The rectangular coordinates of the polar coordinates are as follows:-
a) x = 0 and y = ( -2 )
b) x = ( -√3/2 ) and y = 1
How to convert these polar coordinates to rectangular coordinates?To convert these polar coordinates to rectangular coordinates, we calculate the following,
a) ( 2, 3π/2 ) :
r=2 ; θ = 3π/2
x = r * cosθ
x = 2 * cos( 3π/2 )
x = 2 * 0
Therefore, x = 0
y = r * sinθ
y = 2 * sin( 3π/2 )
y = 2 * ( -1 )
Therefore, y = ( -2 )
b) ( 1, 5π/6 ) :
r = 1 ; θ = 5π/6
x = r * cosθ
x = 1 * cos( 5π/6 )
x = 1 * ( -√3/2 )
Therefore, x = ( -√3/2 )
y = r * sinθ
y = 2 * sin( 5π/6 )
y = 2 * ( 1/2 )
Therefore, y = 1
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A store pays $60 for an item your friend finds the selling price when the markup is20%
Answer:$72 Total
Step-by-step explanation:
20%=1/5
1/5 + 5/5(total) = 6/5
60 * 6/5 = $72
THURSDAY: Use Synthetic Division to divide the polynomial: (x^(4)-5x^(3)+4x-17)-:(x-5)
Synthetic Division of the polynomials (x^(4)-5x^(3)+4x-17) and (x-5) gives answer "x^(3)+4+3/(x-5)."
To divide the given polynomial using synthetic division, we will use the following steps:
Step 1: Write the coefficients of the polynomial in descending order. In this case, the coefficients are 1, -5, 0, 4, -17.
Step 2: Write the value of x from the divisor (x-5) in the left column. In this case, the value of x is 5.
Step 3: Bring down the first coefficient to the bottom row.
Step 4: Multiply the value of x by the number in the bottom row and write the result in the next column.
Step 5: Add the numbers in the second column and write the result in the bottom row.
Step 6: Repeat steps 4 and 5 until you have completed all the columns.
Step 7: The numbers in the bottom row are the coefficients of the quotient, and the last number is the remainder.
The synthetic division will look like this:
5 | 1 -5 0 4 -17
| 5 0 0 20
----------------
1 0 0 4 3
So, the quotient is x^(3)+0x^(2)+0x+4, or simply x^(3)+4, and the remainder is 3.
Therefore, the answer is (x^(4)-5x^(3)+4x-17)÷(x-5) = x^(3)+4+3/(x-5).
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m n
1. Fill in the blank a ÷a = a Where m and n are natural numbers
Answer:
1
Step-by-step explanation:
1 / 1 = 1
I need help with this question can anyone tell me ?
The required value of x for the triangle is 9 units.
What is Thales theorem?The theorem you are referring to is known as the "Parallel Line Theorem" or "Thales' Theorem".
If a line is drawn parallel to one side of a triangle, then the other two sides of the triangle are divided proportionally.
Consider a triangle ABC with a line parallel to one of its sides, say side AB. Let the parallel line intersect sides AC and BC at points D and E, respectively.
Then,
AB/BD = CB/BE
According to question:In triangle;
[tex]$\frac{4x}{x-1} =\frac{27}{6}[/tex]
24x = 27x - 27
= 3x = 27
= x = 9
Thus, required value of x is 9.
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What is the slope of the line represented by the equation: y = 7x
Answer: 7
Step-by-step explanation:
The slope is the number being multiplied by x
what is 1 and 1/7 x 3/5
1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).
What do you mean by decimal?
In mathematics, a decimal is a number that represents a fraction or a part of a whole using a base-ten positional numeral system.
To multiply 1 and 1/7 by 3/5, we can first convert the mixed number to an improper fraction.
1 and 1/7 can be written as:
(7/7 * 1) + 1/7 = 7/7 + 1/7 = 8/7
So, we have:
1 and 1/7 = 8/7
Now, we can multiply 8/7 by 3/5 as follows:
(8/7) * (3/5) = (8 * 3) / (7 * 5) = 24/35
Therefore, 1 and 1/7 multiplied by 3/5 is equal to 24/35 or 0.6857 (rounded to four decimal places).
