The Cartesian Equation of the given parametric equation is written below
: x^2 + y^2=4
To eliminate the parameter t and find the Cartesian equation, we can use the trigonometric identity: cos^2(t) + sin^2(t) = 1. We can rearrange the given equations to get:
x(t) = 2cos(t)
y(t) = 2sin(t)
Dividing both sides of each equation by 2, we get:
cos(t) = x/2
sin(t) = y/2
Squaring both equations and adding them, we get:
cos^2(t) + sin^2(t) = (x/2)^2 + (y/2)^2
Using the identity, we can simplify the right-hand side to get:
1 = (x/2)^2 + (y/2)^2
Multiplying both sides by 4, we get the Cartesian equation:
x^2 + y^2 = 4
This is the equation of a circle with radius 2 centered at the origin. The domain of this equation is all real numbers, and the range is from -2 to 2 (inclusive) in both the x and y directions, represented as [-2,2].
In summary, the Cartesian equation of the given parametric equations is x^2 + y^2 = 4, with domain of all real numbers and range of [-2,2].
To know more about Parametric Equation visit:
https://brainly.com/question/28537985
#SPJ4
What is the slope of the line passing through the points (1, -3) and (1, 0)?
Enter your answer as a number, like this: 42
Or, if the slope is undefined, enter a lowercase letter "u," like this: u
u is the slope of the line passing through the points (1, -3) and (1, 0).
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
We have to find the slope of the line passing through the points (1, -3) and (1, 0)
m=0-(-3)/1-1
=3/0
Undefined
Hence, the slope of the line passing through the points (1, -3) and (1, 0) is u.
To learn more on slope of line click:
https://brainly.com/question/14511992
#SPJ1
A triangle has angles that measure 60°, 70°, and z. What is z?
Answer:
z = 50°
Step-by-step explanation:
We know that,
→ Sum of all sides of triangle is 180°.
Formula we use,
→ <A + <B + <C = 180°
Forming the equation,
→ 60° + 70° + z = 180°
Now the value of z will be,
→ 60° + 70° + z = 180°
→ 130° + z = 180°
→ z = 180° - 130°
→ [ z = 50° ]
Hence, the value of z is 50°.
write an equation of the line passing through the given points. give the final answer in standard form. (1/2,-4) and (3/4,-40
The equation of line in standard form is y = -146x + 69.
A line equation is an algebraic way of representing a set of points that together form a line in a coordinate system. The various points that together form a line of axes can be represented as a set of (x,y) variables and form an algebraic equation, also called the line equation. You can use the straight line equation to determine if a given point lies on a straight line.
Each row of equations is a linear equation of degree 1. Read the entire article to learn about the different forms of linear equations and how to determine them. A line segment can be defined as a connection between two points. Any two points in a 2D geometry can be connected by a line segment or a simple straight line. The equation of a straight line can be obtained in three ways:
slope intercept method
point slope method
Standard method
Given two points lying on a given line, we usually follow the point-gradient method.
The equation for a straight line is y−y1=m(x−x1)
where y1 is the Y-axis coordinate, m is the tilt, x1 Coordinates on the x-axis. m = y2-y1/x2-x1
eqn = y +4 = (-36/1/4)(x-1/2)
y +4 = -146x + 73
y = -146x + 69
know more about equation of line click here;
https://brainly.com/question/19243340
#SPJ4
Grade 5 18 is 15% what number
Answer:
120
Step-by-step explanation:
18 = .15x Divide both sides by .15 (15% as a decimal is .15)
[tex]\frac{18}{.15}[/tex] = [tex]\frac{.15x}{.15\\}[/tex]
120 = x
The triangle on the grid will be translated two units left.
54321
47
C
2 3 4 5 X
Which shows the triangle when it is translated two units left?
The coordinates of the triangle when translated 2 units left are A'(-3, -1), B'(-3, -5) and C'(-1.5, -5).
What is Translation?Translation is a type of transformation of geometrical figures. After translation, the original figure is shifted from a place to another place without affecting it's size.
