leading non-zero entry in the row echelon form of A.
This means that each row of the row echelon form of A has a pivot, and so the matrix has full row rank.
The number of pivot columns in the reduced echelon matrix equals the row rank of A, which is also the same as the column rank of A.
Vector b in R3, the equation Ax = b has a solution if and only if b is a linear combination of the columns of A.
If A has full column rank, then the columns of A span the column space of A, which means that any vector in the column space can be expressed as a linear combination of the columns of A.
If every row of A has a pivot, then A has full row rank, full column rank, and the columns of A span R3.
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Write a matrix that when multiplied by any point [x/y] would have the effect of rotating the point 90 degrees counterclockwise. Also determine the algebraic effect of performing the transformation on the point A.
The result of rotating the point [-5, 2] by 90 degrees counterclockwise is the point [2, -5].
Define rotationRotation is a transformation in geometry that involves turning or spinning an object around a fixed point called the center of rotation. The center of rotation remains stationary while all the points of the object move along circular paths, at the same angle and distance from the center.
To rotate a point [x/y] counterclockwise by 90 degrees, we can use a 2x2 matrix of the form:
[ 0 -1 ]
[ 1 0 ]
To multiply this matrix by a point [x/y], we can use the following matrix multiplication:
[ 0 -1 ] [ x ] [ -y ]
[ 1 0 ] [ y ] = [ x ]
So, if we have a point [x/y] and we want to rotate it 90 degrees counterclockwise, we can multiply it by the matrix [ 0 -1 ; 1 0 ] to get the new point [-y/x].
To rotate a point 90 degrees counterclockwise using a matrix, we need to represent the point as a column matrix and then multiply it by the 2x2 rotation matrix.
Assuming that the point is [x, y] = [-5, 2], we can represent it as a column matrix:
| -5 |
| 2 |
To rotate the point counterclockwise by 90 degrees, we can use the following 2x2 rotation matrix:
| 0 -1 |
| 1 0 |
Multiplying the rotation matrix by the column matrix representing the point gives:
| 0 -1 | | -5 | | 2 |
| 1 0 | ×| 2 | = | -5 |
So the result of rotating the point [-5, 2] by 90 degrees counterclockwise is the point [2, -5].
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What is the definition LCD
Answer:
The definition of LCD is Least Common Denominator.
Step-by-step explanation:
Question 2:
LCD of 7/8 and 3/4 is 8
Question 3 :
7/8 - 3/4
7/8 - 6/8
1/8
Hope this helps :) !!
Answer:
Step-by-step explanation: First of all, the full form of LCD is "Lowest Common Denominator"
LCD is the least common denominator that can be used for all types of fractions.
For example, if we have the fractions 1/2 and 2/3, we can write:
1/2 as 2/4,3/6 or 4/8
and
2/3 as 4/6,6/9 or 8/12
Hence, the least common denominator of 1/2 and 2/3 is 6, since it is present in the denominators of all the fractions.
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In Problems 55–62 write each function in terms of unit step functions. Find the Laplace transform of the given function. = {0, 0<_t<1 f(t) = {t^(2) t>_ 1
The Laplace transform of the given function f(t) = {0, 0 < t < 1, t², t ≥ 1} is 2s⁻³.
Given the function f(t) = {0, 0 < t < 1, t², t ≥ 1}, we need to find its Laplace transform. The Laplace transform of a function f(t) is denoted by F(s) and is defined as follows:
F(s) = L{f(t)} = ∫[0,∞) f(t)e⁻ᵃˣdt,
where s is a complex variable.
To find the Laplace transform of the given function, we need to split the function into two parts: one for 0 < t < 1 and another for t ≥ 1. For 0 < t < 1, f(t) = 0, so the integral becomes:
L{0} = ∫ 0e⁻ᵃˣdt = 0.
For t ≥ 1, f(t) = t², so the integral becomes:
L{t²} = ∫ t²e⁻ᵃˣdt.
To evaluate this integral, we can use integration by parts. Let u = t² and dv = e⁻ᵃˣdt, then du/dt = 2t and v = (-1/s) e⁻ᵃˣ. Using the integration by parts formula, we get:
L{t²} = ∫ t²e⁻ᵃˣdt = (-t²/s)e⁻ᵃˣ∣∣∣[0,∞) + (2/s)∫[0,∞) te⁻ᵃˣdt.