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Four coins are tossed. a. Give an example of a simple event. b. Give an example of a joint event. c. What is the complement of getting a tail on the fourth coin? d. What does the sample space consist of? a. Give an example of a simple event. Which of the following is a simple event? A. Getting a tail on all coins B. Getting a tail on no coins C. Getting a head on the third coin and a head on the fourth coin D. Getting a tail on the fourth coin b. Give an example of a joint event. Which of the following is a joint event? A. Not getting a head on the second coin B. Getting a head or a tail on the first coin C. Getting a head on the first coin and a tail on the third coin D. Not getting a tail on the fourth coin c. What is the complement of getting a tail on the fourth coin? Which of the following is the complement of getting a tail on the fourth coin? A. Getting a head on the fourth coin B. Getting a tail on the fourth coin and a head on all others c. Getting a head on all coins
d. Getting a head on the fourth coin and a tail on all others d. What does the sample space consist of? Which of the following identifies the sample space for the four flipped coins? A. Not getting a head on all coins B. Getting a tail on any of the coins c. Getting a tail on one coin and a head on all others
D. Getting a head or a tail on any of the four coins
The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.
a. A simple event is an event that has only one possible outcome. An example of a simple event is getting a head on the third coin. This is represented by option C: "Getting a head on the third coin and a head on the fourth coin."
b. A joint event is an event that involves two or more simple events happening at the same time. An example of a joint event is getting a head on the first coin and a tail on the third coin. This is represented by option C: "Getting a head on the first coin and a tail on the third coin."
c. The complement of getting a tail on the fourth coin is getting a head on the fourth coin. This is represented by option A: "Getting a head on the fourth coin."
d. The sample space consists of all possible outcomes of the four flipped coins. This is represented by option D: "Getting a head or a tail on any of the four coins." The sample space includes the following outcomes: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.
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A self-tanning lotion advertises that a 4-oz bottle will provide seven applications. The Community Collage theater group needs fake tans for a play they are doing. If the play has cast of 15 how many ounces of self-tanning lotion should the cast purchase
Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.
what is unitary method ?To answer mathematical issues involving proportional relationships between quantities, one uses the unitary method. This approach involves calculating the value of a single unit of a quantity and using that value to calculate the value of any number of additional units. For instance, if we know that five items cost $20 each, we can calculate the price of any other quantity of items by first determining the price of one item, then increasing that figure by the desired quantity of items.
given
The quantity of lotion needed for each application, assuming a 4-oz container of self-tanning lotion yields seven applications, is:
4 oz / 7 = 0.57 oz (rounded to two decimal places) (rounded to two decimal places)
The quantity needed for each application must be multiplied by the number of applications needed by each individual, which equals 1, to determine how many ounces of self-tanning lotion the ensemble of 15 needs to buy:
15. People at 0.57 ounce each equals 8.55 oz.
Therefore, to have enough for one application per person, the ensemble of 15 needs to buy about 8.55 ounces of self-tanning lotion.
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From a point
A
, level with the base of the town hall, the angle of elevation of the topmost point of the building is
35 ∘
. From point B, also at ground level but 30 metres closer to the hall, the same point has an angle of elevation of
60 ∘
. Find how high the topmost point is above ground level. (Give your answer correct to the nearest metre.) 4 A playground roundabout of radius
1.8 m
makes one revolution every five seconds. Find, to the nearest centimetre, the distance travelled by a point on the roundabout in one second if the point is a
1.8 m
from the centre of rotation b
1 m
from the centre of rotation. 5 From a lighthouse, ship
A
is
17.2 km
away on a bearing
S60 ∘
E
and ship
B
is
14.1 km
away on a bearing
N80 ∘
W
. How far, and on what bearing, is B from A? 6 The diagram on the right shows the sketch made by a surveyor after taking measurements for a block of land
ABCD
. Find the area and the perimeter of the block.