The given triangle has the coordinates A(-1, -1), B(-1, -5) and C(0.5, -5).
Translation is done 2 units left.
A(-1, -1) becomes A(-1 - 2, -1) = (-3, -1)
B(-1, -5) becomes B(-1 - 2, -5) = (-3, -5)
C(0.5, -5) becomes C(0.5 - 2, -5) = (-1.5, -5)
Hence the coordinates are A'(-3, -1), B'(-3, -5) and C'(-1.5, -5).
Learn more about Translation here :
https://brainly.com/question/12463306
#SPJ1
i need this ASAP!!!!
In accordance with Side-Angle-Side theorem of congruence, the triangles seen in choices A and C are congruent.
How to find two congruent triangles
In this problem we need to determine what triangles are congruent, that is, two triangles whose sides and angles are the same in measure and distribution. We can prove congruence by means of Side-Angle-Side theorem, where given angle is between two given sides.
Then, by direct inspection, the triangles from choices A and C are congruent, because sides and in-between angle are the same in measure and distribution.
To learn more on congruent triangles: https://brainly.com/question/29116501
#SPJ1
The triangles that are certainly congruent in the diagram are triangle A and triangle C.
What are congruent triangles?Congruent triangles are triangles that have the same size and shape, such that the lengths of the three sides in one triangle are equivalent or have same length as the lengths of the three sides of the other triangle.
The congruent triangles can be obtained by using the congruence rules, or conditions for congruence, which includes; ASA, SAA, SSS, SAS
Where;
ASA is an acronym for Angle-Side-Angle
SAA is an acronym for Side-Angle-Angle
SSS is an acronym for Side-Side-Side
SAS is an acronym for Side-Angle-Side
The lengths of two sides and the measure of the included angle in triangle A are congruent to the length of two sides and the measure of the included side between the two sides in triangle C.
Therefore, the triangle in option A is congruent to the triangle in option C by the Side-Angle-Side, SAS, congruency rule.
The angle 62° is the non included angle to the 14 m and 10 m long side in option B and D, while the 14 m long side is the adjacent side to the 62° angle in option B. In option D the 10 m long side is adjacent to the 62° angle, which the orientation of triangles B and D to be different
The only two congruent triangles are triangles in option A and option C
Learn more on the triangle congruence rules here: https://brainly.com/question/29125567
#SPJ1
determine if the numerical value describes a population parameter or a sample statistic. 79% 79 % of all students at the local university voted in the last election.
"79 % of all students at the local university voted in the last election" the numerical value describes a population parameter.
What is Population parameterPopulation Parameter is a quantity/measurement that can be used as material for follow-up on population management activities.
About populationPopulation is a generalized area consisting of: objects/subjects that have certain qualities and characteristics determined by the researcher to be studied and then drawn conclusions.
The population or universe is the total number of units or individuals whose characteristics are to be studied. And these units are called units of analysis, and can be people, institutions, things, etc.
finite population : population whose number is known with certainty. for example the number of X students. infinite population : population whose number is not known with certainty. for example the number of people in the world. a group of objects that continues to grow (carries out a process due to life or an event process) is an infinite population.Learn more about population parameters at
https://brainly.com/question/29739146
#SPJ4
24% of attempts hit the bull's eye
of attempts hit the bull's eye
25
0.26 of attempts hit the bull's eye
1 out of every 5 attempts hit the bull's eye
00
Which contestant won the competition?
2 of 10 <
1 2 3
4
Four friends entered an archery contest at summer camp. The winner is the contestant who hits the bull's eye the most frequently. Each contestant takes the same number of attempts. The
results are in the table.
5
6 7 8
9
Answer:
Robin.
Step-by-step explanation:
Clinton scored 24% of his attempts which means 24/100.
Then Katniss scored 6/25 of his attempts which means 24/100.
After that Robin scored 0.26 of his attempts which means 26/100.
Finally William scored 1/5 of his attempts which means 20/100.
Therefore, Robin has the highest score and wins the competition.
ken bought a bike that was priced at $450.He was offered a 25% discount what was the discount on the bike.