The first term in the above equation evaluates to zero because e⁻ᵃˣapproaches zero as t approaches infinity. Using integration by parts again for the second term, we get:
L{t²} = (-t²/s)e⁻ᵃˣ∣∣∣[0,∞) + (2/s)(-t/s)e⁻ᵃˣ∣∣∣[0,∞) + (2/s²)∫[0,∞) e⁻ᵃˣdt.
The first two terms in the above equation evaluate to zero for the same reason as before. The integral in the last term evaluates to (1/s²), so we get:
L{t²} = 2/s³.
Therefore, the Laplace transform of the given function f(t) is:
L{f(t)} = L{0} + L{t²} = 0 + 2/s³ = 2s⁻³.
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Find the area of the shaded region for each composite figure. Round your answer to the nearest tenth, if necessary.
The area of the composite figure for this problem is given as follows:
A = 90 in².
How to obtain the area of the composite figure?The area of the composite figure is given by the sum of the areas of all the parts that compose the figure.
The right triangle has a side length of 5 in and an hypotenuse of 13 units, hence the missing side is given as follows:
5² + x² = 13²
25 + x² = 169
x² = 144
x = 12.
Thus the entire base is of:
12 + 4 = 16 in.
Then the figure is composed as follows:
Rectangle of dimensions 3 in and 16 in.Right triangle of sides 12 in and 8 - 3 = 5 in.Rectangle of dimensions 4 in and 3 in.Hence the area is given as follows:
A = 3 x 16 + 0.5 x 12 x 5 + 4 x 3
A = 90 in².
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Ganymede, one of the moons of Jupiter, can be modeled by a sphere with a diameter of approximately 3,388 kilometers. The Moon of Earth, by comparison, can be modeled by a sphere with a diameter of approximately 1,245 kilometers. The volume of Ganymede is approximately how many times the volume of Earth’s Moon?
The volume of Ganymede is approximately 3.48 times the volume of the Moon, based on the given diameters of the two moons and the formula for the volume of a sphere.
What is volume of a sphere?The volume of a sphere is the amount of space occupied by the sphere in three-dimensional space. It can be calculated using the formula V = (4/3)πr³, where V is the volume of the sphere, r is the radius of the sphere, and π (pi) is a mathematical constant approximately equal to 3.14159. This formula states that the volume of a sphere is four-thirds of the product of pi and the cube of the sphere's radius.
In the given question,
The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. Since we are given the diameters of the two moons, we can calculate their radii by dividing the diameters by 2. Therefore, the radius of Ganymede is 3,388/2 = 1,694 km, and the radius of the Moon is 1,245/2 = 622.5 km.
Now, we can calculate the volumes of the two moons:
V_Ganymede = (4/3)π(1694)³ ≈ 7.66 x 10^10 km³
V_Moon = (4/3)π(622.5)³ ≈ 2.2 x 10^10 km³
To find how many times larger Ganymede's volume is compared to the Moon's volume, we can divide the volume of Ganymede by the volume of the Moon:
V_Ganymede / V_Moon ≈ (7.66 x 10¹⁰) / (2.2 x 10¹⁰) ≈ 3.48
Therefore, the volume of Ganymede is approximately 3.48 times the volume of the Moon.
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Which is more, 7,000,001 milligrams or 7 kilograms?
example of definition of derivative:lim as h approaches 0f(a+h) - f(a) / hThe slope of f(x) = -3x^2 +11 at any value of x
The slope of the function f(x) = -3x^2 +11 at any value of x is given by 6x.
The definition of derivative is given by the limit as h approaches 0 of [f(a+h) - f(a)] / h,
where f(x) is a function and a is a fixed value of x.
To find the slope of the function[tex]f(x) = -3x^2 +11[/tex] at any value of x, we need to use this definition of derivative.
So, let's plug in f(x) = -3x^2 +11 and a = x into the formula.
This gives us:
lim as h approaches 0 of [tex][-3(x+h)^2 + 11 - (-3x^2 + 11)] / h[/tex]
Simplifying this expression, we get:
lim as h approaches 0 of [tex][-3x^2 - 6hx - 3h^2 + 11 + 3x^2 - 11] / h[/tex]
Combining like terms, we get:
lim as h approaches 0 of [tex][-6hx - 3h^2] / h][/tex]
Factoring out an h from the numerator, we get:
lim as h approaches 0 of [-h(6x + 3h)] / h
Canceling out the h in the numerator and denominator, we get:
lim as h approaches 0 of 6x + 3h
Now, we can plug in h = 0 to find the slope of the function at any value of x:
6x + 3(0) = 6x.