Ship B is 22.8 km away from ship A on a bearing of N50°W.
The topmost point of the town hall is 28 metres above ground level.
The distance travelled by a point on the roundabout in one second is 113 cm for point A and 63 cm for point B.
Ship B is 22.8 km away from ship A on a bearing of N50°W.
Explanation:
1) For the first question, we can use trigonometry to find the height of the town hall. Let h be the height of the topmost point above ground level, and d be the distance between point A and the base of the town hall. Then we have:
tan(35°) = h/d and tan(60°) = h/(d-30)
Solving for h, we get:
h = d*tan(35°) and h = (d-30)*tan(60°)
Equating the two expressions for h, we get:
d*tan(35°) = (d-30)*tan(60°)
d = 30*tan(60°)/(tan(60°)-tan(35°)) ≈ 48.5 metres
Substituting back into the first equation, we get:
h = 48.5*tan(35°) ≈ 28 metres
Therefore, the topmost point of the town hall is 28 metres above ground level.
2) For the second question, we can use the formula for the circumference of a circle to find the distance travelled by a point on the roundabout in one second. The circumference of a circle is given by C = 2πr, where r is the radius of the circle. The distance travelled by a point in one second is then given by C/5, since the roundabout makes one revolution every five seconds. For point A, which is 1.8 metres from the centre of rotation, we have:
C = 2π*1.8 = 11.31 metres
Distance travelled in one second = 11.31/5 = 2.26 metres ≈ 226 cm ≈ 113 cm to the nearest centimetre
For point B, which is 1 metre from the centre of rotation, we have:
C = 2π*1 = 6.28 metres
Distance travelled in one second = 6.28/5 = 1.26 metres ≈ 126 cm ≈ 63 cm to the nearest centimetre
3) For the third question, we can use the law of cosines to find the distance between ship A and ship B. The law of cosines states that c^2 = a^2 + b^2 - 2ab*cos(C), where a, b, and c are the lengths of the sides of a triangle, and C is the angle opposite side c. Let x be the distance between ship A and ship B, and let θ be the angle between the two ships. Then we have:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(θ)
To find the angle θ, we can use the fact that the sum of the angles in a triangle is 180°. We have:
θ = 180° - 60° - 80° = 40°
Substituting back into the equation, we get:
x^2 = 17.2^2 + 14.1^2 - 2*17.2*14.1*cos(40°) ≈ 520.3
x ≈ 22.8 km
To find the bearing of ship B from ship A, we can use the law of sines. The law of sines states that a/sin(A) = b/sin(B) = c/sin(C), where a, b, and c are the lengths of the sides of a triangle, and A, B, and C are the angles opposite those sides. Let α be the bearing of ship B from ship A. Then we have:
14.1/sin(α) = 22.8/sin(40°)
Solving for α, we get:
α = sin^-1(14.1*sin(40°)/22.8) ≈ 30°
Since ship B is to the west of ship A, the bearing of ship B from ship A is N50°W.
Therefore, ship B is 22.8 km away from ship A on a bearing of N50°W.
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Determine the intervals where the following function is increasing or decreasing. g(x) = 3-1/zx, x<-2 . 4+3/2x, x>-2 Increasing on: (-2,infinity) Decreasing on: (-4,-2)
To determine the intervals where a function is increasing or decreasing, we need to take the derivative of the function and find the critical points. The critical points are where the derivative is equal to zero or undefined.
First, let's take the derivative of the function:
g'(x) = -1/(zx)^2, x<-2 . 3/2, x>-2
Now, let's find the critical points:
-1/(zx)^2 = 0 -> There are no values of x that will make this equation true, so there are no critical points for x<-2.
3/2 = 0 -> There are no values of x that will make this equation true, so there are no critical points for x>-2.
Since there are no critical points, the function is either always increasing or always decreasing. To determine which one it is, we can pick a value of x in each interval and plug it into the derivative:
For x<-2, let's pick x=-3:
g'(-3) = -1/(-3z)^2 = 1/(9z^2) > 0
Since the derivative is positive, the function is increasing on the interval (-infinity, -2).