Answer:
You just want to find what 25% of 450 is. To do so, you can change 25% into a decimal, and then multiply:
25% as decimal is 0.25.
0.25 * 450 = 112.5
The discount is $112.50.
Step-by-step explanation:
Hope it helps! =D
The average of four numbers is 10.three of the numbers are 17, 13 and 6.find the fourth numberthe average of four numbers is 10 three of the numbers are 1713 and 6 find the fourth number
The fourth number is equal to -6.
How to calculate the mean?Mathematically, the arithmetic average for a given data set can be calculated by using this formula:
Average = [F(x)]/n
Next, we would calculate the arithmetic average for the four numbers (data set) as follows:
Let the fourth number be represented by the variable n.
Total numbers = 17 + 13 + 16 + n
Total numbers = 46 + n
10 = 46 + n/4
40 = 46 + n
n = 40 - 46
n = -6
Read more on arithmetic average here: https://brainly.com/question/28329938
#SPJ1
I need help solving x
Answer:
x = 12
Step-by-step explanation:
The exterior angle is equal to the sum of the opposite interior angles.
5x+20 = 20 + 6x-12
Combine like terms.
5x+20 = 6x +8
Subtract 5x from each side.
5x+20-5x = 6x+8-5x
20 = x+8
Subtract 8 from each side.
20-8 = x+8-8
12 =x
Let g be the function given by g(x)=limh→0sin(x+h)−sinx/h. What is the instantaneous rate of change of g with respect to x at x=π3?
Answer:
The function g(x) is the derivative of the function sin(x). So, g(x) = cos(x).
Therefore, the instantaneous rate of change of g with respect to x at x = π/3 is cos(π/3), which is approximately equal to 0.5.
Step-by-step explanation:
np and pls mark brainlist:)
the mass of the part of a rod that lies between its left end and a point x meters to the right is 8 x 3 kg. the linear density of the rod at 1 meters is
The rod's linear density at 1 metre from the left end and a point x meters to the right is 8 x 3 kg is = 8(1)2 = 8 kg/m.
We can start by using the definition of linear density, which is the mass per unit length of a rod or wire. Let's assume that the linear density of the rod at a distance of 1 meter from the left end is ρ kg/m.
Then, the mass of the part of the rod that lies between 0 meters and 1 meter to the right is:
m1 = ρ * 1 kg
Similarly, the mass of the part of the rod that lies between 0 meters and x meters to the right is:
m2 = 8x^3 kg
The mass of the part of the rod that lies between 1 meter and x meters to the right is:
m2 - m1 = (8x^3 - ρ) kg
Since we know that this mass is equal to ρ times the length of the rod segment, which is (x-1) meters, we can write:
ρ * (x-1) = 8x^3 - ρ
By condensing this solution, we obtain:
ρ * (x-1+1) = 8x^3
ρ * x = 8x^3
ρ = 8x^2 kg/m
As a result, the rod's linear density at 1 metre from the left end is = 8(1)2 = 8 kg/m.
Learn more about density here:
https://brainly.com/question/29775886
#SPJ4
Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v, and w.u=i+jv=j+kw=i+k
The volume of the parallelepiped formed by the adjacent edges u=i+j, v=j+k, and w=i+k is: 1.
The triple scalar product can be used to find the volume of the parallelepiped formed by three non-coplanar vectors u, v, and w. The triple scalar product is defined as:
u · (v × w)where × represents the cross product and · represents the dot product.
Given that u=i+j, v=j+k, and w=i+k, we can first find the cross product of v and w as:
v × w = (j+k) x (i+k)= i x (j+k) + k x (j+k)
= i x j + i x k + k x j + k x k
= -k + i -j
Next, we can find the dot product of u with the cross product of v and w as:
u · (v × w) = (i+j) · (-k+i-j)= i · (-k) + i · i + i · (-j) + j · (-k) + j · i + j · (-j)
= -ik + i^2 - ij - jk + ji - j^2
= i^2 - j^2 - k(i+j)
= 1 - 0 - (1+0)
= -1
The absolute value of this result gives the volume of the parallelepiped, so the volume is:
|u · (v × w)| = |-1| = 1Therefore, the volume of the parallelepiped formed by the adjacent edges u=i+j, v=j+k, and w=i+k is 1.