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The vertices of a figure are given. Rotate the figure as described. Find the coordinates of the image.
E(−4, 1), F(−3, 3), G(−2, 3), H(−1, 1)
180°
about the origin
answer these questions please
The difference in the two figure's areas is 96 square units.
The height of the horse trough is 4 feet.
How to find the difference in the two figure's areas?No. 11
The area of Figure A is:
A = l * w = 12 * 8 = 96 square units
The area of Figure B is:
B = l * w = 16 * 12 = 192 square units
The difference in the two figure's areas is:
B - A = 192 - 96 = 96 square units
Therefore, the difference in the two figure's areas is 96 square units.
No. 12
We are given that the base area is 24 square feet, and the volume is 96 cubic feet.
Volume = base area * height
96 = 24 * height
height = 96/24
height = 4 feet
Therefore, the height of the horse trough is 4 feet.
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Answer:...............
Step-by-step explanation:
...................
Question
The table of values shows the height of a windmill blade from the ground over time. This can be modeled by a sinusoidal function.
The table of values is attached
Which statements are true?
Select each correct answer.
The frequency of the function is 4.
The minimum value of the function is 5.8.
The amplitude of the function is 1.5.
The period of the function is 1/4.
Answer:
the amplíetude of the function is 1.5
Step-by-step explanation:
.
Developers determine the specifics of how to build a system in the _____ phase.
a. analysis
b. design
c. implementation
d. testing
Developers determine the specifics of how to build a system in the design phase. b
The design phase and its importance:
Define Objectives:
The design phase begins after the completion of the analysis phase, where developers have gathered and evaluated requirements.
The main objective of the design phase is to create a blueprint for the system that addresses all identified requirements.
Create Architectural Design:
During this step, developers establish the overall structure of the system, including its components, their relationships, and the interactions between them.
This architectural design provides a high-level view of the system's organization.
Design System Components:
With the architectural design in place, developers focus on designing individual components or modules of the system. They create detailed specifications, which include the functionality, inputs, outputs, and processes for each component.
Design User Interface:
The user interface design involves creating a user-friendly and efficient way for end-users to interact with the system. This can include designing screen layouts, menus, buttons, and other interface elements.
Validate Design:
The final step in the design phase is to ensure that the proposed design meets all the requirements identified during the analysis phase.
Developers may use various methods, such as reviews and simulations, to validate their design before proceeding to the implementation phase.
The design phase is a crucial part of the system development process, as it defines how the system will be built, ensuring it meets the needs of its users and the project's objectives.
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please be correct :/
Question 12
A recent conference had 750 people in attendance. In one exhibit room of 70 people, there were 18 teachers and 52 principals. What prediction can you make about the number of principals in attendance at the conference?
There were about 193 principals in attendance.
There were about 260 principals in attendance.
There were about 557 principals in attendance.
There were about 680 principals in attendance.
Question 13
A college cafeteria is looking for a new dessert to offer its 4,000 students. The table shows the preference of 225 students.
Ice Cream Candy Cake Pie Cookies
81 9 72 36 27
Which statement is the best prediction about the number of cookies the college will need?
The college will have about 480 students who prefer cookies.
The college will have about 640 students who prefer cookies.
The college will have about 1,280 students who prefer cookies.
The college will have about 1,440 students who prefer cookies.
Question 14
A random sample of 100 middle schoolers were asked about their favorite sport. The following data was collected from the students.
Sport Basketball Baseball Soccer Tennis
Number of Students 17 12 27 44
Which of the following graphs correctly displays the data?
histogram with the title favorite sport and the x axis labeled sport and the y axis labeled number of students, with the first bar labeled basketball going to a value of 17, the second bar labeled baseball going to a value of 12, the third bar labeled soccer going to a value of 27, and the fourth bar labeled tennis going to a value of 44
histogram with the title favorite sport and the x axis labeled sport and the y axis labeled number of students, with the first bar labeled baseball going to a value of 17, the second bar labeled basketball going to a value of 12, the third bar labeled tennis going to a value of 27, and the fourth bar labeled soccer going to a value of 44
bar graph with the title favorite sport and the x axis labeled sport and the y axis labeled number of students, with the first bar labeled basketball going to a value of 17, the second bar labeled baseball going to a value of 12, the third bar labeled soccer going to a value of 27, and the fourth bar labeled tennis going to a value of 44
bar graph with the title favorite sport and the x axis labeled sport and the y axis labeled number of students, with the first bar labeled baseball going to a value of 17, the second bar labeled basketball going to a value of 12, the third bar labeled tennis going to a value of 27, and the fourth bar labeled soccer going to a value of 44
Question 15
The line plots represent data collected on the travel times to school from two groups of 15 students.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 4, 6, 14, and 28. There are two dots above 10, 12, 18, and 22. There are three dots above 16. The graph is titled Bus 47 Travel Times.