For x>-2, let's pick x=0:
g'(0) = 3/2 > 0
Since the derivative is positive, the function is also increasing on the interval (-2, infinity).
Therefore, the function is increasing on the entire domain of the function, which is (-infinity, infinity).
Answer: Increasing on: (-infinity, infinity) Decreasing on: None
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Multiply the polynomials using a special product formula. Express your answer as a single polynomial in standard form. (4x-7)^(2)
The product of the polynomials (4x-7)^(2) is 16x^(2)-56x+49. This is the final answer expressed as a single polynomial in standard form.
To multiply the polynomials using a special product formula, we can use the formula (a-b)^(2)=a^(2)-2ab+b^(2). In this case, a=4x and b=7. Plugging these values into the formula gives us:
(4x-7)^(2)=(4x)^(2)-2(4x)(7)+(7)^(2)
Simplifying the terms on the right side of the equation gives us:
(4x-7)^(2)=16x^(2)-56x+49
Simplifying, we have (4x-7)2 = 16x2 - 56x + 49, which is our answer expressed as a single polynomial in standard form.
Multiplying polynomials require only three steps.
First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
Add the powers of the same variables using the exponent rule.
Then, simplify the resulting polynomial by adding or subtracting the like terms.
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how many car number plates can be made if each plate contians 3 different letters followed by 3 different digits?
There are a total of 26 letters in the English alphabet and 10 digits (0-9). Therefore, the total number of car number plates that can be made if each plate contains 3 different letters followed by 3 different digits can be calculated as follows:
- For the first letter, there are 26 options.
- For the second letter, there are 25 options (since the first letter has already been used).
- For the third letter, there are 24 options (since the first and second letters have already been used).
- For the first digit, there are 10 options.
- For the second digit, there are 9 options (since the first digit has already been used).
- For the third digit, there are 8 options (since the first and second digits have already been used).
Therefore, the total number of car number plates that can be made is:
26 × 25 × 24 × 10 × 9 × 8 = 11,232,000
So, the answer is 11,232,000 car number plates can be made if each plate contains 3 different letters followed by 3 different digits.
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1. Calculate the compound interest on N2,000 for 2years at the rate of 10% per annum
The compound interest on N2,000 for 2 years at the rate of 10% per annum is N441.00.
Compound interest is the interest earned not only on the principal amount but also on the accumulated interest. To calculate the compound interest, we use the formula:
A = P[tex](1 + r/n)^{(nt)[/tex]where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.
In this case, P = N2,000, r = 0.1 (10%), n = 1 (compounded annually), and t = 2 years.
So, A = 2000[tex](1 + 0.1/1)^{(1*2)[/tex] = N2,441.00
Therefore, the compound interest is N2,441.00 - N2,000.00 = N441.00.
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Ronen has 1,003 marbles. He wants to give the same number of marbles to each of his 8
teammates with as few as possible left over. How many marbles did each teammate receive?
How many marbles were left?
Enter the correct value to complete each sentence.
Blank marbles were shared with each teammate.
Blank marbles were left.
Answer:
3 Marbles were left.
Step-by-step explanation:
To find the number of marbles Ronen gave to each teammate, we can divide the total number of marbles by the number of teammates:
1003 ÷ 8 = 125 with a remainder of 3
Therefore, each teammate received 125 marbles, and there were 3 marbles left over.
So, 125 marbles were shared with each teammate, and 3 marbles were left.
The volume of a rectangular prism is 900 cubic meters. Its width is 12 meters, and its height is 3 meters shorter than its length.
To the nearest tenth of a meter, what is the length of the prism?