Learn more about volume of the parallelepiped here:
brainly.com/question/29140066
#SPJ4
mike tyson is high no ee aiqidj
Answer:
Jakaur7889 on inta
Step-by-step explanation:
Babahaja
The cost of going to the movie theater for Elissa and 11 of her friends is shown in the table. The total cost is divided equally among each person. How much does each person pay?
The cost for each person is $12.
So, each person paid $12 for a movie ticket.
What is division?The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
The cost of going to the movie theater for Elissa and 11 of her friends is $144.
The total cost is divided equally among each person.
The number of total people is 12.
So, the cost of each person,
= total cost/ the number of total people
= 144/12
= 12
Therefore, 12 is the required value.
To learn more about the division;
https://brainly.com/question/13263114
#SPJ1
The average of any three odd integers is an odd integer. What proof method would you use to show that the statement is true, or false? Prove true by exhaustion. Prove true by contraposition. O Prove false by counterexample. O Prove true by direct proof. Theorem: The difference between any odd integer and any even integer is odd. In an attempted proof of this statement a student wrote the following lines: Suppose n is any odd integer, and m is any even integer. By definition of odd integer, n = 2k + 1 where k is an integer. By definition of even integer, m = 2k where k is an integer. Then n m = (2k + 1) – 2k = 1 and 1 is odd. Therefore, the difference between any odd integer and any even integer is odd. Which one of the following best describes the reason the proof is not correct? O The proof does not prove that 1 is odd. O The proof violates the parity property. The proof violates the zero-product property. The proof assumes m and n are consecutive integers.
The proof method to show that the statement "The average of any three odd integers is an odd integer" is true is "Prove true by direct proof."
To prove that the statement is true by direct proof, we can assume that the three odd integers are a, b, and c, where a = 2k + 1, b = 2m + 1, and c = 2n + 1 for some integers k, m, and n. Then, the average of these three odd integers is:
(a + b + c)/3 = (2k + 1 + 2m + 1 + 2n + 1)/3
= (2k + 2m + 2n + 3)/3
= 2(k + m + n) + 1
Since k, m, and n are integers, k + m + n is also an integer. Therefore, the average of any three odd integers is of the form 2x + 1, where x is an integer, which is an odd integer.
The reason the attempted proof of the statement "The difference between any odd integer and any even integer is odd" is not correct is "The proof violates the parity property." The student incorrectly subtracted an even integer from an odd integer and concluded that the result is odd. However, the difference between an odd integer and an even integer is always odd, which can be proven directly as follows:
Let n be any odd integer and m be any even integer. Then, by definition, n = 2k + 1 and m = 2j for some integers k and j. The difference between n and m is:
n - m = (2k + 1) - 2j
= 2k - 2j + 1
= 2(k - j) + 1
Since k and j are integers, k - j is also an integer. Therefore, the difference between any odd integer and any even integer is of the form 2x + 1, where x is an integer, which is an odd integer.
You can learn more about how to prove method at
https://brainly.com/question/11465297
#SPJ4
What is 4 to the 4th power
Answer:
4 to the 4th power is 256
Explanation:
4 to the 4th power is a mathematical expression indicating that the number 4 is multiplied by itself four times. Mathematically, it is represented as 4^4.
To calculate 4^4, you would multiply 4 by itself four times: 4 x 4 x 4 x 4.
The result of this multiplication is 256, which means that 4 to the 4th power is equal to 256.
The number 4 to the 4th power is equal to 256.
When you raise a number to a power, it means multiplying that number by itself the specified number of times.
In the case of 4⁴, you are multiplying 4 by itself four times:
4⁴ = 4 × 4 × 4 × 4
When you perform the multiplication, you get:
4 × 4 = 16
16 × 4 = 64
64 × 4 = 256
Therefore, 4 to the 4th power is equal to 256.