A horizontal line starting at 0, with tick marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 8, 9, 18, 20, and 22. There are two dots above 6, 10, 12, 14, and 16. The graph is titled Bus 18 Travel Times.
Compare the data and use the correct measure of center to determine which bus typically has the faster travel time. Round your answer to the nearest whole number, if necessary, and explain your answer.
Bus 18, with a median of 13
Bus 47, with a median of 16
Bus 18, with a mean of 13
Bus 47, with a mean of 16
Answer:
I already did this go check the answers on the other one
Step-by-step explanation:
What is the area of this tile?
And it isn’t 12 because I did that for the other one and got it wrong.
The area of this tile is 12 cm2.
What is area?Area is a two-dimensional measure of space, referring to the total surface area of a shape. It can be measured in terms of square units, such as square feet, meters, or kilometers. Area is an important concept in geometry and can be used to calculate the perimeter, circumference, and volume of a shape. Area calculations are also used in fields such as engineering, architecture, and construction.
Area is the size of a two-dimensional shape or the amount of space within a three-dimensional figure. It is a measure of how much space an object occupies.
The area of this tile can be calculated using the formula:
Area = Length x Width
In this case, the length of the tile is 6 cm and the width is 2 cm. Therefore, the area of the tile is 12 cm2.
To calculate the area, we multiplied the length and width of the tile. This formula is applicable to all rectangles, as the area of any rectangle is determined by multiplying the length and width.
It is important to note that the units used to measure the length and width must be the same in order to get an accurate area measurement. For example, if the length was measured in inches and the width in centimeters, then the area would be incorrect.
In conclusion, the area of this tile is 12 cm2.
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can yall help me with this plsss?
Violet rents a bike in a city she is visiting. She pays an initial fee of $2.50 plus $3.75 per hour she rents the bike. Which equation can be used to find y, the total cost of renting a bike, if x represents the number of hours Violet rents the bike?
Answer: y = 3.75x + 2.50
Step-by-step explanation:
The equation that can be used to find y, the total cost of renting a bike, if x represents the number of hours Violet rents the bike is:
y = 3.75x + 2.50
Explanation:
Violet pays an initial fee of $2.50, which is a fixed cost that does not depend on the number of hours she rents the bike. Additionally, she pays $3.75 per hour she rents the bike, which is a variable cost that depends on the number of hours. Therefore, the total cost of renting a bike (y) can be expressed as the sum of the fixed cost ($2.50) and the variable cost ($3.75x), which gives the equation:
3. (07.05 MC) An equation is shown below: 12x + 3(x - 5) = 6x + 9x - 15
Part A: Solve the equation and write the number of solutions. Show all the steps. (6 points)
Part B: Name one property you used to solve this equation. (4 points)
The equation 12x + 3(x - 5) = 6x + 9x - 15 has infinite many solutions
Solving the equationPart A:
Starting with the given equation:
12x + 3(x - 5) = 6x + 9x - 15
Simplify by distributing the 3:
12x + 3x - 15 = 6x + 9x - 15
Combine like terms on both sides:
15x - 15 = 15x - 15
Add 15 to both sides:
0 = 0
Therefore, the equation has infinite many solutions
Part B:
The property used to solve this equation is the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. In this case, we used the distributive property to simplify 3(x - 5) to 3x - 15.
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have you ever been frustrated because you could not get a container of some sort to release the last bit of its contents? an article reported on an investigation of this issue for various consumer products. suppose five 6.0 oz tubes of toothpaste of a particular brand are randomly selected and squeezed until no more toothpaste will come out. then each tube is cut open and the amount remaining is weighed, resulting in the following data: 0.51, 0.63, 0.42, 0.5, 0.37. does it appear that the true average amount left is less than 10% of the advertised net contents?
The null hypothesis is that the true average amount left is equal to 10% of the advertised net contents. In In another case of true average amount is less than 10% of the advertised net contents.