Rearranging this equation gives a quadratic equation[tex]length^2 - 3 length - 75 = 0[/tex]
What is the volume of a rectangular prism?Let's start by using the formula for the volume of a rectangular prism:
Volume = length x width x height
We know that the volume is 900 cubic meters and the width is 12 meters. Let's substitute these values into the formula:
[tex]900 = length \times 12 \times height[/tex]
Now we need to use the information about the height. We know that the height is 3 meters shorter than the length, so we can write:
height = length - 3
Substituting this into the formula gives:
[tex]900 = length \times 12 \times (length - 3)[/tex]
Simplifying this equation gives:
[tex]900 = 12 length^2 - 36 length[/tex]
Dividing both sides by 12 gives:
[tex]75 = length^2 - 3 length[/tex]
Therefore, Rearranging this equation gives a quadratic equation:
[tex]length^2 - 3 length - 75 = 0[/tex]
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Can someone plss help me with this? they're linear equations 2x2
0.16x+0.20y=11
x+y=60
Answer:
Step-by-step explanation:
0.16x + 0.20y = 11
we can write 0.16 = 16/100 = 4/25 and 0.20 = 20/100 = 1/5
[tex]\frac{4}{25}x + \frac{1}{5}y = 11[/tex] --------(1)
[tex]x + y = 60 \\[/tex] --------(2)
multiply both side by 5 in equation (1) we get
[tex]\frac{4}{5}x + y =55[/tex] --------(3)
Subtract equation 3 from 2
[tex]x - \frac{4}{5}x + y - y = 60 - 55\\\frac{x}{5} = 5\\ x = 25[/tex]
put the value of x in equation 2
25 + y = 60
y = 35
Each door in the hotel is locked with probability 1/3 independently of the others. An arriving guest is placed in Room 0 and can then wander freely (insofar as the locked doors allow). Show that the guest’s chance of escape is about - 9 − √27/4
The probability of escape for the guest is approximately 9 - √27/4.
This can be calculated using the principle of inclusion-exclusion. To calculate the probability of escape, we can calculate the probability of the guest being unable to move from Room 0 to any of the other rooms.
This is the probability of all doors from Room 0 being locked, which is equal to 1/3 x 1/3 = 1/9. Therefore, the probability of escape is 1 - 1/9 = 8/9.
Using the inclusion-exclusion principle, we can calculate the probability of the guest being able to move from Room 0 to any of the other rooms. The probability of the guest being able to move from Room 0 to any of the other rooms is equal to 1 - (1/3 + 1/3 - 1/9) = 9 - √27/4.
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Ffx) iš a third degree polynomial function, how many distinct complex roots are possible?
O 0 or 2
O 0. 1. 2 or 3
O 1 or 2
O 1. 2. Or 3
The possible number of distinct complex roots for a third-degree polynomial function are 0, 1, 2, or 3, making option (B) the correct answer.
A polynomial function of degree n can have at most n distinct roots. For a third-degree polynomial, n=3, so it can have at most 3 roots.
Complex roots of a polynomial come in conjugate pairs. So if a third-degree polynomial has one real root, then the other two roots must be a conjugate pair of complex roots. If the polynomial has two real roots, then the third root must be a complex root that is not real.
Therefore, the possible number of distinct complex roots for a third-degree polynomial function is 0, 1, 2, or 3, making option (B) the correct answer.
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How mary solutions' does this equation have? 4(5r-15)=13+11+8r no solution one soluthn infinitely many solutions
The equation 4(5r-15)=13+11+8r has a unique i.e., one solution.
The given equation is: 4(5r-15) = 13+11+8r.
Simplifying the left-hand side, we get 20r-60 = 32+8r.
Bringing all the r terms to the left-hand side and the constants to the right-hand side, we get:
20r - 8r = 32 + 60
12r = 92
r = 7.67
Therefore, we get a unique solution for the given equation, which is r = 7.67.
There are a few possible reasons why an equation may not have any solution or may have infinitely many solutions. In this case, since we obtained a unique solution, neither of these situations applies.
One common reason why an equation may not have any solution is that the equation may be inconsistent, meaning that the left-hand side and right-hand side of the equation cannot be made equal for any value of the variable. For example, the equation 2x + 3 = 2x + 4 has no solution since the two sides of the equation can never be equal for any value of x.