To learn more on Multiplication click:
https://brainly.com/question/5992872
#SPJ6
You start driving west for 3 miles, turn right, and drive north for
another 13 miles. At the end of driving, what is your straight line
distance from your starting point? Round to the nearest tenth of a
mile.
Answer:
13.3 miles
Step-by-step explanation:
Picture a right triangle with legs of 3 and 13. The hypotenuse is the straight line distance back to the starting point. Use the Pythagorean theorem to find the hypotenuse (c):
c² = 13² + 3² = 178
c = √178 = 13.3 miles
determine the amount of cash collected at the time of making a cash sale of $9,220 worth of merchandise subject to 13 % HST
The total amount of cash collected is $10,418.6.
What is a percentage?A percentage is expressed as a number or ratio that can be expressed as fraction of 100. It is denoted by % symbol.
The amount of cash collected at the time of making a cash sale of $9,220 worth of merchandise subject to 13 % HST is,
Rate of interest on the sales amount is 13%
13% of 9220 = [tex]\frac{13}{100}*9220=1198.6[/tex]
Here, sales price = $9,220
The total amount of cash collected = 1198.6+9220
= 10,418.6
Hence, the total amount of cash collected is $10,418.6.
Learn more about percentage here:
https://brainly.com/question/16797504
#SPJ9
An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the last 3 years. The pmf of Y is the following.
y 0 1 2 3
p(y) 0.50 0.25 0.20 0.05
(a) Compute E(Y).
E(Y) =
(b) Suppose an individual with Y violations incurs a surcharge of $110Y2. Calculate the expected amount of the surcharge.
$
Expert A
(a) The value of E(Y) will be = 0.8
(b) The expected amount of the surcharge is $165.
(a) The expected value of Y is given by:
E(Y) = 0(0.5) + 1(0.25) + 2(0.2) + 3(0.05)
= 0.25 + 0.4 + 0.15
= 0.8
Therefore, the expected number of moving violations for which the individual was cited during the last 3 years is 0.8.
(b) The amount of the surcharge for an individual with Y violations is $110[tex]y^{2}[/tex]. The expected surcharge is given by:
E($110[tex]y^{2}[/tex]) = $110 * E([tex]y^{2}[/tex])
To calculate E([tex]y^{2}[/tex]), we use the formula:
E([tex]y^{2}[/tex]) = Σ [tex]y^{2}[/tex] p(y)
where the sum is taken over all possible values of Y.
E([tex]y^{2}[/tex]) = [tex]0^{2}[/tex](0.5) + [tex]1^{2}[/tex](0.25) + [tex]2^{2}[/tex](0.2) + [tex]3^{2}[/tex](0.05)
= 0 + 0.25 + 0.8 + 0.45
= 1.5
Therefore, the expected surcharge is:
E($110[tex]y^{2}[/tex]) = $110 * E([tex]y^{2}[/tex])
= $110 * 1.5
= $165
So, the expected amount of the surcharge for an individual with moving violations is $165.
For a similar question on the surcharge,
https://brainly.com/question/18102056
#SPJ4
Discuss what can be done to raise a credit score, as well as what you might do to lower it.
Use the following key points to start your discussion:
- factors affecting a person's credit score
- ways people can improve their credit history
a) The factors that impact a person's credit score and history include:
Credit mixNew creditType of creditPayment historyLength of credit historyAmounts of indebtedness.b) To raise or improve a credit score and history, you must undertake the following actions:
Ensuring timely paymentsRegular review of credit reportsLimiting opening new credit accountsMaintaining a low credit utilization rateKeeping old credit accounts operational.What is a credit score?A credit score is a numerical rating or expression of an individual's credit history to determine their creditworthiness.
Credit scores are presented on credit reports, which are regularly prepared by credit bureaus.
On the other hand, a credit history is an official record of an individual's credit management.
Thus, one must understand the credit history and meet payment obligations to improve a person's credit score.
Learn more about credit scores and histories at https://brainly.com/question/13375913.