Hence, we will utilize one-tailed t-test with a significance level of 0.05 to test this hypothesis.
The sample mean of the five tubes is (0.51+0.63+0.42+0.5+0.37)/5 = 0.486 ounces.
The sample standard deviation is √(((0.51-0.486)² + (0.63-0.486)² + (0.42-0.486)² + (0.5-0.486)² + (0.37-0.486)²)/4) = 0.107 ounces.
The t-statistic is calculated as
(mean sample - hypothe -sized mean)/( standard deviation sample/√( size sample))
= (0.486 - 0.6)/(0.107/√(5))
= -3.06.
The p-value for this test is 0.0249 which is less than our significance level of 0.05.
Therefore, we reject the null hypothesis and conclude that there is evidence that the true average amount left in a tube of toothpaste is less than 10% of the advertised net contents.
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Going to Market - Basics Math
QUESTION 5 of 10: Advertisers want to create "awareness," which is whether a consumer even knows about a business. If your business
starts with 3% awareness and grows awareness by 5% per week (compounded), how long will it take to reach 75% awareness? The formula
for compound growth is E= B x (1 + R), where E is ending value, B is beginning value, R is rate of growth, and T is the amount of time.
a) 12 weeks
b) 66 weeks
c) 83 weeks
d) 94 weeks
If your business starts with 3% awareness and grows awareness by 5% per week (compounded), how long will it take to reach 75% awareness is b) 66 weeks
What is Compound Growth?Compound growth denotes the procedure of generating profits not only on the investment's principal amount, but also on the accumulated interest.
It is essentially reallocating its earnings where the investor's interest is integrated with the beginning sum, while additional interest is contingent on this amplified amount.
The purposeful act of compounding can yield noteworthy accomplishments over time, since interest earned from pre-existing income inflates drastically in value due to accruing interest outlays, and hence boosts the final outcome substantially and rapidly- resulting in exponential growth of the given input.
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Are the ratios 6:7 and 1:2 equivalent?
Answer:
No
Step-by-step explanation:
If you divide 6 by 7, it is not equal to the number you get when you divide 1 by 2, therfore making them not equavalent.
~~~Harsha~~~
Answer:
No, the are not equivalent
Step-by-step explanation:
The ratios 6:7 & 1:2 are already in their simplest form, meaning we can't simple them anymore.
6:7 ≠ 1:2
The function f given by f(x)= 3x^5 -4x^3 -3x has a relative maximum at x=a) -1b) - root 5/5c) 0d) root 5/5e) 1
The function f has a relative maximum at x = -√[(2 + sqrt(2))/5], so the answer is (b) - root 5/5.
To find the relative maximum of a function, we need to find the critical points of the function, where the derivative of the function is zero or undefined.
Let's start by finding the derivative of f(x)
f'(x) = 15x⁴ - 12x² - 3
Now, we can find the critical points by solving the equation f'(x) = 0
15x⁴ - 12x² - 3 = 0
We can factor this equation by substituting y = x²
15y² - 12y - 3 = 0
Solving for y, we get
y = (2 ± √(2))/5
Substituting back x² for y, we get
x² = (2 ± √(2))/5
Taking the square root of both sides, we get
x = ± √[(2 ± √(2))/5]
Therefore, the critical points are
x = -√[(2 + √(2))/5], √[(2 - √(2))/5]
Now, we need to determine whether these critical points are relative maximum or minimum.
We can use the second derivative test to determine the nature of the critical points. The second derivative of f(x) is
f''(x) = 60x³ - 24x
Plugging in the critical points, we get
f''(-√[(2 + √(2))/5]) = -48√(2) < 0 (relative maximum)
f''(√[(2 - √(2))/5]) = 48√(2) > 0 (relative minimum)
Therefore, the relative maximum of the function f(x) is at
x = -√[(2 + √(2))/5]
So, the correct answer is (b) - root 5/5.
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Find the entire perimeter of the entire floor
Mendel is back at it with another genetics experiment. This time he needs to partition a
rectangular garden into 4 parts by constructing three additional fences parallel to one of the
sides, in addition to fencing the perimeter. What is the minimum amount of fencing (in
meters) he will need to enclose 2500 m2?
The minimum amount of fencing Mendel will need to enclose 2500 m² = 317 meters
Let's assume that x represents the length of and y represents the width of the rectangular garden.