On the other hand, an equation may have infinitely many solutions if it is an identity, meaning that the left-hand side and right-hand side of the equation are always equal, regardless of the value of the variable. For example, the equation x + 1 = x + 1 is an identity since the left-hand side and right-hand side are always equal for any value of x.
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PLEASE ANSWER!!!! Jeremiah has collected 8 U.S. stamps and 12 international stamps. He wants to display them in identical groups of U.S. and international stamps, with no stamps left over. What is the greatest number of groups Jeremiah can display them in?
i will give brainliest.
Answer:
the greatest number of groups you can make is 4
Step-by-step explanation:
First we have to find the greatest number which can divide 8 and 12 exactly.
there is none but G.C.F is 4
That is, 8 U.S stamps can be displayed in 4 groups at 2 groups.
And 12 international stamps can be displayed in 4 groups at 3 groups
In this way, each of the 4 groups would have 2 U.S stamps and 3 international stamps. And all the 4 groups would be identical.
Classify each angle pair, then find the value of X
Please explain your work!!
Answer: The angles are classified as interior angles, where x=9
Step-by-step explanation:
The angle measure of a straight line is 180°, meaning that 7x-10 and 12x+19 must add up to 180.
It is clear from the visual that the former is an acute angle, and the latter is an obtuse angle. This means that when solving each, 7x-10 must be less than 90, and 12x+19 must be greater than 90, but less than 180.
If you set up the equation to solve for x; (7x-10) + (12x+19) = 180, you must combine like terms.
19x+9=180 is the new equation. Since x needs to be isolated, subtract 9 from each side, to get 19x=171. Dividing each side by 19 results in x=9.
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A boat is heading towards a lighthouse, where Feng is watching from a vertical
distance of 127 feet above the water. Feng measures an angle of depression to the
boat at point A to be 5°. At some later time, Feng takes another measurement and
finds the angle of depression to the boat (now at point B) to be 57°. Find the distance
from point A to point B. Round your answer to the nearest tenth of a foot if
necessary.
Answer:
1369.1
Step-by-step explanation:
Answer:
1369.1 feet
Step-by-step explanation:
your family is driving on the highway. The number of miles y you travel in x minutes is approximated by the equation y=1.1x approximately how far do you travel in 80 minutes?
Answer:88 miles
Step-by-step explanation:
1.1x80+88
Which graph represents the function f(x)=√x+3-1?
The required, graph of the function f(x) = √[x+3]-1 is a curve that starts at (-3,-1) and extends to the right, increasing in value but at a decreasing rate due to the square root function.
What are functions?Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.
Here,
The graph of the function f(x) = √[x+3]-1 is a curve that starts at the point (-3,-1) and extends to the right indefinitely. The square root function √x has a domain of x ≥ 0, so in this case, the domain of the function is x ≥ -3.
The graph is always above the x-axis, as the square root function can only output non-negative values. The graph also approaches but never touches the horizontal line y = 0 as x increases without bound, since the -1 term in the function only shifts the graph downward by one unit.
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"What are the corner points?
What is the solution to the linear programming problem?"
Minimize: C = 3x – 3y Subject to: 3x – y >= 2 x + y <= 5 x >= 0, y >= 0
The minimum value of the objective function is -15, which occurs at the corner point (0, 5).
The corner points of a linear programming problem are the points where the constraints intersect. These points can be found by solving the system of inequalities for each pair of constraints.
For this problem, we can find the corner points by solving the system of inequalities for each pair of constraints:
3x – y >= 2 and x + y <= 5:
- Add y to both sides of the first inequality: 3x >= 2 + y
- Subtract 2 from both sides of the first inequality: 3x - 2 >= y
- Substitute 3x - 2 for y in the second inequality: x + (3x - 2) <= 5
- Simplify: 4x <= 7
- Divide by 4: x <= 7/4
- Substitute 7/4 for x in the first inequality: 3(7/4) - 2 >= y
- Simplify: 5/4 >= y
The first corner point is (7/4, 5/4).