#SPJ1
The rate of change of the temperature during the hour and a half after the beginning of a thunderstorm is given by T(h) = 9.43 h^3 − 15.41h^2 + 17.34h − 9.83T(h) = 9.43h^3- 15.41h^2 + 17.34h − 9.83where output is measured in per hour and h is the number of hours since the storm began.(a) Evaluate ∫1.5 T(h)dh. ∫0 Tb(h)dh0 1.5(Round your answer to three decimal places.)(b) Interpret the answer to part (a)
∫1.5 T(h) dh = 9.573, rounded to three decimal places. (a) To evaluate ∫1.5 T(h) dh, we need to integrate T(h) with respect to h from 0 to 1.5: ∫1.5 T(h) dh = ∫0.1.5 (9.43h^3 - 15.41h^2 + 17.34h - 9.83) dh
Using the power rule of integration, we can integrate each term of the polynomial: ∫0.1.5 (9.43h^3 - 15.41h^2 + 17.34h - 9.83) dh
= (9.43/4)h^4 - (15.41/3)h^3 + (17.34/2)h^2 - 9.83h |0.1.5
= [(9.43/4)(1.5)^4 - (15.41/3)(1.5)^3 + (17.34/2)(1.5)^2 - 9.83(1.5)] - [(9.43/4)(0)^4 - (15.41/3)(0)^3 + (17.34/2)(0)^2 - 9.83(0)] = 9.573
Therefore, ∫1.5 T(h) dh = 9.573, rounded to three decimal places.
(b) The answer to part (a) represents the total change in temperature during the hour and a half after the beginning of the thunderstorm. This means that the temperature increased by an average of 9.573 degrees per hour during this time period.
This integral gives us a measure of the total effect of the thunderstorm on the temperature, which is useful for predicting the weather and understanding the behavior of natural phenomena.
You can read more about changes in temperature at https://brainly.com/question/25677592
#SPJ4
How do you find the volume of the parallelepiped with adjacent edges pq, pr, and ps where p(3,0,1), q(-1,2,5), r(5,1,-1) and s(0,4,2)?
The volume of the parallelepiped is 190.
Parallelepiped VolumeThe volume V of a parallelepiped with adjacent edges pq, pr, and ps can be found using the scalar triple product as:
V = |pq · (pr x ps)|
where · denotes the dot product and x denotes the cross product.
First, we need to find the vectors pq, pr, and ps:pq = q - p = (-1-3, 2-0, 5-1) = (-4, 2, 4)
pr = r - p = (5-3, 1-0, -1-1) = (2, 1, -2)
ps = s - p = (0-3, 4-0, 2-1) = (-3, 4, 1)
Next, we need to find the cross product of pr and ps:pr x ps = det([i, j, k; 2, 1, -2; -3, 4, 1]) = i(-4) - j(7) + k(10) = (-4, -7, 10)
Finally, we can find the volume of the parallelepiped:V = |(-4, 2, 4) · (-4, -7, 10)| = 190
Learn more about Parallelepiped Volume here:
https://brainly.com/question/27953025
#SPJ4
Determine if //,
I
y+3=
-
F
or neither.
4
(x − 1)
y=-x-1
Select the correct response:
o Perpendicular
Parallel
O Neither
If y+3=-4/3(x-1) and y=-4/3x-1 are two lines then these are parallel.
What is Slope of Line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
The two equations are
y+3=-4/3(x-1)
y=-4/3x-1
When we compare both the equations to slope intercept form.
we get slope -4/3 for both the equations.
If the lines have same slope then the lines are parallel.
Hence, the two lines are parallel lines.
To learn more on slope of line click:
https://brainly.com/question/14511992
#SPJ1
find the value of x 2×+× +x+2×+3×+30+(2×-10)
Answer:
= 11x + 20.
Step-by-step explanation:
2x + x + x+2x+3x+30+(2x-10)
= 2x+x+x+2x+3x+30+2x-10
=2x + x + x + 2x + 3x + 2x + 30 - 10
=11x + 20.
What is the difference of 8/3x+5 -2x/3x+5
The difference of 8/3x+5 -2x/3x+5 is (8 - 2x) / (3x+ 5).