Using the formula for tha area of rectangle, the area of the garden would be,
A = length × width
A = xy
Here, A = 2500 m²
so, 2500 = xy
⇒ y = 2500/x
Using the formula for the perimeter of the rectangle, the perimeter of the garden would be,
P = 2x + 2y
But a rectangular garden is partitioned into 4 parts by constructing three additional fences parallel to one of the side.
So, the perimeter would be,
P = 2x + 5y
sides, in addition to fencing the perimeter.
P = 2x + 5(2500/x)
P = 2x + (12500/x)
Consider the derivative of P(x) with respect to x.
P'(x) = 2 - 12500/x²
To find the minimum value consider P'(x) = 0
2 - 12500/x² = 0
2 = 12500/x²
x² = 12500/2
x² = 6250
x = 79.05 m
and y = 2500/79.05
y = 31.63 m
So, the minimum amount of fencing would be,
P = 2x + 5y
P = 2(79.05) + 5( 31.63 )
P = 316.25 meters
P ≈ 317 meters
The minimum amount of fencing need to enclose 2500 m² = 316.25 meters
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Which of the following gives the volume of the solid created by revolving a curve f(x) around the y-axis?V=2Π∫ xf(x)dxV=Π∫ xf(x)dxV=2Πx2f(x)dxV=Π2∫ xf(x)dx
The volume of the solid created by revolving a curve f(x) around the y-axis is 2π [tex]\int\limits^a_b {xf(x)} \, dx[/tex]
What is the cylindrical shell?
A region enclosed by two identically sized cylinders with the same central axis is known as a cylindrical shell. We often denote the height of the cylinders by H, the inner cylinder's radius by r, and the thickness of the shell by t, resulting in a larger cylinder's radius equal to r + t.
Here,
We consider a strip of height y = f(x), and width dx; it is x units away from the axis of rotation.
Now, we rotate this strip about the y-axis; we obtain a cylindrical shell with
radius = x
height = f(x)
thickness = dx
The volume of cylindrical shell = (circumference)×(height)×thickness
= (2πx)×(f(x))×dx
The volume of revolution = 2π [tex]\int\limits^a_b {xf(x)} \, dx[/tex]
Hence, the volume of the solid created by revolving a curve f(x) around the y-axis is 2π [tex]\int\limits^a_b {xf(x)} \, dx[/tex]
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The standard form of the equation of a circle is (x−5)2+(y−6)2=1.
What is the general form of the equation?
Responses
x2+y2+10x+12y+62=0
x squared plus y squared plus 10 x plus 12 y plus 62 equals 0
x2+y2−10x−12y+60=0
x squared plus y squared minus 10 x minus 12 y plus 60 equals 0
x2+y2−10x−12y−62=0
x squared plus y squared minus 10 x minus 12 y minus 62 equals 0
x2+y2+10x+12y+60=0
The general form of the equation is x² + y² - 10x - 12y + 60 = 0.
How find the general form of the equation of a circle?The general form of the equation of a circle is given by:
x² + y² + Cx + Dy + E = 0
Where C, D and E are constant
Given the standard form of the equation of a circle as (x−5)² + (y−6)² = 1, we can rewrite it as follow:
(x−5)² + (y−6)² = 1
(x² - 10x + 25) + (y² - 12y + 36) = 1 (Expand the brackets)
x² + y² - 10x - 12y + 25 + 36 - 1 = 0
x² + y² - 10x - 12y + 60 = 0
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If three squares have sides that make an acute triangle, then the sum of the areas of the two small squares...
If three squares have sides that make an obtuse triangle, then the sum of the areas of the two small squares...
If three squares have sides that make a right triangle, then the sum of the areas of the two small squares...
The Pythagorean Theorem can be utilized in any situation to establish a connection between the side lengths of the squares and the sort of triangle that results.
What is the Pythagorean Theorem?Let's use the notation "a, b, and c" to indicate the three squares' side lengths, where "a" stands for the shortest side, "b" for the midpoint, and "c" for the longest side.
The area of the largest square (c), which is created by the hypotenuse of the triangle, is equal to the sum of the areas of the two smaller squares when three squares' sides form an acute triangle.
The areas of the two smaller squares (a and b) formed by the two larger squares (a and b) are equal when three squares have sides that form an obtuse triangle.
If three squares have sides that make a right triangle, then the sum of the areas of the two small squares is equal to the area of the largest square (c) formed by the hypotenuse of the right triangle.
In all cases, the Pythagorean Theorem can be used to determine the relationship between the side lengths of the squares and the type of triangle formed.
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