3x – y >= 2 and x >= 0:
- Set x = 0 and solve for y: 3(0) - y >= 2, y <= -2
- Set y = 0 and solve for x: 3x - 0 >= 2, x >= 2/3
The second corner point is (2/3, 0).
x + y <= 5 and x >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5
The third corner point is (0, 5).
x + y <= 5 and y >= 0:
- Set x = 0 and solve for y: 0 + y <= 5, y <= 5
- Set y = 0 and solve for x: x + 0 <= 5, x <= 5
The fourth corner point is (5, 0).
Now we can plug each corner point into the objective function to find the minimum value:
C = 3x – 3y
C = 3(7/4) - 3(5/4) = 3
C = 3(2/3) - 3(0) = 2
C = 3(0) - 3(5) = -15
C = 3(5) - 3(0) = 15
Therefore, the solution to the linear programming problem is (0, 5).
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Use the Integral Test to show that the series ∑_(k=1)^[infinity]▒(k^2/e^k ) converges. Hint: use integration by parts
As per the concept of integral, the series ∑ne⁻ⁿ converges.
Let's consider the series ∑ne⁻ⁿ. Since each term of the series is positive, the first condition of the Integral Test is satisfied. To check the second condition, we need to determine whether the function f(x) = xe⁻ˣ is decreasing for x ≥ 1.
To do this, we can take the derivative of f(x) with respect to x:
f'(x) = e⁻ˣ - xe⁻ˣ
Setting f'(x) = 0, we get:
e⁻ˣ - xe⁻ˣ = 0
x = 1
So f(x) has a maximum value at x = 1. Since f'(x) < 0 for x ≥ 1, f(x) is decreasing for x ≥ 1.
Now, we can set up the corresponding integral:
∫₁^∞ xe⁻ˣ dx
To evaluate this integral, we can use integration by parts:
u = x, dv = e⁻ˣ dx
du = dx, v = -e⁻ˣ
∫₁^∞ xe⁻ˣ dx = -xe⁻ˣ │₁^∞ + ∫₁^∞ e⁻ˣ dx
= 0 + e⁻ˣ │₁^∞
= 1
Since the integral converges, the series ∑ne⁻ⁿ also converges by the Integral Test.
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Complete Question:
How do you use the Integral Test to determine convergence or divergence of the series: ∑ne⁻ⁿ from n=1 to infinity?
How do you do c) really need help!!
how do i graph 3/5x
Answer:
Rise and run
Step-by-step explanation:
Lets have it in slope intercept form
y=3/5x
Therefore our slope is 3/5 and y-intercept is (0,0)
Now we can start to graph. Start on our Y-intercept which is zero. Let's count 3 spaces up, which is our rise, then lets count 5 spaces to our right, which is our positive run, and plot a dot on our final point. You can do the same negative to graph our opposing points. Start on our 0,0 y-intercept, and count down 3 spaces. Then count 5 spaces to the left, and plot a dot on your final point. Now, let's draw a line through our points (-5, -3), (0,0), and (5,3)!
6. Consider the expansion of x^2 (3x^2 + k/x)^8. The constant term is 16128. Find k. [7 marks]
The value of k is ∛√199.
Consider the expansion of x^2(3x^2 + k/x)^8. The constant term is 16128. We need to find the value of k.
The expansion of (3x^2 + k/x)^8 will have terms of the form (3x^2)^a(k/x)^b, where a + b = 8. The constant term will be the term where the powers of x cancel out, so we need to find a and b such that 2a - b = 0.
Solving for a and b, we get a = 4 and b = 8. So the constant term will be (3x^2)^4(k/x)^8 = 81x^8(k^8/x^8) = 81k^8.
Setting this equal to 16128 and solving for k, we get:
81k^8 = 16128
k^8 = 16128/81
k^8 = 199
k = ∛√199
Therefore, the value of k is ∛√199.
k = ∛√199.
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87 oz= how many lb and oz
Step-by-step explanation:
How many Oz make 1 Pound
Answer is 16
Divide 87 by 16
Answer is 5Lb 7oz