The correct option is (A).
What is Expression?Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)
We have the Expression as:
8/3x+5 -2x/3x+5
Now, Simplifying the expression as:
= 8/3x+5 -2x/3x+5
= (8 - 2x) / (3x+ 5)
= 8/(3x+ 5) - 2x/ (3x+ 5)
Learn more about expression here:
https://brainly.com/question/14083225
#SPJ9
A) In the sequence for which the general term is t_(n)=(n^(2)-1)/(3n^(2)),
find the value of n, if any, for which t_(n)=0.3325.
B) Find S_(n) for the arithmetic series 16+13+10+cdots and determine the value of n for which S_(n)=-6.
S_(n)=. The value of n for which S_(n)=-6 is .
we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4 and the value of n for which [tex]S_n[/tex]= -6 is 6.
How to find value of n and sum of sequence?A) To find the value of n for which [tex]t_n[/tex] = 0.3325, we can use trial and error method.
Starting with n=1, we have:
[tex]t_1 = (1^2 - 1)/(3 * 1^2) = 0[/tex]
For n=2, we have:
[tex]t_2 = (2^2 - 1)/(3 * 2^2) = (3)/(12) = 0.25[/tex]
For n=3, we have:
[tex]t_3 = (3^2 - 1)/(3 * 3^2) = (8)/(27) = 0.296296...[/tex]
Continuing in this way, we find that t_3 < 0.3325 < t_4, where
[tex]t_4 = (4^2 - 1)/(3 * 4^2) = (15)/(48) = 0.3125[/tex]
Thus, we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4, and it can be found more accurately by using numerical methods such as bisection or Newton's method.
B) To find the sum of the arithmetic series 16 + 13 + 10 + ..., we can use the formula for the sum of an arithmetic series:
[tex]S_n = n/2 * (a_1 + a_n),[/tex]
where n is the number of terms in the series, [tex]a_1[/tex] is the first term (16 in this case), and a_n is the nth term.
Let's assume that the nth term is represented by [tex]a_n = 16 - 3 * (n - 1),[/tex] so that a_1 = 16 and a_n = 16 - 3 * (n - 1).
Then, we have:
S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n
We want to find the value of n for which S_n = -6, so we can substitute the value of S_n into the equation and solve for n:
[tex]S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n[/tex]
[tex]-6 = 16n - (3/2)n^2 + (3/2)n[/tex]
Expanding the equation and solving for n, we find that n = 6.
Thus, the value of n for which [tex]S_n[/tex]= -6 is 6.
To know more about Sequence visit:
brainly.com/question/30262438
#SPJ1
use the fundamental theorem of calculus to find the exact value of the definite integral `\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx`. a correct solution must: - clearly state an antiderivative of the integrand, - show where the fundamental theorem of calculus is used and - give an exact value (do not approximate).
By using fundamental theorem of calculus The exact value of the definite integral [tex]\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx[/tex] is 6(√(π) - 1).
First, we can find an antiderivative of the integrand by using the power rule and the constant multiple rule of integration:
∫(3x^(-1/2)) dx = 3 ∫(x^(-1/2)) dx = 3 (2x^(1/2)) + C = 6√(x) + C
where C is the constant of integration.
Now, we can use the fundamental theorem of calculus to evaluate the definite integral \int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx. The theorem states that if f(x) is continuous on the interval [a, b] and F(x) is an antiderivative of f(x), then
∫[a, b] f(x) dx = F(b) - F(a)
In this case, f(x) = 3x^(-1/2) and F(x) = 6√(x) + C. Therefore,
∫{1}^{π} (3x^(-1/2)) dx = F(π) - F(1) = 6√(π) + C - 6√(1) - C = 6√(π) - 6√(1) = 6(√(π) - 1)
So the exact value of the definite integral [tex]\int {1}^{\pi}\left(3x^{-1} \sqrt{x}\right)\ dx[/tex] is 6(√(π) - 1).
To know more about fundamental theorem of calculus:
https://brainly.com/question/21403462
#SPJ4
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).